# The Feynman–Kac Formula: A Measure-Theoretic, Analytic, and Probabilistic Synthesis

Sourangshu Ghosh

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Sourangshu Ghosh. The Feynman–Kac Formula: A Measure-Theoretic, Analytic, and Probabilistic Synthesis. 2025. hal-05378000

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# The Feynman–Kac Formula:

A Measure-Theoretic, Analytic, and
Probabilistic Synthesis

Sourangshu Ghosh

*sourangshug@iisc.ac.in, Department of Civil Engineering,Indian Institute of Science Bangalore
# Contents

<table><tr><td>Preface</td><td>vi</td></tr><tr><td>The Triad: Probability, Analysis, and Physics</td><td>vi</td></tr><tr><td>Scope and Philosophy</td><td>vii</td></tr><tr><td>Structure of the Book</td><td>vii</td></tr><tr><td>Intended Audience</td><td>viii</td></tr><tr><td>Acknowledgments</td><td>viii</td></tr><tr><td>A Final Word</td><td>viii</td></tr></table>

<table><tr><td>I Mathematical Foundations</td><td>1</td></tr></table>

<table><tr><td>1 Measure-Theoretic Probability and Stochastic Processes</td><td>2</td></tr><tr><td>1.1 Introduction</td><td>2</td></tr><tr><td>1.2 Probability Spaces and Random Variables</td><td>2</td></tr><tr><td>1.3 Independence and Conditional Expectation</td><td>4</td></tr><tr><td>1.4 Filtrations and Adapted Processes</td><td>5</td></tr><tr><td>1.5 Martingales and Stopping Times</td><td>7</td></tr><tr><td>1.6 Stochastic Processes and Finite-Dimensional Distributions</td><td>8</td></tr><tr><td>1.7 Markov Processes and Transition Kernels</td><td>11</td></tr><tr><td>1.8 Generators and Semigroups</td><td>13</td></tr><tr><td>1.9 Conclusion</td><td>14</td></tr></table>

<table><tr><td>2 Brownian Motion and Stochastic Integrals</td><td>15</td></tr><tr><td>2.1 Introduction</td><td>15</td></tr><tr><td>2.2 Definition and Basic Properties of Brownian Motion</td><td>15</td></tr><tr><td>2.3 Quadratic Variation and Semimartingale Structure</td><td>17</td></tr><tr><td>2.4 Simple Integrands and the Itô Integral</td><td>19</td></tr><tr><td>2.5 Itô's Formula</td><td>23</td></tr><tr><td>2.5.1 Itô's Lemma for Jump-Diffusion (Poisson) Processes</td><td>26</td></tr><tr><td>2.5.2 Itô's Lemma for Discontinuous Semimartingales</td><td>29</td></tr><tr><td>2.5.3 Itô's Lemma and Explicit Solution for Geometric Brownian Motion</td><td>30</td></tr><tr><td>2.6 Martingale Representation and Predictable Projection</td><td>32</td></tr><tr><td>2.7 Stochastic Differential Equations and Strong Solutions</td><td>35</td></tr><tr><td>2.8 Girsanov Transform (Informal Statement)</td><td>37</td></tr><tr><td>2.9 Conclusion</td><td>39</td></tr></table>

<table><tr><td>3 Generators and Semigroups</td><td>40</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>3.1</td>
      <td>Infinitesimal generator of the semigroup</td>
      <td>40</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Generator of a Diffusion Process</td>
      <td>43</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Kolmogorov Equations</td>
      <td>48</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Generator as the Infinitesimal Limit of the Expectation Semigroup Acting on Path Functionals</td>
      <td>50</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Conclusion</td>
      <td>51</td>
    </tr>
    <tr>
      <td>II</td>
      <td>The Feynman-Kac Formula</td>
      <td>53</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Stochastic Differential Equations and Martingale Problems</td>
      <td>54</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Stochastic Differential Equations: Strong and Weak Formulations</td>
      <td>54</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>The Martingale Problem Formulation</td>
      <td>56</td>
    </tr>
    <tr>
      <td>4.2.1</td>
      <td>Stroock-Varadhan uniqueness condition</td>
      <td>58</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Connection between Generators and SDEs</td>
      <td>62</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Weak Convergence and Stability</td>
      <td>63</td>
    </tr>
    <tr>
      <td>4.5</td>
      <td>Conclusion</td>
      <td>66</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Statement and Proof of the Classical Feynman-Kac Formula</td>
      <td>67</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Introduction and hypotheses</td>
      <td>67</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Statement of the classical theorem</td>
      <td>68</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Proof of existence: martingale argument and verification</td>
      <td>68</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Uniqueness (analytic semigroup argument)</td>
      <td>69</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Remarks and extensions</td>
      <td>70</td>
    </tr>
    <tr>
      <td>5.6</td>
      <td>Feynman-Kac formula with unbounded potentials (Kato class): rigorous proof</td>
      <td>70</td>
    </tr>
    <tr>
      <td>5.7</td>
      <td>Regularity and boundary conditions</td>
      <td>74</td>
    </tr>
    <tr>
      <td>5.8</td>
      <td>Conclusion</td>
      <td>77</td>
    </tr>
    <tr>
      <td>6</td>
      <td>The Feynman-Kac Semigroup</td>
      <td>79</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Definition of the Feynman-Kac semigroup</td>
      <td>80</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Strong continuity</td>
      <td>83</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Infinitesimal generator of the Feynman-Kac semigroup</td>
      <td>84</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Semigroup representation of parabolic PDEs</td>
      <td>85</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>Approximation procedure for unbounded Kato-class potentials</td>
      <td>87</td>
    </tr>
    <tr>
      <td>6.6</td>
      <td>Conservativeness and sub-Markov property</td>
      <td>89</td>
    </tr>
    <tr>
      <td>6.7</td>
      <td>Sub-Markovian semigroups and killing measures: Dirichlet form perspective</td>
      <td>90</td>
    </tr>
    <tr>
      <td>6.8</td>
      <td>Analytic characterization</td>
      <td>91</td>
    </tr>
    <tr>
      <td>6.9</td>
      <td>Kernel estimates and small-time asymptotics</td>
      <td>92</td>
    </tr>
    <tr>
      <td>6.10</td>
      <td>Spectral and resolvent theory of the Feynman-Kac semigroup</td>
      <td>95</td>
    </tr>
    <tr>
      <td>6.11</td>
      <td>Functional integration and the Feynman-Kac formula in L<sup>2</sup>(R<sup>d</sup>)</td>
      <td>97</td>
    </tr>
    <tr>
      <td>6.12</td>
      <td>Spectral interpretation</td>
      <td>100</td>
    </tr>
    <tr>
      <td>6.13</td>
      <td>Conclusion</td>
      <td>103</td>
    </tr>
    <tr>
      <td>III</td>
      <td>Analytical and Physical Interpretations</td>
      <td>104</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Potential Theory and Dirichlet Forms</td>
      <td>105</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>7.1</td>
      <td>Dirichlet Forms: Definitions and Basic Properties</td>
      <td>105</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Capacity, Excessive Functions, and Fine Topology</td>
      <td>113</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Quasi-Continuity, Revuz Correspondence and Smooth Measures</td>
      <td>119</td>
    </tr>
    <tr>
      <td>7.4</td>
      <td>Beurling-Deny Decomposition and Nonlocal Forms</td>
      <td>124</td>
    </tr>
    <tr>
      <td>7.5</td>
      <td>Applications to Feynman-Kac Semigroups and Schrödinger Operators</td>
      <td>127</td>
    </tr>
    <tr>
      <td>7.6</td>
      <td>Dirichlet Form Perturbations and Feynman-Kac Semigroups</td>
      <td>129</td>
    </tr>
    <tr>
      <td>7.7</td>
      <td>Conclusion</td>
      <td>134</td>
    </tr>
    <tr>
      <td><b>8</b></td>
      <td><b>Path Integral Representation</b></td>
      <td><b>135</b></td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Wiener Measure and Configuration Space of Paths</td>
      <td>135</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Feynman-Kac Functional and Schrödinger Semigroup Representation</td>
      <td>137</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Analytic Justification and Domain Correspondence</td>
      <td>139</td>
    </tr>
    <tr>
      <td>8.4</td>
      <td>Path Integral as Infinite-Dimensional Limit</td>
      <td>141</td>
    </tr>
    <tr>
      <td>8.5</td>
      <td>Connection to Dirichlet Forms and Potential Theory</td>
      <td>143</td>
    </tr>
    <tr>
      <td>8.6</td>
      <td>Conclusion</td>
      <td>144</td>
    </tr>
    <tr>
      <td><b>9</b></td>
      <td><b>Functional Analytic Properties</b></td>
      <td><b>145</b></td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>Strongly Continuous Semigroups and Their Generators</td>
      <td>145</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>Self-Adjoint Operators and Spectral Theorem</td>
      <td>153</td>
    </tr>
    <tr>
      <td>9.2.1</td>
      <td>Self-adjointness of the generator</td>
      <td>154</td>
    </tr>
    <tr>
      <td>9.2.2</td>
      <td>Spectral representation.</td>
      <td>154</td>
    </tr>
    <tr>
      <td>9.2.3</td>
      <td>Quadratic form and fractional powers</td>
      <td>155</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>Compactness and Spectral Discreteness</td>
      <td>155</td>
    </tr>
    <tr>
      <td>9.3.1</td>
      <td>Compactness criterion and its consequences</td>
      <td>155</td>
    </tr>
    <tr>
      <td>9.3.2</td>
      <td>Examples and sufficient conditions for compactness</td>
      <td>156</td>
    </tr>
    <tr>
      <td>9.3.3</td>
      <td>Summary of implications</td>
      <td>157</td>
    </tr>
    <tr>
      <td>9.4</td>
      <td>Form Methods and the Friedrichs Extension</td>
      <td>157</td>
    </tr>
    <tr>
      <td>9.4.1</td>
      <td>First Representation Theorem (Kato-Friedrichs Theorem)</td>
      <td>158</td>
    </tr>
    <tr>
      <td>9.4.2</td>
      <td>Lower Bounded Forms and Friedrichs Extension</td>
      <td>160</td>
    </tr>
    <tr>
      <td>9.4.3</td>
      <td>Connection with Dirichlet Forms</td>
      <td>160</td>
    </tr>
    <tr>
      <td>9.5</td>
      <td>Analyticity, Sectorial Forms, and Holomorphic Semigroups</td>
      <td>160</td>
    </tr>
    <tr>
      <td>9.6</td>
      <td>Spectral Measures, Functional Calculus, and Resolvent Identities</td>
      <td>162</td>
    </tr>
    <tr>
      <td>9.7</td>
      <td>Spectral Decomposition of Schrödinger Operators</td>
      <td>164</td>
    </tr>
    <tr>
      <td>9.8</td>
      <td>Conclusion</td>
      <td>167</td>
    </tr>
    <tr>
      <td><b>IV</b></td>
      <td><b>Generalizations and Modern Extensions</b></td>
      <td><b>168</b></td>
    </tr>
    <tr>
      <td><b>10</b></td>
      <td><b>Extensions to Lévy and Jump Processes</b></td>
      <td><b>169</b></td>
    </tr>
    <tr>
      <td>10.1</td>
      <td>Motivation and General Framework.</td>
      <td>169</td>
    </tr>
    <tr>
      <td>10.2</td>
      <td>Lévy-Khintchine Representation</td>
      <td>169</td>
    </tr>
    <tr>
      <td>10.3</td>
      <td>Nonlocal Dirichlet Forms.</td>
      <td>171</td>
    </tr>
    <tr>
      <td>10.4</td>
      <td>Fractional Laplacian as a Paradigm.</td>
      <td>173</td>
    </tr>
    <tr>
      <td>10.5</td>
      <td>Feynman-Kac Representation with Jumps</td>
      <td>175</td>
    </tr>
    <tr>
      <td>10.6</td>
      <td>Generators with Drift and Jump-Diffusion Operators.</td>
      <td>176</td>
    </tr>
    <tr>
      <td>10.7</td>
      <td>Analytic and Probabilistic Duality</td>
      <td>178</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>10.8 Spectral and Regularity Properties</td>
      <td>180</td>
    </tr>
    <tr>
      <td>10.9 Conclusion</td>
      <td>182</td>
    </tr>
    <tr>
      <td><b>11 Nonlocal PDEs and Fractional Diffusions</b></td>
      <td><b>183</b></td>
    </tr>
    <tr>
      <td>11.1 Motivation and Overview</td>
      <td>183</td>
    </tr>
    <tr>
      <td>11.2 Fractional Laplacian: Analytic Definitions</td>
      <td>183</td>
    </tr>
    <tr>
      <td>11.3 Fractional Heat Equation and Its Kernel</td>
      <td>186</td>
    </tr>
    <tr>
      <td>11.4 Cauchy Problems and Well-Posedness</td>
      <td>188</td>
    </tr>
    <tr>
      <td>11.5 Fractional Schrödinger Operators</td>
      <td>189</td>
    </tr>
    <tr>
      <td>11.6 Boundary Value Problems and Extension Technique</td>
      <td>192</td>
    </tr>
    <tr>
      <td>11.7 Nonlocal Variational Problems and Energy Minimization</td>
      <td>194</td>
    </tr>
    <tr>
      <td>11.8 Spectral Properties and Regularity</td>
      <td>195</td>
    </tr>
    <tr>
      <td>11.9 Conclusion</td>
      <td>197</td>
    </tr>
    <tr>
      <td><b>12 Infinite-Dimensional Diffusions and Stochastic Partial Differential Equations</b></td>
      <td><b>198</b></td>
    </tr>
    <tr>
      <td>12.1 Motivation and Overview</td>
      <td>198</td>
    </tr>
    <tr>
      <td>12.2 Stochastic Integration in Hilbert Spaces</td>
      <td>198</td>
    </tr>
    <tr>
      <td>12.3 Mild and Weak Solutions</td>
      <td>200</td>
    </tr>
    <tr>
      <td>12.3.1 Mild solution</td>
      <td>201</td>
    </tr>
    <tr>
      <td>12.3.2 Weak solution</td>
      <td>201</td>
    </tr>
    <tr>
      <td>12.3.3 Equivalence between mild and weak solutions</td>
      <td>201</td>
    </tr>
    <tr>
      <td>12.4 The Linear Stochastic Heat Equation</td>
      <td>202</td>
    </tr>
    <tr>
      <td>12.5 Nonlinear SPDEs and Monotone Operators</td>
      <td>205</td>
    </tr>
    <tr>
      <td>12.6 Invariant Measures and Ergodicity</td>
      <td>207</td>
    </tr>
    <tr>
      <td>12.7 Malliavin Calculus and Regularity of Laws</td>
      <td>209</td>
    </tr>
    <tr>
      <td>12.8 Spectral and Functional Analytic Structure</td>
      <td>212</td>
    </tr>
    <tr>
      <td>12.9 Connection to Dirichlet Forms</td>
      <td>214</td>
    </tr>
    <tr>
      <td>12.10Conclusion</td>
      <td>216</td>
    </tr>
    <tr>
      <td><b>13 Stochastic Analysis on Manifolds and Geometric Diffusions</b></td>
      <td><b>217</b></td>
    </tr>
    <tr>
      <td>13.1 Brownian Motion on Riemannian Manifolds</td>
      <td>217</td>
    </tr>
    <tr>
      <td>13.2 Geometric Interpretation via Horizontal Lifts</td>
      <td>220</td>
    </tr>
    <tr>
      <td>13.3 Heat Semigroup and Heat Kernel on Manifolds</td>
      <td>221</td>
    </tr>
    <tr>
      <td>13.4 Gradient Operators and Stochastic Parallel Transport</td>
      <td>223</td>
    </tr>
    <tr>
      <td>13.5 Bochner Identity and Curvature Effects</td>
      <td>226</td>
    </tr>
    <tr>
      <td>13.5.1 Preliminary definitions</td>
      <td>226</td>
    </tr>
    <tr>
      <td>13.5.2 Derivation of the Bochner-Weitzenböck identity</td>
      <td>226</td>
    </tr>
    <tr>
      <td>13.5.3 Interpretations and geometric consequences</td>
      <td>227</td>
    </tr>
    <tr>
      <td>13.6 Diffusions with Drift and Divergence Form Operators</td>
      <td>228</td>
    </tr>
    <tr>
      <td>13.6.1 Stochastic Differential Representation</td>
      <td>228</td>
    </tr>
    <tr>
      <td>13.6.2 Invariant Measures and Reversibility</td>
      <td>229</td>
    </tr>
    <tr>
      <td>13.6.3 Gradient Drift and Self-Adjoint Realization</td>
      <td>229</td>
    </tr>
    <tr>
      <td>13.6.4 Dirichlet Form Representation</td>
      <td>230</td>
    </tr>
    <tr>
      <td>13.6.5 Summary of the Analytic and Probabilistic Correspondence</td>
      <td>230</td>
    </tr>
    <tr>
      <td>13.7 Geometric Dirichlet Forms</td>
      <td>230</td>
    </tr>
  </tbody>
</table>
<table><tr><td>13.7.1 Definition and Domain of the Dirichlet Form</td><td>230</td></tr><tr><td>13.7.2 Fundamental Analytic Properties</td><td>231</td></tr><tr><td>13.7.3 Associated Self-Adjoint Operator</td><td>231</td></tr><tr><td>13.7.4 Functional Analytic Representation.</td><td>231</td></tr><tr><td>13.7.5 Geometric and Probabilistic Interpretations</td><td>232</td></tr><tr><td>13.7.6 Summary</td><td>232</td></tr><tr><td>13.8 Feynman-Kac Formula on Manifolds</td><td>232</td></tr><tr><td>13.8.1 The Schrödinger-type Operator</td><td>233</td></tr><tr><td>13.8.2 The Feynman-Kac Representation</td><td>233</td></tr><tr><td>13.8.3 Analytic Consequences</td><td>236</td></tr><tr><td>13.8.4 Curvature and Potential Effects</td><td>241</td></tr><tr><td>13.8.5 Summary</td><td>243</td></tr><tr><td>13.9 Heat Kernel Estimates and Geometric Bounds</td><td>244</td></tr><tr><td>13.9.1 Gaussian-Type Estimates under Ricci Curvature Bounds.</td><td>244</td></tr><tr><td>13.9.2 Analytic Consequences of Heat Kernel Bounds</td><td>246</td></tr><tr><td>13.9.3 Relation to Curvature-Dimension Conditions</td><td>254</td></tr><tr><td>13.9.4 Summary of the Geometric-Probabilistic Correspondence</td><td>254</td></tr><tr><td>13.9.5 Conclusion</td><td>254</td></tr><tr><td>13.10 Connections to Geometry, Analysis, and Probability</td><td>254</td></tr><tr><td>13.11 Conclusion</td><td>257</td></tr></table>
# Preface

> "In mathematics, one does not understand things. One just gets used to them."

> — John von Neumann

The **Feynman–Kac formula** stands among the rare mathematical results that elegantly bridge distinct conceptual worlds — analysis, probability, and physics — by asserting that the evolution of analytic structures such as parabolic partial differential equations can be represented through the expectations of random paths. It is a synthesis that makes rigorous Feynman’s intuitive vision of quantum propagation via path integrals and Kac’s probabilistic representation of parabolic equations.

The purpose of this monograph is to present the Feynman–Kac formula and its far-reaching generalizations within a **rigorous measure-theoretic and functional-analytic framework**. While many excellent expositions introduce the formula as a computational tool or as a heuristic bridge between stochastic processes and partial differential equations, few texts aim to *unify its analytical, probabilistic, and physical interpretations* within a single, fully self-contained narrative.

This book takes that unification as its central theme.

## The Triad: Probability, Analysis, and Physics

At its heart, the Feynman–Kac formula expresses a deep correspondence:

(Stochastic Process) $\leftrightarrow$ (Semigroup of Operators) $\leftrightarrow$ (Parabolic PDE).

Each of these viewpoints offers its own language:

* The **probabilist** sees the formula as an expectation of functionals of diffusion processes;

* The **analyst** views it as the integral kernel representation of a strongly continuous semi-group;

* The **physicist** recognizes it as the Euclidean-time formulation of the path integral.
Bringing these languages into harmony requires machinery from measure-theoretic probability, semigroup theory, potential theory, and spectral analysis. The result is not only a proof of the formula itself but also a conceptual architecture connecting Markov processes, Schrödinger operators, and heat kernels.

## Scope and Philosophy

The treatment here is **rigorous, self-contained, and comprehensive**. Every theorem is proved from first principles, beginning with the construction of probability spaces and Wiener measure, proceeding through Itô's stochastic calculus, and culminating in the analysis of Feynman-Kac semigroups, spectral representations, and modern extensions to infinite-dimensional and stochastic partial differential equations.

The presentation seeks not only to *prove* the formula but also to *illuminate* it — to show why it is inevitable once one understands the interplay between randomness and linear evolution. To that end, analytic rigor is complemented by geometric and probabilistic intuition, and wherever appropriate, by remarks that connect the results to their origins in physics and to their modern applications in analysis and probability.

## Structure of the Book

The book is organized into four major parts:

**Part I: Mathematical Foundations** develops the measure-theoretic and analytic background required to understand the Feynman-Kac formula in its full generality. Topics include probability spaces, stochastic processes, Brownian motion, Itô calculus, and the theory of strongly continuous semigroups.

**Part II: The Feynman-Kac Formula** presents the classical and generalized statements of the Feynman-Kac theorem, together with detailed proofs. This part develops the correspondence between stochastic differential equations and the generators of parabolic PDEs, leading to the precise probabilistic representation of Schrödinger semigroups.

**Part III: Analytical and Physical Interpretations** explores the deep structural implications of the formula — its manifestation in potential theory, its realization as a rigorous Euclidean path integral, and its role in the spectral theory of Schrödinger operators.

**Part IV: Generalizations and Modern Extensions** surveys the frontier: extensions to Lévy and jump processes, fractional operators, infinite-dimensional systems, stochastic PDEs, and contemporary applications in quantum field theory and statistical physics.

Each chapter is accompanied by remarks, examples, and, where appropriate, historical notes tracing the intellectual evolution of the subject from Wiener's construction of Brownian motion (1923) and Kac's probabilistic approach to diffusion (1949) to modern analytic developments in semigroup and potential theory.
## Intended Audience

This monograph is addressed to **graduate students, researchers, and mathematically mature readers** in analysis, probability theory, and mathematical physics. A reader familiar with measure theory, functional analysis, and the basics of stochastic calculus will be fully prepared to engage with the material, though the presentation strives to develop the necessary background systematically and self-sufficiently.

For physicists, the book provides a rigorous foundation for the intuitive path-integral formalism. For probabilists, it demonstrates how analytic and operator-theoretic techniques enrich the study of stochastic processes. For analysts, it shows how probabilistic representations yield deep insights into PDEs and spectral theory.

## Acknowledgments

This book owes its existence to the enduring beauty of the Feynman-Kac formula itself — a work of mathematical art born from the collaboration between probability and physics. The author expresses profound gratitude to those who inspired and sustained this exploration — to the mentors who nurtured a deep appreciation of mathematical structure, to colleagues for their stimulating discussions, and to the community of mathematicians and physicists whose works have laid the foundations on which this book stands.

## A Final Word

If there is a single message this book aims to convey, it is that the **Feynman-Kac formula is not merely a tool — it is a philosophy**. It embodies the idea that randomness can reveal structure, that expectation can replace determinism, and that, in the language of mathematics, the probabilistic and the analytic are but two sides of the same truth.

Sourangshu Ghosh

Indian Institute of Science, Bangalore

November 22, 2025
# Part I

## Mathematical Foundations
# Measure-Theoretic Probability and Stochastic Processes

> “Probability is the measure theory of ignorance; its objects are sets of possible worlds, its morphisms are filtrations of information.”

> — Adapted from Joseph L. Doob (1953)

## 1.1 Introduction

The purpose of this chapter is to lay the rigorous measure-theoretic foundations upon which the Feynman-Kac formula is built. The formula, which links stochastic processes with parabolic partial differential equations and quantum mechanics, presupposes a precise understanding of probability as a measure space, of stochastic processes as measurable mappings, and of martingales, filtrations, and conditional expectations as the structural backbone of probabilistic dynamics.

We shall work entirely within the modern framework of measure-theoretic probability, viewing stochastic processes as measurable functions on product spaces endowed with suitable filtrations. This framework enables the use of tools from analysis, topology, and functional analysis to treat stochastic evolution, which is essential for the analytic continuation into semigroup and path-integral formulations in later chapters.

## 1.2 Probability Spaces and Random Variables

A probability space is the fundamental object in modern probability theory, providing a rigorous framework for the formalization of uncertainty. Formally, a probability space is defined as a triple $(\Omega, \mathcal{F}, \mathbb{P})$, where $\Omega$ is a non-empty set called the sample space, $\mathcal{F}$ is a σ-algebra of subsets of $\Omega$, and $\mathbb{P}: \mathcal{F} \to [0, 1]$ is a probability measure satisfying $\mathbb{P}(\Omega) = 1$ and countable additivity.

**Definition 1.1** (Probability Space). A *probability space* is a triple $(\Omega, \mathcal{F}, \mathbb{P})$, where:

1. $\Omega$ is the sample space, the set of all possible outcomes;
2. $\mathcal{F}$ is a $\sigma$-algebra of subsets of $\Omega$;

3. $\mathbb{P}: \mathcal{F} \to [0, 1]$ is a countably additive measure with $\mathbb{P}(\Omega) = 1$.

The $\sigma$-algebra $\mathcal{F}$ encodes the collection of events whose probabilities are well-defined, and countable additivity ensures that if $A_{n=1}^\infty$ is a sequence of pairwise disjoint events in $\mathcal{F}$, then

$$ \mathbb{P}\left(\bigcup_{n=1}^{\infty} A_n\right) = \sum_{n=1}^{\infty} \mathbb{P}(A_n). \qquad (1.1) $$

This axiom captures the intuitive notion that probabilities of mutually exclusive outcomes add linearly. Within this framework, events with probability zero are negligible in the sense that they do not affect the almost sure properties of random variables and stochastic processes.

A random variable is a measurable function from the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ to a measurable space $(E, \mathcal{E})$, most commonly $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$.

**Definition 1.2 (Random Variable).** A random variable $X$ is a measurable function

$$ X: (\Omega, \mathcal{F}) \to (\mathbb{R}, \mathcal{B}(\mathbb{R})), \qquad (1.2) $$

where $\mathcal{B}(\mathbb{R})$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$.

The pushforward measure $\mathbb{P}_X(B) := \mathbb{P}(X^{-1}(B))$ defines the distribution of $X$. Expectations are Lebesgue integrals with respect to $\mathbb{P}$, denoted $\mathbb{E}[X] = \int_{\Omega} X d\mathbb{P}$. Formally, a function $X: \Omega \to \mathbb{R}$ is a random variable if, for every $B \in \mathcal{B}(\mathbb{R})$, the preimage

$$ X^{-1}(B) = \omega \in \Omega: X(\omega) \in B \in \mathcal{F} \qquad (1.3) $$

This requirement ensures that probabilistic statements about $X$ can be expressed in terms of the underlying measure $\mathbb{P}$. For instance, the probability that $X$ falls within a set $B$ is defined as

$$ \mathbb{P}(X \in B) := \mathbb{P}(X^{-1}(B)). \qquad (1.4) $$

The distribution of a random variable $X$ is the measure $\mathbb{P}_X$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ given by $\mathbb{P}_X(B) = \mathbb{P}(X \in B)$, which completely characterizes the probabilistic behavior of $X$. In this context, the expectation of an integrable random variable $X \in L^1(\Omega, \mathcal{F}, \mathbb{P})$ is defined as the Lebesgue integral

$$ \mathbb{E}[X] := \int_{\Omega} X(\omega), d\mathbb{P}(\omega), \qquad (1.5) $$

providing a linear functional on the space of integrable random variables. Higher moments, variances, and covariance structures are defined analogously, and they form the building blocks for the study of convergence, limit theorems, and stochastic calculus.

Random variables can be combined to form vectors or stochastic processes. If $X_1, X_2, \dots, X_n$ are random variables on the same probability space, their joint distribution is defined on the product measurable space $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ by

$$ \mathbb{P}_{X_1, \dots, X_n}(B_1 \times \dots \times B_n) := \mathbb{P}(X_1 \in B_1, \dots, X_n \in B_n), \qquad (1.6) $$
for all $B_1, \dots, B_n \in \mathcal{B}(\mathbb{R})$. This definition generalizes naturally to infinite-dimensional settings, leading to the theory of stochastic processes where each $X_t$ is a random variable indexed by time $t$. The concept of measurability ensures that all probabilistic constructions, including expectations, conditional distributions, and integrals with respect to probability measures, are mathematically well-defined.

Independence of random variables is also formalized within this framework. A collection $X_i * i \in I$ is said to be independent if, for any finite subcollection $X * i_1, \dots, X_{i_n}$ and any sets $B_1, \dots, B_n \in \mathcal{B}(\mathbb{R})$, one has

$$ \mathbb{P}(X_{i_1} \in B_1, \dots, X_{i_n} \in B_n) = \prod_{k=1}^{n} \mathbb{P}(X_{i_k} \in B_k). \qquad (1.7) $$

This rigorous, measure-theoretic definition provides the foundation for the development of martingales, Markov processes, and Brownian motion, and ultimately allows the extension to continuous-time stochastic calculus, which is the core framework underlying the Feynman-Kac formula.

## 1.3 Independence and Conditional Expectation

Independence is one of the central concepts in probability theory, formalizing the notion that the occurrence of one event or the value of one random variable does not influence another. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, and let $\mathcal{A}, \mathcal{B} \subset \mathcal{F}$ be two sub-$\sigma$-algebras.

**Definition 1.3 (Independence).** Two $\sigma$-algebras $\mathcal{A}, \mathcal{B} \subset \mathcal{F}$ are said to be *independent* if

$$ \mathbb{P}(A \cap B) = \mathbb{P}(A)\mathbb{P}(B), \quad \forall A \in \mathcal{A}, B \in \mathcal{B}. \qquad (1.8) $$

This property implies that the probability of the intersection of events factorizes as the product of individual probabilities. Random variables $X, Y$ are independent if the $\sigma$-algebras they generate, $\sigma(X)$ and $\sigma(Y)$, are independent.

A collection of $\sigma$-algebras $\mathcal{F}_i, i \in I$ is independent if every finite subcollection is independent in this sense. Random variables $X$ and $Y$ are independent if the $\sigma$-algebras they generate, $\sigma(X)$ and $\sigma(Y)$, are independent. Equivalently, for all Borel sets $B_1, B_2 \subset \mathbb{R}$,

$$ \mathbb{P}(X \in B_1, Y \in B_2) = \mathbb{P}(X \in B_1), \mathbb{P}(Y \in B_2). \qquad (1.9) $$

Independence extends naturally to families of random variables and to stochastic processes, allowing the decomposition of joint distributions into products of marginal distributions, which is crucial for the construction of product measures and for establishing limit theorems such as the law of large numbers and the central limit theorem.

Conditional expectation generalizes the classical notion of expectation by incorporating partial information represented by a sub-$\sigma$-algebra. Let $X \in L^1(\Omega, \mathcal{F}, \mathbb{P})$ be an integrable random variable, and let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
**Definition 1.4** (Conditional Expectation). Let $X \in L^1(\Omega, \mathcal{F}, \mathbb{P})$ and $\mathcal{G} \subset \mathcal{F}$ a sub-$\sigma$-algebra. The conditional expectation of $X$ given $\mathcal{G}$ is the unique (a.s.) $\mathcal{G}$-measurable random variable $\mathbb{E}[X|\mathcal{G}]$ satisfying

$$ \int_G \mathbb{E}[X|\mathcal{G}] d\mathbb{P} = \int_G X d\mathbb{P}, \quad \forall G \in \mathcal{G}. \qquad (1.10) $$

Conditional expectations generalize the notion of best approximation in $L^2$ to the $\sigma$-algebra $\mathcal{G}$. They play a central role in defining martingales and Markov processes. This definition ensures that $\mathbb{E}[X|\mathcal{G}]$ preserves the integral of $X$ over all events measurable with respect to $\mathcal{G}$, and it can be interpreted as the "best prediction" of $(X)$ given the information in $\mathcal{G}$. When $\mathcal{G} = \emptyset, \Omega$ is the trivial $\sigma$-algebra, $\mathbb{E}[X|\mathcal{G}] = \mathbb{E}[X]$, whereas if $\mathcal{G} = \mathcal{F}$, then $\mathbb{E}[X|\mathcal{G}] = X$.

Conditional expectation satisfies linearity, monotonicity, and the tower property: for $\mathcal{H} \subset \mathcal{G} \subset \mathcal{F}$,

$$ \mathbb{E}[\mathbb{E}[X | \mathcal{G}] | \mathcal{H}] = \mathbb{E}[X | \mathcal{H}]. \qquad (1.11) $$

If $X$ is independent of $\mathcal{G}$, then $\mathbb{E}[X | \mathcal{G}] = \mathbb{E}[X]$ almost surely. For two integrable random variables $X$ and $Y$, the conditional expectation $\mathbb{E}[X | Y]$ is defined as $\mathbb{E}[X | \sigma(Y)]$, providing a measurable function $g(Y)$ such that

$$ \mathbb{E}[X \mathbf{1}_{Y \in B}] = \mathbb{E}[g(Y) \mathbf{1}_{Y \in B}], \quad \forall B \in \mathcal{B}(\mathbb{R}). \qquad (1.12) $$

Conditional expectation also interacts naturally with integrals and limits, allowing the application of dominated convergence and monotone convergence theorems in a conditional setting. This machinery underpins martingale theory, stochastic integration, and the rigorous development of Markov processes, forming the analytic foundation necessary for the Feynman-Kac formula, where conditional expectations over Brownian paths produce solutions to parabolic partial differential equations.

## 1.4 Filtrations and Adapted Processes

In the study of stochastic processes, the concept of a filtration formalizes the evolution of information over time. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. A filtration is a family of sub-$\sigma$-algebras $(\mathcal{F}_t)_{t \ge 0}$ such that

$$ \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}, \quad \forall, 0 \le s \le t. \qquad (1.13) $$

Each $\mathcal{F}_t$ represents the collection of events whose outcomes are "known" at time $(t)$. The increasing nature of the filtration ensures that no information is lost as time progresses; rather, knowledge accumulates, which is essential for the precise definition of conditional expectations and martingales in continuous time.

**Definition 1.5** (Filtration). A filtration $(\mathcal{F}_t)_{t \ge 0}$ on $(\Omega, \mathcal{F})$ is an increasing family of sub-$\sigma$-algebras:

$$ \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}, \quad \forall s \le t. \qquad (1.14) $$

The pair $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ is called a filtered probability space.

Filtrations provide the formal backbone for modeling temporal dependencies in stochastic processes, particularly when dealing with adapted and progressively measurable processes.
A stochastic process $(X_t)_{t \ge 0}$ is said to be adapted to a filtration $(\mathcal{F}_t)$ if, for every $t \ge 0$, the random variable $X_t$ is $\mathcal{F}_t$-measurable, i.e.,

$$X_t^{-1}(B) \in \mathcal{F}_t, \quad \forall B \in \mathcal{B}(\mathbb{R}). \qquad (1.15)$$

Adaptedness ensures that the value of the process at time $(t)$ depends only on the information available up to that time, reflecting the natural causality in stochastic dynamics. This condition is fundamental in defining stochastic integrals, where integrands must be adapted to guarantee that they do not "anticipate" future information.

Progressive measurability is a stronger regularity condition. A process $(X_t)_{t \ge 0}$ is progressively measurable with respect to $(\mathcal{F}_t)$ if, for every $t \ge 0$, the mapping $(s, \omega) \mapsto X_s(\omega)$ restricted to $[0, t] \times \Omega$ is measurable with respect to the product $\sigma$-algebra $\mathcal{B}([0, t]) \otimes \mathcal{F}_t$. Formally, for all Borel sets $B \subset \mathbb{R}$,

$$(s, \omega) \in [0, t] \times \Omega : X_s(\omega) \in B \in \mathcal{B}([0, t]) \otimes \mathcal{F}_t. \qquad (1.16)$$

Progressive measurability guarantees that the process can be integrated in the Itô sense, and it ensures the existence of modifications with continuous paths when such regularity is required.

**Definition 1.6** (Adapted and Progressive Processes). A stochastic process $(X_t)_{t \ge 0}$ is said to be:

* Adapted if $X_t$ is $\mathcal{F}_t$-measurable for all $t$.

* Progressively measurable if $(t, \omega) \mapsto X_t(\omega)$ is measurable with respect to $\mathcal{B}([0, t]) \otimes \mathcal{F}_t$ for every $t$.

Importantly, every progressively measurable process is adapted, but the converse is not necessarily true. In practice, most processes of interest in stochastic analysis, such as Brownian motion and solutions to stochastic differential equations, are constructed to be progressively measurable.

Filtrations can also be augmented to satisfy the usual conditions: completeness and right-continuity. A filtration $(\mathcal{F}_t)$ is complete if all $\mathbb{P}$-null sets of $\mathcal{F}$ are included in each $\mathcal{F}_t$. Right-continuity means that $\mathcal{F}_t = \bigcap_{s>t} \mathcal{F}_s$. These conditions are essential in the formulation of the strong Markov property and in ensuring that stopping times are well-behaved. In particular, the right-continuous augmented filtration guarantees that stopping times $\tau$ satisfy $\tau \le t \in \mathcal{F}_t$ for all $t$, enabling rigorous applications of optional sampling theorems and martingale convergence results.

The interplay between filtrations and adapted processes forms the structural backbone for modern stochastic calculus. For a stochastic integral $\int_0^t X_s, dB_s$ with respect to a Brownian motion $B_t$, the integrand $X_s$ must be adapted to the natural filtration generated by $(B)$ to prevent anticipating future increments. Likewise, the conditional expectation $\mathbb{E}[X_t | \mathcal{F}_s]$ for $s \le t$ is meaningful only in the context of an adapted process relative to the filtration $(\mathcal{F}_t)$. In the context of the Feynman-Kac formula, the adaptedness of functionals of Brownian motion
guarantees that stochastic integrals and exponential functionals are well-defined and measurable with respect to the natural filtration, allowing a pathwise representation of solutions to parabolic partial differential equations.

## 1.5 Martingales and Stopping Times

Martingales form one of the cornerstones of modern probability theory, capturing the notion of a "fair game" in a rigorous measure-theoretic framework. Let $(\Omega, \mathcal{F}, (\mathcal{F} * t) * t \ge 0, \mathbb{P})$ be a filtered probability space satisfying the usual conditions, and let $(M_t)_{t \ge 0}$ be an adapted stochastic process with $M_t \in L^1(\Omega)$ for all $t \ge 0$.

**Definition 1.7 (Martingale).** Let $(\mathcal{F}_t)$ be a filtration. A process $(M_t)_{t \ge 0}$ is a martingale if:

1. $M_t \in L^1(\Omega)$ for all $t$;

2. $\mathbb{E}[M_t|\mathcal{F}_s] = M_s$ a.s. for all $s < t$.

If $\mathbb{E}[M_t|\mathcal{F}_s] \ge M_s$, it is a submartingale; if $\le$, a supermartingale.

This property expresses the intuitive idea that, given all information available up to time $s$, the conditional expectation of future values equals the present value, implying that there is no predictable trend or drift in the process. If the inequality

$$ \mathbb{E}[M_t | \mathcal{F}_s] \ge M_s \qquad (1.17) $$

holds, the process is called a submartingale, representing a process with a nonnegative expected drift, while the reverse inequality defines a supermartingale. Martingales satisfy several fundamental properties, including linearity, preservation under stopping times (under appropriate conditions), and the martingale convergence theorem, which asserts that certain bounded or integrable martingales converge almost surely and in $L^1$ as time tends to infinity.

Stopping times are random times at which one may choose to observe or "stop" a stochastic process based only on past and present information. Formally, a random variable $\tau: \Omega \to [0, \infty]$ is a stopping time with respect to a filtration $(\mathcal{F}_t)$ if, for each $t \ge 0$, the event $\tau \le t \in \mathcal{F}_t$.

**Definition 1.8 (Stopping Time).** A random variable $\tau: \Omega \to [0, \infty]$ is a stopping time if

$$ \{\omega: \tau(\omega) \le t\} \in \mathcal{F}_t, \quad \forall t \ge 0. \qquad (1.18) $$

Stopping times encode random observation times determined by the evolution of the process itself, and are indispensable in the optional sampling and strong Markov properties of diffusions.

This definition ensures that the decision to stop at time $\tau$ depends only on information available up to that time, prohibiting any anticipation of future events. Classic examples include the first exit time from a set $D \subset \mathbb{R}^n$ by a stochastic process $X_t$:

$$ \tau_D := \inf t \ge 0: X_t \notin D. \qquad (1.19) $$
Stopping times are central to optional sampling and the strong Markov property. The optional sampling theorem states that if $M_t$ is a martingale and $\tau$ is a bounded stopping time, then

$$ \mathbb{E}[M_{\tau}] = \mathbb{E}[M_0]. \qquad (1.20) $$

More generally, for two stopping times $\sigma \le \tau$, one has

$$ \mathbb{E}[M_{\tau} | \mathcal{F}_{\sigma}] = M_{\sigma} \quad \text{a.s.,} \qquad (1.21) $$

under suitable integrability conditions. This theorem formalizes the notion that a martingale retains its "fairness" even when stopped at a random time. Stopping times also facilitate the construction of stopped processes $M_{t \wedge \tau}$, which remain martingales with respect to the original filtration.

Martingales and stopping times interact intimately in the analysis of Markov processes, Brownian motion, and solutions to stochastic differential equations. For example, in the context of a Brownian motion $B_t$ and a sufficiently smooth function $f$, the process

$$ M_t := f(B_t) - \int_0^t \frac{1}{2} \Delta f(B_s), ds \qquad (1.22) $$

is a martingale, and stopping it at the first exit time from a domain $D$ yields a powerful probabilistic representation of solutions to the Dirichlet problem. These concepts underpin the Feynman-Kac formula, where exponential functionals of martingales stopped at random times represent solutions of parabolic partial differential equations with boundary conditions.

Finally, martingale theory provides convergence and decomposition results, such as Doob's martingale convergence theorem and the Doob-Meyer decomposition, which allow any sub-martingale to be expressed as the sum of a martingale and a predictable, increasing process. These results form the analytic foundation for continuous-time stochastic integration, stochastic calculus, and the rigorous derivation of the Feynman-Kac representation, establishing a deep connection between probabilistic and analytic formulations of evolution equations.

## 1.6 Stochastic Processes and Finite-Dimensional Distributions

A stochastic process is a collection of random variables indexed by a parameter, typically interpreted as time. Formally, let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, and let $T \subseteq \mathbb{R}$ denote the index set, often $T = [0, \infty)$. A stochastic process is a family $(X_t)_{t \in T}$ of random variables $X_t: \Omega \to \mathbb{R}^d$, each measurable with respect to $\mathcal{F}$ and the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^d)$. The process may be viewed equivalently as a single measurable mapping from $\Omega$ into the product space $\mathbb{R}^{d \times T}$, endowed with the product $\sigma$-algebra generated by cylinder sets. This viewpoint is particularly useful when constructing measures on spaces of paths, such as $C([0, \infty), \mathbb{R}^d)$ or $D([0, \infty), \mathbb{R}^d)$.

The law of a stochastic process is determined by its finite-dimensional distributions. For any finite collection of times $t_1 < t_2 < \dots < t_n$ in $T$, the corresponding finite-dimensional distribution
is the measure on $(\mathbb{R}^{dn}, \mathcal{B}(\mathbb{R}^{dn}))$ defined by

$$ \mu_{t_1, \dots, t_n}(B_1 \times \dots \times B_n) := \mathbb{P}(X_{t_1} \in B_1, \dots, X_{t_n} \in B_n), \quad B_i \in \mathcal{B}(\mathbb{R}^d). \quad (1.23) $$

These distributions describe the joint behavior of the process at finitely many times and must satisfy consistency conditions: if $\pi_{1,...,n}$ denotes the projection from $\mathbb{R}^{d(m)}$ to $\mathbb{R}^{dn}$ with $m > n$, then

$$ \mu_{t_1, \dots, t_n} = \mu_{t_1, \dots, t_m} \circ \pi_{1, \dots, n}^{-1}. \quad (1.24) $$

Consistency guarantees that the collection of finite-dimensional distributions defines a unique measure on the path space, a result formalized in the Kolmogorov Extension Theorem. Explicitly, if $\mu_{t_1, \dots, t_n} * n \in \mathbb{N}$, $t_1 < \dots < t_n \in T$ is a consistent family, there exists a probability space $(\Omega', \mathcal{F}', \mathcal{P}')$ and a stochastic process $(X_t) * t \in T$ on this space such that the finite-dimensional distributions of $X$ coincide with the given measures $\mu_{t_1, \dots, t_n}$.

This construction allows the rigorous definition of canonical processes on path spaces. For instance, consider the canonical space $\Omega = C([0, \infty), \mathbb{R}^d)$ of continuous functions, equipped with the Borel $\sigma$-algebra generated by the topology of uniform convergence on compact intervals. The coordinate mappings $X_t(\omega) = \omega(t)$ define a process whose law is entirely determined by the finite-dimensional distributions. In this context, conditions such as continuity, Hölder regularity, or bounded variation can be imposed pathwise, and Kolmogorov's continuity theorem ensures the existence of a continuous modification if certain moment conditions on the finite-dimensional distributions are satisfied: if there exist constants $\alpha, \beta, C > 0$ such that

$$ \mathbb{E}[|X_t - X_s|^\alpha] \le C|t - s|^{1+\beta}, \quad \forall s, t \in T, \quad (1.25) $$

then there exists a modification of the process with almost surely Hölder continuous paths of order $\gamma$ for any $\gamma < \beta/\alpha$.

The notion of finite-dimensional distributions also allows the precise definition of independence and stationarity for stochastic processes. A process $(X_t) * t \in T$ has independent increments if, for any finite sequence $0 \le t_0 < t_1 < \dots < t_n$, the increments $X * t_1 - X_{t_0}, \dots, X_{t_n} - X_{t_{n-1}}$ are independent random variables. Stationarity of the finite-dimensional distributions, i.e., $\mathbb{P}(X_{t_1+h} \in B_1, \dots, X_{t_n+h} \in B_n) = \mathbb{P}(X_{t_1} \in B_1, \dots, X_{t_n} \in B_n)$ for all $h$, ensures temporal homogeneity, which is crucial in the construction of Markov processes and Brownian motion.

In summary, finite-dimensional distributions provide the probabilistic skeleton of a stochastic process, encoding the joint behavior at any finite set of times. Through consistency and extension theorems, they enable the rigorous definition of processes on function spaces, control over path properties, and the foundation for the development of Markovian and martingale structures that underpin stochastic calculus and the Feynman-Kac formula.

A stochastic process is a collection of random variables $(X_t)_{t \in T}$ indexed by time. Its law is determined by the family of finite-dimensional distributions:

$$ \mu_{t_1, \dots, t_n}(A_1 \times \dots \times A_n) := \mathbb{P}(X_{t_1} \in A_1, \dots, X_{t_n} \in A_n), \quad (1.26) $$
subject to consistency conditions ensuring the existence of a process via the Kolmogorov Extension Theorem.

**Theorem 1.9 (Kolmogorov Extension).** Given a family of consistent finite-dimensional distributions on $\mathbb{R}^T$, there exists a stochastic process with these finite-dimensional marginals.

*Proof.* Let (T) be a non-empty index set and consider the product space $\Omega := \mathbb{R}^T$, whose elements are functions $\omega : T \to \mathbb{R}$. Let $\mathcal{C}$ denote the collection of cylinder sets, that is, sets of the form

$$C = \omega \in \Omega : (\omega(t_1), \dots, \omega(t_n)) \in B, \qquad (1.27)$$

where $n \in \mathbb{N}$, $(t_1, \dots, t_n \in T$ are distinct, and $B \in \mathcal{B}(\mathbb{R}^n)$ is a Borel set. The collection $\mathcal{C}$ generates the cylinder $\sigma$-algebra $\mathcal{F}_{\text{cyl}} := \sigma(\mathcal{C})$, which will serve as the initial domain for defining a measure corresponding to the finite-dimensional distributions.

Suppose we are given a family of finite-dimensional distributions $\mu_{t_1, \dots, t_n} * n \in \mathbb{N}, t_i \in T$ on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ satisfying the consistency condition: for any $m > n$ and any selection of times $t_1, \dots, t_m$, the marginal on the first $n$ coordinates coincides with $\mu * t_1, \dots, t_n$, i.e.,

$$\mu_{t_1, \dots, t_m} \circ \pi^{-1} * 1, \dots, n = \mu * t_1, \dots, t_n, \qquad (1.28)$$

where $\pi_{1, \dots, n} : \mathbb{R}^m \to \mathbb{R}^n$ is the canonical projection $(x_1, \dots, x_m) \mapsto (x_1, \dots, x_n)$. This consistency ensures that the family is projectively compatible, a necessary condition for the extension to a measure on the full product space.

Define a set function $\mathbb{P}_0$ on the algebra $\mathcal{C}$ by

$$\mathbb{P} * 0(\omega \in \Omega : (\omega(t_1), \dots, \omega(t_n)) \in B) := \mu * t_1, \dots, t_n(B), \quad B \in \mathcal{B}(\mathbb{R}^n). \qquad (1.29)$$

The consistency of the finite-dimensional distributions guarantees that $\mathbb{P}_0$ is well-defined; that is, if the same cylinder set can be represented using different coordinate sets, the assigned probability is independent of the representation. It is straightforward to verify that $\mathbb{P}_0$ is finitely additive on $\mathcal{C}$ and satisfies $\mathbb{P}_0(\Omega) = 1$.

Next, we invoke **Carathéodory's extension theorem**. Since $\mathcal{C}$ is an algebra and $\mathbb{P} * 0$ is countably additive on increasing sequences of cylinder sets (which follows from the continuity of measures in finite dimensions), there exists a unique measure $\mathbb{P}$ on $\sigma(\mathcal{C}) = \mathcal{F} * \text{cyl}$ such that

$$\mathbb{P}|_{\mathcal{C}} = \mathbb{P}_0. \qquad (1.30)$$

This measure $\mathbb{P}$ now lives on the product space $\Omega = \mathbb{R}^T$ with the cylinder $\sigma$-algebra, and by construction, the coordinate projections $X_t : \Omega \to \mathbb{R}$, defined by $X_t(\omega) := \omega(t)$, are random variables with the property that, for any finite set of times $t_1, \dots, t_n \in T$ and any Borel set $B \in \mathcal{B}(\mathbb{R}^n)$,

$$\mathbb{P}((X_{t_1}, \dots, X_{t_n}) \in B) = \mu_{t_1, \dots, t_n}(B). \qquad (1.31)$$

This demonstrates that the process $(X_t) * t \in T$ has the prescribed finite-dimensional distributions. The uniqueness of $\mathbb{P}$ on $(\mathcal{F} * \text{cyl})$ guarantees that any other process with the same
finite-dimensional distributions is equivalent in law, i.e., their pushforward measures on the product space coincide.

If the index set (T) is uncountable, one may refine the construction by considering the Kolmogorov consistency conditions and applying extensions to the product σ-algebra generated by all cylinder sets. In particular, one can construct a version of the process on a path space Ω = ℝ^T or on C([0, ∞), ℝ^d) using tightness arguments and continuity theorems such as Kolmogorov's continuity criterion, which ensures the existence of a modification with continuous paths under suitable moment bounds on the finite-dimensional distributions: if there exist constants α, β, C > 0 such that

$$ \mathbb{E}[|X_t - X_s|^\alpha] \le C|t-s|^{1+\beta} \quad \forall s, t \in T \quad (1.32) $$

then a continuous modification exists almost surely. This completes the construction of a stochastic process consistent with any given family of finite-dimensional distributions, yielding the rigorous statement of the Kolmogorov Extension Theorem. □

## 1.7 Markov Processes and Transition Kernels

A stochastic process $(X_t)_{t \ge 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ is called a Markov process if it satisfies the **Markov property**, which formalizes the notion that the conditional distribution of the future, given the past and present, depends only on the present. Specifically, for all $0 \le s \le t$ and all Borel sets $B \subset \mathbb{R}^d$,

$$ \mathbb{P}(X_t \in B | \mathcal{F}_s) = \mathbb{P}(X_t \in B | X_s) \quad \text{almost surely.} \quad (1.33) $$

Equivalently, for any finite collection of times $0 \le t_0 < t_1 < \dots < t_n$ and any Borel sets $B_1, \dots, B_n \subset \mathbb{R}^d$,

$$ \mathbb{P}(X_{t_n} \in B_n | X_{t_0}, \dots, X_{t_{n-1}}) = \mathbb{P}(X_{t_n} \in B_n | X_{t_{n-1}}), \quad (1.34) $$

which expresses the memoryless property of the process. The Markov property ensures that the evolution of the process can be described entirely in terms of **transition probabilities** from the current state to future states.

These transition probabilities are formalized using **transition kernels**. A transition kernel is a function $P: \mathbb{R}^d \times \mathcal{B}(\mathbb{R}^d) \to [0, 1]$ satisfying two conditions:

1. For each fixed $x \in \mathbb{R}^d$, the mapping $B \mapsto P(x, B)$ is a probability measure on $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$

2. For each fixed $B \in \mathcal{B}(\mathbb{R}^d)$, the mapping $x \mapsto P(x, B)$ is measurable.

If $(X_t)$ is a Markov process, its finite-dimensional distributions can be expressed recursively via these transition kernels. For $s < t$, let $P_{s,t}(x, B)$ denote the conditional probability

$$ P_{s,t}(x, B) := \mathbb{P}(X_t \in B | X_s = x), \quad (1.35) $$
so that for any measurable function $f: \mathbb{R}^d \to \mathbb{R}$, the conditional expectation can be written as

$$ \mathbb{E}[f(X_t) | X_s] = \int_{\mathbb{R}^d} f(y), P_{s,t}(X_s, dy) \quad \text{a.s.} \qquad (1.36) $$

In the **time-homogeneous case**, the kernel depends only on the time difference $\tau = t-s$, so that $P_{\tau}(x, B) := P_{s,s+\tau}(x, B)$. The Chapman-Kolmogorov equation then governs the composition of these kernels: for $0 \le s < u < t$,

$$ P_{s,t}(x, B) = \int_{\mathbb{R}^d} P_{s,u}(x, dy), P_{u,t}(y, B), \qquad (1.37) $$

which expresses the semigroup property of the transition kernels and is fundamental for the analytic characterization of Markov processes via generators and semigroups.

Transition kernels allow one to construct Markov processes explicitly through **finite-dimensional distributions**. Given an initial distribution $\mu_0$ on $\mathbb{R}^d$ and a family of transition kernels $(P_{s,t})_{0 \le s \le t}$ satisfying consistency and measurability, the joint distribution of $(X * t_0, \dots, X_{t_n})$ is given recursively by

$$ \mathbb{P}(X_{t_0} \in B_0, \dots, X_{t_n} \in B_n) = \int_{B_0} \mu_0(dx_0) \int_{B_1} P_{t_0, t_1}(x_0, dx_1) \cdots \int_{B_n} P_{t_{n-1}, t_n}(x_{n-1}, dx_n), \quad (1.38) $$

which guarantees consistency with the finite-dimensional marginals. This construction, combined with the Kolmogorov Extension Theorem, yields a rigorous realization of the Markov process on a suitable path space.

**Definition 1.10 (Markov Process).** A process $(X_t)_{t \ge 0}$ adapted to $(\mathcal{F}_t)$ is Markov if

$$ \mathbb{P}(X_t \in A | \mathcal{F}_s) = \mathbb{P}(X_t \in A | X_s), \quad s < t. \qquad (1.39) $$

For a Markov process, $p_t(x, A) = \mathbb{P}(X_t \in A | X_0 = x)$ forms a semigroup of transition probabilities satisfying the Chapman-Kolmogorov equation:

$$ p_{s+t}(x, A) = \int_E p_s(x, dy) p_t(y, A) \qquad (1.40) $$

In the case of continuous-time processes with continuous paths, such as diffusion processes or Brownian motion, transition kernels can be described in terms of **densities** $p(t, x, y)$ with respect to the Lebesgue measure, satisfying

$$ P_{s,t}(x, B) = \int_B p(t - s, x, y), dy. \qquad (1.41) $$

The Chapman-Kolmogorov equation then becomes

$$ p(t + s, x, z) = \int_{\mathbb{R}^d} p(t, x, y), p(s, y, z), dy, \qquad (1.42) $$

which is a crucial analytic identity connecting probabilistic evolution with semigroup theory. This representation underlies the derivation of infinitesimal generators $\mathcal{L}$ and the associated Kolmogorov forward (Fokker-Planck) and backward equations, which in turn lead directly to the Feynman-Kac formula when coupled with suitable potential terms.
Thus, transition kernels provide a rigorous probabilistic structure for Markov processes, enabling recursive construction of finite-dimensional distributions, formulation of semigroup properties, and analytic characterization through generators, all of which are foundational in stochastic analysis and the study of parabolic partial differential equations via probabilistic methods.

## 1.8 Generators and Semigroups

Let $(X_t) * t \ge 0$ be a time-homogeneous Markov process with state space $\mathbb{R}^d$ and associated transition semigroup $(P_t) * t \ge 0$, where for any bounded measurable function $f: \mathbb{R}^d \to \mathbb{R}$,

$$P_t f(x) := \mathbb{E}[f(X_t) | X_0 = x] = \int_{\mathbb{R}^d} f(y), P_t(x, dy). \quad (1.43)$$

The family $(P_t)_{t \ge 0}$ satisfies the **semigroup property**: for all $s, t \ge 0$,

$$P_{t+s} = P_t P_s, \quad P_0 = I, \quad (1.44)$$

where $I$ denotes the identity operator, and $P_t P_s f(x) := P_t(P_s f)(x)$. The semigroup property is a direct consequence of the Chapman-Kolmogorov equation for the transition kernels and encodes the memoryless evolution of the Markov process.

The infinitesimal generator $\mathcal{L}$ of the semigroup $(P_t)$ is a linear operator defined on a domain $\mathcal{D}(\mathcal{L}) \subset C_b(\mathbb{R}^d)$, the space of bounded continuous functions, by the strong limit

$$\mathcal{L}f := \lim_{t \downarrow 0} \frac{P_t f - f}{t}, \quad f \in \mathcal{D}(\mathcal{L}), \quad (1.45)$$

where the limit is taken in the supremum norm. Intuitively, $\mathcal{L}f(x)$ measures the instantaneous rate of change of the expected value of $f(X_t)$ at time zero, conditioned on $X_0 = x$. For many diffusion processes with smooth coefficients, $\mathcal{L}$ takes the form of a second-order differential operator

$$\mathcal{L}f(x) = \sum_{i=1}^{d} b_i(x) \frac{\partial f}{\partial x_i}(x) + \frac{1}{2} \sum_{i,j=1}^{d} a_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j}(x), \quad (1.46)$$

where $b: \mathbb{R}^d \to \mathbb{R}^d$ is the drift vector and $a(x) = \sigma(x)\sigma(x)^T$ is the diffusion matrix associated with the stochastic differential equation

$$dX_t = b(X_t), dt + \sigma(X_t), dB_t. \quad (1.47)$$

The connection between semigroups and generators is expressed rigorously through the **Kolmogorov backward equation**: for $f \in \mathcal{D}(\mathcal{L})$,

$$\frac{\partial}{\partial t} P_t f = \mathcal{L} P_t f, \quad P_0 f = f \quad (1.48)$$

which is an abstract Cauchy problem in the Banach space $C_b(\mathbb{R}^d)$. Conversely, given a closed linear operator $\mathcal{L}$ satisfying appropriate conditions (such as the Hille-Yosida theorem), one can construct a strongly continuous semigroup $(P_t)$ solving this equation. This correspondence pro-
vides a rigorous analytic foundation for representing the evolution of Markov processes in terms of their generators.

In addition, the generator characterizes the martingale property of functionals of the process. If $f \in \mathcal{D}(\mathcal{L})$, the process

$$M_t^f := f(X_t) - f(X_0) - \int_0^t \mathcal{L}f(X_s), ds \quad (1.49)$$

is a martingale with respect to the natural filtration of $X_t$. This representation is fundamental for deriving the **Feynman-Kac formula**, where one modifies the generator by a potential term $V(x)$ and considers exponential functionals of the resulting martingale. Specifically, if $\mathcal{L} - V$ denotes the generator with killing, the Feynman-Kac representation of the solution $u(t, x)$ to the parabolic PDE

$$\frac{\partial u}{\partial t} = \mathcal{L}u - V(x)u, \quad u(0, x) = f(x) \quad (1.50)$$

is given probabilistically by

$$u(t, x) = \mathbb{E}_x \left[ \exp \left( - \int_0^t V(X_s), ds \right) f(X_t) \right] \quad (1.51)$$

where the expectation is taken over the paths of the Markov process generated by $\mathcal{L}$ starting from $X_0 = x$. Thus, the semigroup and generator formalism provides a rigorous bridge between stochastic processes, martingales, and the analytic theory of partial differential equations, forming the backbone of the Feynman-Kac methodology.

In later chapters, we shall show that the Feynman-Kac formula identifies $P_t$ with the semigroup generated by the Schrödinger-type operator $A + V$, where $V$ acts as a potential.

## 1.9 Conclusion

This chapter established the rigorous measure-theoretic framework for stochastic processes. The machinery of filtrations, conditional expectations, and semigroup generators forms the mathematical substrate on which the Feynman-Kac formula rests.

In the next chapter, we shall construct the Brownian motion as the canonical continuous Markov process on $C([0, \infty), \mathbb{R}^n)$, develop Itô integration and stochastic calculus, and demonstrate how these lead naturally to the Feynman-Kac representation of parabolic PDEs.

**References for this chapter:** Karatzas & Shreve (2014) [1], Øksendal (2013) [2], Revuz & Yor (1999) [3], Stroock (2010) [4], Doob (1984) [5], Billingsley (2017) [6] and Billingsley (2013) [7], Durrett (2019) [8].
# Brownian Motion and Stochastic Integrals

> “Brownian motion is an inexhaustible source of beautiful mathematics and useful models.”

> — Adapted from P. Mörters & Y. Peres

## 2.1 Introduction

This chapter constructs Brownian motion as the canonical continuous Gaussian process, develops its fundamental pathwise and martingale properties, introduces the Itô stochastic integral for non-anticipative integrands, and proves Itô’s formula together with basic consequences such as the martingale representation and quadratic variation. These tools are indispensable for the probabilistic representation of linear and semilinear parabolic equations and form the analytic backbone of the Feynman–Kac correspondence treated in subsequent chapters.

## 2.2 Definition and Basic Properties of Brownian Motion

**Definition 2.1** (Brownian Motion / Wiener Process). A stochastic process $(B_t)_{t \ge 0}$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with values in $\mathbb{R}^d$ is called a (standard, $d$-dimensional) *Brownian motion* if:

1. $B_0 = 0$ almost surely;

2. $B$ has independent increments: for $0 \le t_0 < t_1 < \cdots < t_n$, the increments $B_{t_1} - B_{t_0}, \dots, B_{t_n} - B_{t_{n-1}}$ are independent;

3. For $0 \le s < t$, the increment $B_t - B_s$ is Gaussian with mean 0 and covariance $(t-s)I_d$;

4. $B$ has almost surely continuous paths: $\omega \mapsto t \mapsto B_t(\omega)$ is continuous.

From the Gaussian increment property it follows that for each fixed $t$, $B_t \sim \mathcal{N}(0, tI_d)$, and that all finite-dimensional distributions are multivariate normal with covariances $\mathbb{E}[B_s B_t^\top] = (s \wedge t)I_d$. The finite-dimensional covariance structure and consistency of the Gaussian family guarantee existence via the Kolmogorov extension theorem; continuity of paths is furnished by Kolmogorov’s continuity criterion (see below).
**Proposition 2.2 (Existence and Continuity).** Given the Gaussian family with mean zero and covariance $E[B_s^{(i)} B_t^{(j)}] = \delta_{ij}(s \wedge t)$, there exists a probability measure on $\Omega = C([0, \infty), \mathbb{R}^d)$ making the coordinate process a Brownian motion. Moreover, the process admits almost surely Hölder continuous paths of any order $\gamma \in (0, 1/2)$.

*Proof.* Let $(\mu_{t_1, \dots, t_n})$ denote the consistent family of $dn$-dimensional Gaussian measures corresponding to the finite-dimensional distributions with mean zero and covariance

$$E[B_{t_k}^{(i)} B_{t_\ell}^{(j)}] = \delta_{ij}(t_k \wedge t_\ell), \quad 1 \le i, j \le d, \; 1 \le k, \ell \le n. \qquad (2.1)$$

By the **Kolmogorov Extension Theorem**, there exists a probability measure $P_0$ on $(\mathbb{R}^{d \times T}, \mathcal{B}(\mathbb{R}^{d \times T}))$ such that the coordinate projections $B_t(\omega) := \omega(t)$ have these finite-dimensional distributions. This provides a process with the correct Gaussian marginals but not yet with continuous paths.

Define for each $n \in \mathbb{N}$ the measure $P_n$ on the space $D([0, n], \mathbb{R}^d)$ of càdlàg functions (right-continuous with left limits) restricted to $[0, n]$ equipped with the **Skorokhod topology**, induced by the metric $d_S$. Consider the sequence of probability measures $(P_n)$ induced by $P_0$ on the projections to $[0, n]$. For $\delta > 0$, define the modulus of continuity

$$w(B, \delta) := \sup_{s,t \in [0,n] \mid |t-s| \le \delta} |B_t - B_s|. \qquad (2.2)$$

From the Gaussian moment estimates, for any $p \ge 2$ and $s, t \in [0, n]$

$$E[|B_t - B_s|^p] = C_p |t - s|^{p/2}, \quad C_p = 2^{p/2} \Gamma\left(\frac{p+1}{2}\right) / \sqrt{\pi}. \qquad (2.3)$$

Hence, by **Markov's inequality** and the union bound over dyadic partitions of $[0, n]$, for $\varepsilon > 0$ and sufficiently small $\delta > 0$,

$$P_n(w(B, \delta) > \varepsilon) \le \sum_k \frac{E[|B_{t_{k+1}} - B_{t_k}|^p]}{\varepsilon^p} \le C_p \frac{n \delta^{p/2}}{\delta \varepsilon^p} = C_p n \varepsilon^{-p} \delta^{p/2-1}. \qquad (2.4)$$

Choosing $p > 2$ ensures $p/2 - 1 > 0$. Therefore, for every $\eta > 0$, there exists $\delta > 0$ such that $P_n(w(B, \delta) > \varepsilon) < \eta$. This establishes tightness of $(P_n)$ in the Skorokhod space.

By **Prokhorov's theorem**, there exists a weakly convergent subsequence $P_{n_k} \Rightarrow P$ on $D([0, n], \mathbb{R}^d)$. The limiting measure $P$ is concentrated on the subset of continuous paths because the modulus-of-continuity estimates imply that for almost all $\omega$,

$$\lim_{\delta \to 0} w(\omega, \delta) = 0. \qquad (2.5)$$

Thus, $P$ is supported on $C([0, n], \mathbb{R}^d)$. By consistency across $n$, we can define a unique probability measure on $C([0, \infty), \mathbb{R}^d)$ such that the canonical projections $B_t(\omega) := \omega(t)$ have the desired Gaussian finite-dimensional distributions and are continuous almost surely.

Finally, the **Hölder continuity** of order $\gamma \in (0, 1/2)$ follows from the Kolmogorov continuity
criterion. Indeed, for $p$ large enough,

$$ \mathbb{E}[|B_t - B_s|^p] \leq C_p |t-s|^{p/2} \qquad (2.6) $$

so choosing $p > 1/(1/2 - \gamma)$ ensures that

$$ \exists K(\omega) < \infty : |B_t(\omega) - B_s(\omega)| \leq K(\omega)|t-s|^{\gamma}, \quad \forall s, t \in [0, T], \qquad (2.7) $$

almost surely. This shows that the paths are almost surely Hölder continuous of any order $\gamma < 1/2$, completing the construction of (d)-dimensional Brownian motion on $\Omega = C([0, \infty), \mathbb{R}^d)$. □

**Remark 2.3.** Brownian motion is a mean-zero martingale with respect to its natural (completed, right-continuous) filtration ($\mathcal{F}_t^B$). It has stationary and independent increments, is self-similar of index $1/2$ i.e.,

$$ (c^{-1/2}B_{ct})_{t \geq 0} \stackrel{d}{=} (B_t)_{t \geq 0} \qquad (2.8) $$

and is recurrent in dimension $d = 1, 2$ and transient for $d \geq 3$.

## 2.3 Quadratic Variation and Semimartingale Structure

For a continuous path $\omega \in C([0, T], \mathbb{R})$ and a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = T\}$, define the quadratic variation along $\Pi$ by

$$ Q(\Pi, \omega) := \sum_{k=1}^{n} (\omega(t_k) - \omega(t_{k-1}))^2. \qquad (2.9) $$

For Brownian motion $B$, the following fundamental result holds.

**Theorem 2.4 (Quadratic Variation of Brownian Motion).** Let $(\Pi^m)$ be a sequence of partitions of $[0, T]$ with mesh $|\Pi^m| \to 0$. Then, almost surely,

$$ \lim_{m \to \infty} Q(\Pi^m, B) = T. \qquad (2.10) $$

More generally, for $d$-dimensional Brownian motion,

$$ \lim_{|\Pi| \to 0} \sum_k |B_{t_k} - B_{t_{k-1}}|^2 = dT \quad \text{a.s.} \qquad (2.11) $$

*Proof.* Let $B = (B_t)_{t \in [0, T]}$ be a standard one-dimensional Brownian motion on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ satisfying the usual conditions. Consider a sequence of partitions $\Pi^m = 0 = t_0^m < t_1^m < \cdots < t_{n(m)}^m = T$ with mesh $|\Pi^m| = \max_k(t_k^m - t_{k-1}^m) \to 0$ as $m \to \infty$. Define the quadratic sum along the partition as

$$ Q(\Pi^m, B) := \sum_{k=1}^{n(m)} (B_{t_k^m} - B_{t_{k-1}^m})^2. \qquad (2.12) $$

We aim to prove that $Q(\Pi^m, B) \to T$ almost surely as $m \to \infty$. Begin by computing the first two moments of $Q(\Pi^m, B)$. Using the independence and Gaussianity of increments, for each $k$,

$$ \mathbb{E}[(B_{t_k^m} - B_{t_{k-1}^m})^2] = t_k^m - t_{k-1}^m, \qquad (2.13) $$
so by linearity of expectation,

$$ \mathbb{E}[Q(\Pi^m, B)] = \sum_{k=1}^{n(m)} (t_k^m - t_{k-1}^m) = T. \qquad (2.14) $$

Next, we compute the variance. Again using independence of increments and the fact that for a Gaussian $X \sim N(0, \sigma^2)$, $\mathrm{Var}(X^2) = 2\sigma^4$, we obtain

$$ \mathrm{Var}(Q(\Pi^m, B)) = \sum_{k=1}^{n(m)} \mathrm{Var}((B_{t_k^m} - B_{t_{k-1}^m})^2) = \sum_{k=1}^{n(m)} 2(t_k^m - t_{k-1}^m)^2 = 2 \sum_{k=1}^{n(m)} (t_k^m - t_{k-1}^m)^2. \quad (2.15) $$

Since $|\Pi^m| \to 0$, we have

$$ \sum_{k=1}^{n(m)} (t_k^m - t_{k-1}^m)^2 \le |\Pi^m| \sum_{k=1}^{n(m)} (t_k^m - t_{k-1}^m) = |\Pi^m|, \quad T \to 0 \quad \text{as } m \to \infty. \qquad (2.16) $$

Thus, $\mathrm{Var}(Q(\Pi^m, B)) \to 0$. By Chebyshev's inequality, for any $\varepsilon > 0$,

$$ \mathbb{P}(|Q(\Pi^m, B) - T| > \varepsilon) \le \frac{\mathrm{Var}(Q(\Pi^m, B))}{\varepsilon^2} \to 0. \qquad (2.17) $$

This shows convergence in probability:

$$ Q(\Pi^m, B) \xrightarrow{\mathbb{P}} T. \qquad (2.18) $$

To upgrade to almost sure convergence, apply a standard **subsequence argument**. Choose a subsequence $(\Pi^{m_j})$ with $|\Pi^{m_j}| \le 2^{-j}$. Then

$$ \sum_{j=1}^{\infty} \mathbb{P}(|Q(\Pi^{m_j}, B) - T| > \varepsilon) \le \sum_{j=1}^{\infty} \frac{2T|\Pi^{m_j}|}{\varepsilon^2} \le \sum_{j=1}^{\infty} \frac{2T}{\varepsilon^2} 2^{-j} = \frac{4T}{\varepsilon^2} < \infty. \qquad (2.19) $$

By the **Borel–Cantelli lemma**, almost surely $|Q(\Pi^{m_j}, B) - T| \to 0$ along this subsequence. Since the partitions are nested or can be refined arbitrarily, one can pass to the full sequence $\Pi^m$ and conclude that

$$ \lim_{m \to \infty} Q(\Pi^m, B) = T \quad \text{a.s.} \qquad (2.20) $$

This establishes the one-dimensional case. For $d$-dimensional Brownian motion $B = (B^{(1)}, \dots, B^{(d)})$ with independent components, we have

$$ \sum_k |B_{t_k} - B_{t_{k-1}}|^2 = \sum_{i=1}^{d} \sum_k (B_{t_k}^{(i)} - B_{t_{k-1}}^{(i)})^2. \qquad (2.21) $$

By the previous argument, for each (i), $\sum_k (B_{t_k}^{(i)} - B_{t_{k-1}}^{(i)})^2 \to T$ almost surely. Since a finite union of almost sure events is almost sure, we obtain

$$ \lim_{|\Pi| \to 0} \sum_k |B_{t_k} - B_{t_{k-1}}|^2 = \sum_{i=1}^{d} T = dT \quad \text{a.s.} \qquad (2.22) $$

Hence, the quadratic variation of $d$-dimensional Brownian motion along any sequence of parti-
tions with vanishing mesh exists almost surely and equals $dT$. □

This nontrivial pathwise property implies that Brownian motion is not of bounded variation and that stochastic calculus must be developed in the semimartingale framework; Brownian motion is a continuous local martingale with predictable quadratic variation $\langle B \rangle_t = t$.

## 2.4 Simple Integrands and the Itô Integral

To define stochastic integration, begin with elementary (simple) adapted processes. Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in [0,T]}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions (completeness, right-continuity), and let $B = (B_t)_{t \in [0,T]}$ be a standard Brownian motion adapted to $(\mathcal{F}_t)$. To construct the stochastic integral rigorously, one begins with simple (elementary) adapted processes. These serve as building blocks, because the integral can be extended by continuity to a wider class of integrands.

A simple adapted process $H$ on $[0, T]$ is defined by

$$H_t(\omega) = \sum_{k=0}^{n-1} H_k(\omega) \mathbf{1}_{(t_k, t_{k+1}]}(t), \qquad (2.23)$$

where $0 = t_0 < t_1 < \dots < t_n = T$ is a partition of the interval $[0, T]$, and each $H_k$ is $\mathcal{F}_{t_k}$-measurable and square-integrable, that is, $\mathbb{E}[H_k^2] < \infty$. Progressive measurability of $H$ is automatically satisfied for such processes, since each $H_k$ depends only on the past up to time $t_k$. The simple structure ensures that the stochastic integral can be defined in a **Riemann–Stieltjes-like manner**, using the Brownian increments over each subinterval of the partition.

Specifically, the **stochastic integral** of a simple process $H$ with respect to $B$ is defined as

$$\int_{0}^{T} H_{s} dB_{s} := \sum_{k=0}^{n-1} H_{k}(B_{t_{k+1}} - B_{t_{k}}). \qquad (2.24)$$

Here, each increment $B_{t_{k+1}} - B_{t_k}$ is independent of $\mathcal{F}_{t_k}$, which allows one to rigorously compute moments of the integral. In particular, the **Itô isometry** asserts that

$$\mathbb{E}\left[(\int_{0}^{T} H_s dB_s)^2\right] = \sum_{k=0}^{n-1} \mathbb{E}[H_k^2](t_{k+1} - t_k) = \mathbb{E}\left[\int_{0}^{T} H_s^2 ds\right]. \qquad (2.25)$$

The Itô isometry is a **fundamental identity**: it shows that the map $H \mapsto \int_0^T H_s dB_s$ is a linear isometry from the space of simple adapted processes, equipped with the norm

$$|H|_{\mathcal{H}^2} := \left( \mathbb{E} \left[ \int_0^T H_s^2 ds \right] \right)^{1/2}, \qquad (2.26)$$

into $L^2(\Omega)$. This isometry provides the rigorous foundation to extend the definition of the stochastic integral from simple processes to the full Hilbert space

$$\mathcal{H}^2([0, T]) := \left\{ H \text{ progressively measurable} : \mathbb{E}\left[\int_0^T H_s^2 ds\right] < \infty \right\}. \qquad (2.27)$$
By approximating an arbitrary $H \in \mathcal{H}^2([0, T])$ in the $\mathcal{H}^2$-norm by a sequence of simple processes $(H^{(n)}),$ and using the Itô isometry to pass to the limit in $L^2(\Omega),$ one obtains a **well-defined,** continuous linear operator

$$I: \mathcal{H}^2([0, T]) \to L^2(\Omega), \quad H \mapsto \int_0^T H_s dB_s \qquad (2.28)$$

which preserves the isometry property. Moreover, for each $t \in [0, T]$, the process

$$M_t := \int_0^t H_s, dB_s \qquad (2.29)$$

is a continuous square-integrable martingale with respect to $(\mathcal{F}_t)$, and its predictable quadratic variation is given by

$$\langle M \rangle_t = \int_0^t H_s^2, ds. \qquad (2.30)$$

Thus, starting from simple adapted processes, the Itô isometry allows a rigorous extension of stochastic integration to all square-integrable progressively measurable processes, producing a continuous martingale with well-defined quadratic variation.

**Proposition 2.5 (Itô Isometry).** For any simple adapted $H$ as above,

$$\mathbb{E}\left[(\int_{0}^{T} H_{s} dB_{s})^{2}\right] = \mathbb{E}\left[\int_{0}^{T} H_{s}^{2} ds\right]. \qquad (2.31)$$

*Proof.* Let $B = (B_t)_{t \in [0,T]}$ be a standard one-dimensional Brownian motion on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ satisfying the usual conditions. Consider a simple adapted process $H$ of the form

$$H_t(\omega) = \sum_{k=0}^{n-1} H_k(\omega) \mathbf{1}_{(t_k, t_{k+1}]}(t), \qquad (2.32)$$

where $0 = t_0 < t_1 < \dots < t_n = T$ is a partition of $[0, T]$ and each $H_k$ is $\mathcal{F}_{t_k}$-measurable and square-integrable: $\mathbb{E}[H_k^2] < \infty$. The stochastic integral of $H$ with respect to $B$ is defined as

$$\int_{0}^{T} H_{s}, dB_{s} := \sum_{k=0}^{n-1} H_{k}(B_{t_{k+1}} - B_{t_{k}}). \qquad (2.33)$$

We aim to compute $\mathbb{E}[(\int_0^T H_s, dB_s)^2]$. Expanding the square gives

$$(\int_0^T H_s, dB_s)^2 = \sum_{k=0}^{n-1} H_k^2 (B_{t_{k+1}} - B_{t_k})^2 + 2 \sum_{0 \le k < \ell \le n-1} H_k H_\ell (B_{t_{k+1}} - B_{t_k}) (B_{t_\ell+1} - B_{t_\ell}). \qquad (2.34)$$

Taking expectations, we have

$$\mathbb{E}\left[(\int_{0}^{T} H_{s}, dB_{s})^{2}\right] = \sum_{k=0}^{n-1} \mathbb{E}[H_{k}^{2}(B_{t_{k+1}} - B_{t_{k}})^{2}] + 2 \sum_{0 \le k < \ell \le n-1} \mathbb{E}[H_{k} H_{\ell}(B_{t_{k+1}} - B_{t_{k}})(B_{t_{\ell+1}} - B_{t_{\ell}})]. \qquad (2.35)$$

Since $H_k$ is $\mathcal{F}_{t_k}$-measurable and $B_{t_{\ell+1}} - B_{t_{\ell}}$ is independent of $\mathcal{F}_{t_\ell} \supset \mathcal{F}_{t_k}$ for $\ell > k$, the cross
terms vanish:

$$ \mathbb{E}[H_k H_\ell (B_{t_{k+1}} - B_{t_k})(B_{t_{\ell+1}} - B_{t_\ell})] = \mathbb{E}[H_k H_\ell (B_{t_{k+1}} - B_{t_k}) \mathbb{E}[B_{t_{\ell+1}} - B_{t_\ell} | \mathcal{F}_{t_\ell}]] = 0. \quad (2.36) $$

Hence, only the diagonal terms remain:

$$ \mathbb{E}\left[(\int_{0}^{T} H_s, dB_s)^2\right] = \sum_{k=0}^{n-1} \mathbb{E}[H_k^2(B_{t_{k+1}} - B_{t_k})^2]. \quad (2.37) $$

For each $k$, since $B_{t_{k+1}} - B_{t_k} \sim \mathcal{N}(0, t_{k+1} - t_k)$ and is independent of $H_k$, we have

$$ \mathbb{E}[H_k^2(B_{t_{k+1}} - B_{t_k})^2] = \mathbb{E}[H_k^2] \mathbb{E}[(B_{t_{k+1}} - B_{t_k})^2] = \mathbb{E}[H_k^2](t_{k+1} - t_k). \quad (2.38) $$

Summing over $k$, we obtain

$$ \mathbb{E}\left[(\int_{0}^{T} H_s, dB_s)^2\right] = \sum_{k=0}^{n-1} \mathbb{E}[H_k^2](t_{k+1} - t_k) = \mathbb{E}\left[\sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} H_k^2 ds\right] = \mathbb{E}\left[\int_{0}^{T} H_s^2 ds\right]. \quad (2.39) $$

This completes the proof: the Itô isometry holds for all simple adapted processes. It provides a natural Hilbert space isometry between the space of square-integrable adapted processes and the space of $L^2(\Omega)$ martingales generated by stochastic integrals with respect to Brownian motion. $\square$

Define the space $\mathcal{H}^2([0, T])$ as the set of progressively measurable processes $H$ satisfying

$$ \mathbb{E}[\int_{0}^{T} H_{s}^{2} ds] < \infty \quad (2.40) $$

By Itô isometry, simple processes are dense in $\mathcal{H}^2$ (in the norm induced by the right-hand side), so one extends $\int_0^T H_s dB_s$ by isometry to all $H \in \mathcal{H}^2$. The resulting integral is a continuous linear map from $\mathcal{H}^2$ to $L^2(\Omega)$ and produces a continuous square-integrable martingale $(\int_0^t H_s dB_s)_{t \in [0, T]}$ with predictable quadratic variation $\int_0^t H_s^2 ds$. Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in [0, T]}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions, and let $B = (B_t)_{t \in [0, T]}$ be a standard Brownian motion adapted to $(\mathcal{F}_t)$. Define the Hilbert space

$$ \mathcal{H}^2([0, T]) := \{ H : [0, T] \times \Omega \to \mathbb{R} \mid H \text{ is progressively measurable, } |H|_{\mathcal{H}^2}^2 := \mathbb{E}[\int_0^T H_s^2 ds] < \infty \}. \quad (2.41) $$

Here, progressive measurability means that for each $t \in [0, T]$, the map $(s, \omega) \mapsto H_s(\omega)\mathbf{1}_{[0,t]}(s)$ is $\mathcal{B}([0, t]) \otimes \mathcal{F}_t$-measurable. The norm

$$ |H|_{\mathcal{H}^2}^2 := \mathbb{E}[\int_{0}^{T} H_{s}^{2} ds] \quad (2.42) $$

makes $\mathcal{H}^2([0, T])$ a complete Hilbert space. Let $\mathcal{S}$ denote the set of **simple adapted processes**, i.e., processes of the form

$$ H_t(\omega) = \sum_{k=0}^{n-1} H_k(\omega) \mathbf{1}_{[t_k, t_{k+1}]}(t), \quad (2.43) $$

where $(t_k)*k = 0^n$ is a partition of $[0, T]$ and each $H_k$ is $\mathcal{F} \cdot t_k$-measurable and square-integrable.
The Itô isometry, proved for simple processes, states that for $H \in \mathcal{S}$

$$ \mathbb{E}[(\int_{0}^{T} H_s, dB_s)^2] = |H|_{\mathcal{H}^2}^2. \qquad (2.44) $$

This equality shows that the map

$$ I : \mathcal{S} \to L^2(\Omega), \quad H \mapsto \int_{0}^{T} H_s, dB_s, \qquad (2.45) $$

is a linear isometry. Hence $\mathcal{S}$ is a dense subspace of $\mathcal{H}^2([0, T])$ with respect to the $\mathcal{H}^2$ norm: for any $H \in \mathcal{H}^2([0, T])$ and $\varepsilon > 0$, there exists a simple process $H^\varepsilon \in \mathcal{S}$ such that

$$ |H - H^\varepsilon|_{\mathcal{H}^2} < \varepsilon. \qquad (2.46) $$

Using the completeness of $L^2(\Omega)$ and the isometry, we extend (I) uniquely to all of $\mathcal{H}^2([0, T])$ by setting

$$ \int_{0}^{T} H_s, dB_s := \lim_{n \to \infty} \int_{0}^{T} H_{s}^{(n)}, dB_s, \qquad (2.47) $$

where $H^{(n)} \in \mathcal{S}$ is a sequence of simple processes converging to $H$ in $\mathcal{H}^2$. The limit exists in $L^2(\Omega)$ and is independent of the approximating sequence due to the isometry. This shows that the stochastic integral

$$ I : \mathcal{H}^2([0, T]) \to L^2(\Omega), \quad H \mapsto \int_{0}^{T} H_s, dB_s, \qquad (2.48) $$

is a continuous linear operator, with operator norm (1).

For each $t \in [0, T]$, define the process

$$ M_t := \int_{0}^{t} H_s, dB_s := I(H1_{[0,t]}). \qquad (2.49) $$

Then $(M_t)_{t \in [0,T]}$ is a square-integrable martingale with respect to $(\mathcal{F}_t)$, because for $s < t$,

$$ \mathbb{E}[M_t | \mathcal{F}_s] = \mathbb{E}[\int_0^s H_u, dB_u + \int_s^t H_u, dB_u, | \mathcal{F}_s] = \int_0^s H_u, dB_u = M_s, \qquad (2.50) $$

using the independence of the increment $B_t - B_s$ from $\mathcal{F}_s$ and the $\mathcal{F}_s$-measurability of $H_u$ for $u \le s$. Moreover, the quadratic variation of $M$ is given by the predictable process

$$ \langle M \rangle_t = \int_0^t H_s^2, ds, \qquad (2.51) $$

because for simple processes it holds by definition, and the continuity of the quadratic variation under $L^2$-limits ensures that it extends to all $H \in \mathcal{H}^2$. Consequently, the stochastic integral produces a continuous, square-integrable martingale with predictable quadratic variation $\int_0^t H_s^2 ds$, and the Itô isometry holds in full generality:

$$ \mathbb{E}[M_T^2] = \mathbb{E}[\int_0^T H_s^2, ds]. \qquad (2.52) $$
## 2.5 Itô's Formula

Itô's formula generalizes the classical chain rule to Itô processes and is the cornerstone of stochastic calculus.

**Theorem 2.6 (Itô's Formula).** Let $X$ be an Itô process in $\mathbb{R}^d$:

$$X_t = X_0 + \int_0^t b_s \, ds + \int_0^t \sigma_s \, dB_s, \qquad (2.53)$$

where $b$ and $\sigma$ are progressively measurable with suitable integrability, $b: \Omega \times [0, T] \to \mathbb{R}^d$, $\sigma: \Omega \times [0, T] \to \mathbb{R}^{d \times m}$, and $B$ is $m$-dimensional Brownian motion. Let $f: \mathbb{R}^d \to \mathbb{R}$ be $C^2$ with bounded derivatives up to second order. Then, almost surely for all $t \in [0, T]$,

$$ \begin{aligned} f(X_t) &= f(X_0) + \sum_{i=1}^{d} \int_{0}^{t} \frac{\partial f}{\partial x_i}(X_s) dX_s^{(i)} + \frac{1}{2} \sum_{i,j=1}^{d} \int_{0}^{t} \frac{\partial^2 f}{\partial x_i \partial x_j}(X_s) d\langle X^{(i)}, X^{(j)} \rangle_s \\ &= f(X_0) + \int_{0}^{t} \nabla f(X_s) \cdot b_s \, ds + \int_{0}^{t} \nabla f(X_s) \sigma_s \, dB_s + \frac{1}{2} \int_{0}^{t} \text{tr}(\sigma_s^\top D^2 f(X_s) \sigma_s) \, ds. \end{aligned} \qquad (2.54) $$

*Proof.* Let $X = (X_t)_{t \in [0, T]}$ be an Itô process in $\mathbb{R}^d$ defined by

$$X_t = X_0 + \int_0^t b_s \, ds + \int_0^t \sigma_s \, dB_s, \qquad (2.55)$$

where $b: \Omega \times [0, T] \to \mathbb{R}^d$ and $\sigma: \Omega \times [0, T] \to \mathbb{R}^{d \times m}$ are progressively measurable and satisfy the integrability conditions

$$ \mathbb{E} \left[ \int_{0}^{T} |b_s| \, ds \right] < \infty, \quad \mathbb{E} \left[ \int_{0}^{T} |\sigma_s|^2 \, ds \right] < \infty. \qquad (2.56) $$

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a twice continuously differentiable $C^2$ function with bounded first and second derivatives. Our goal is to show that almost surely, for every $t \in [0, T]$,

$$ \begin{aligned} f(X_t) &= f(X_0) + \sum *i = 1^d \int_0^t \frac{\partial f}{\partial x_i}(X_s) dX_s^{(i)} + \frac{1}{2} \sum_{i,j=1}^d \int_0^t \frac{\partial^2 f}{\partial x_i \partial x_j}(X_s) d\langle X^{(i)}, X^{(j)} \rangle_s \\ &= f(X_0) + \int_0^t \nabla f(X_s) \cdot b_s \, ds + \int_0^t \nabla f(X_s) \sigma_s \, dB_s + \frac{1}{2} \int_0^t \text{tr}(\sigma_s^\top D^2 f(X_s) \sigma_s) \, ds. \end{aligned} \qquad (2.57) $$

We begin with the one-dimensional case $d=1$, which captures the essential ideas. Suppose

$$X_t = X_0 + \int_0^t b_s \, ds + \int_0^t \sigma_s \, dB_s, \qquad (2.58)$$

and let $f \in C^2(\mathbb{R})$ with bounded $f'$ and $f''$. Let $\Pi^m = 0 = t_0^m < t_1^m < \dots < t_{nm}^m = t$ be a sequence of partitions of $[0, t]$ such that the mesh

$$ |\Pi^m| = \max_k(t_k^m - t_{k-1}^m) \to 0 \qquad (2.59) $$
as $m \to \infty$. For each $m$, we write the telescoping sum:

$$f(X_t) - f(X_0) = \sum_{k=1}^{n_m} (f(X_{t_k^m}) - f(X_{t_{k-1}^m})). \qquad (2.60)$$

Using Taylor's theorem with remainder for each term, for some $\xi_k^m$ between $X_{t_{k-1}^m}$ and $X_{t_k^m}$, we have

$$f(X_{t_k^m}) - f(X_{t_{k-1}^m}) = f'(X_{t_{k-1}^m})\Delta X_k^m + \frac{1}{2}f''(\xi_k^m)(\Delta X_k^m)^2, \qquad (2.61)$$

where $\Delta X_k^m := X_{t_k^m} - X_{t_{k-1}^m}$. Hence,

$$f(X_t) - f(X_0) = \sum_{k=1}^{n_m} f'(X_{t_{k-1}^m}) \Delta X_k^m + \frac{1}{2} \sum_{k=1}^{n_m} f''(\xi_k^m) (\Delta X_k^m)^2. \qquad (2.62)$$

We decompose each increment $\Delta X_k^m$ using the Itô process definition:

$$\Delta X_k^m = \int_{t_{k-1}^m}^{t_k^m} b_s \, ds + \int_{t_{k-1}^m}^{t_k^m} \sigma_s \, dB_s. \qquad (2.63)$$

We analyze each term separately.

## Step 1: The first sum

The first term

$$\sum_{k=1}^{n_m} f'(X_{t_{k-1}^m}) \Delta X_k^m \qquad (2.64)$$

is a Riemann–Stieltjes-type sum which, as $|\Pi^m| \to 0$, converges in $L^2(\Omega)$ and almost surely to

$$\int_0^t f'(X_s), dX_s. \qquad (2.65)$$

By the definition of the stochastic integral,

$$\int_0^t f'(X_s), dX_s = \int_0^t f'(X_s)b_s, ds + \int_0^t f'(X_s)\sigma_s, dB_s. \qquad (2.66)$$

## Step 2: The second sum

We now study the sum

$$\frac{1}{2} \sum_{k=1}^{n_m} f''(\xi_k^m) (\Delta X_k^m)^2. \qquad (2.67)$$

We note that $(\Delta X_k^m)^2$ can be expanded as

$$(\Delta X_k^m)^2 = \left(\int_{t_{k-1}^m}^{t_k^m} b_s \, ds\right)^2 * 2 \left(\int_{t_{k-1}^m}^{t_k^m} b_s \, ds\right) \left(\int_{t_{k-1}^m}^{t_k^m} \sigma_s \, dB_s\right) * \left(\int_{t_{k-1}^m}^{t_k^m} \sigma_s \, dB_s\right)^2. \qquad (2.68)$$
The first term satisfies

$$ \sum_{k=1}^{n_m} f''(\xi_k^m) \left( \int_{t_{k-1}^m}^{t_k^m} b_s \, ds \right)^2 \to 0 \quad \text{in } L^1(\Omega), \qquad (2.69) $$

because the mesh tends to zero and $b_s$ is integrable. Similarly, the mixed term with $b_s$ and $\sigma_s$ converges to zero in $L^1(\Omega)$ since the Itô integral has zero mean and bounded variance.

The only term contributing in the limit is the last one:

$$ \sum_{k=1}^{n_m} f''(\xi_k^m) \left( \int_{t_{k-1}^m}^{t_k^m} \sigma_s \, dB_s \right)^2 . \qquad (2.70) $$

By Itô isometry,

$$ \mathbb{E} \left[ \left( \int_{t_{k-1}^m}^{t_k^m} \sigma_s \, dB_s \right)^2 \right] = \mathbb{E} \left[ \int_{t_{k-1}^m}^{t_k^m} \sigma_s^2 \, ds \right]. \qquad (2.71) $$

Thus, using the quadratic variation property of Brownian motion and the continuity of $f''$,

$$ \sum_{k=1}^{n_m} f''(\xi_k^m) \left( \int_{t_{k-1}^m}^{t_k^m} \sigma_s \, dB_s \right)^2 \to \int_0^t f''(X_s) \sigma_s^2 \, ds \quad \text{a.s. and in } L^1(\Omega). \qquad (2.72) $$

### Step 3: Combining limits

Putting everything together and passing to the limit $|\Pi^m| \to 0$, we obtain almost surely

$$ f(X_t) - f(X_0) = \int_0^t f'(X_s)b_s \, ds + \int_0^t f'(X_s)\sigma_s \, dB_s + \frac{1}{2} \int_0^t f''(X_s)\sigma_s^2 \, ds. \qquad (2.73) $$

This completes the proof for the one-dimensional case.

### Step 4: The multidimensional case

Now let $X_t = (X_t^{(1)}, \dots, X_t^{(d)})^\top$ be an Itô process in $\mathbb{R}^d$ satisfying

$$ X_t^{(i)} = X_0^{(i)} + \int_0^t b_s^{(i)} \, ds + \sum_{\alpha=1}^m \int_0^t \sigma_s^{(i\alpha)} \, dB_s^{(\alpha)}. \qquad (2.74) $$

The covariance structure of $X$ is described by its quadratic covariation matrix process

$$ \langle X^{(i)}, X^{(j)} \rangle_t = \int_0^t \sum_{\alpha=1}^m \sigma_s^{(i\alpha)} \sigma_s^{(j\alpha)} \, ds. \qquad (2.75) $$

Applying the one-dimensional Itô formula to $f(X_t)$ as a function of multiple variables and using the multidimensional Taylor expansion, we obtain

$$ f(X_t) - f(X_0) = \sum_{i=1}^{d} \int_{0}^{t} \frac{\partial f}{\partial x_i}(X_s) \, dX_s^{(i)} + \frac{1}{2} \sum_{i,j=1}^{d} \int_{0}^{t} \frac{\partial^2 f}{\partial x_i \partial x_j}(X_s) \, d\langle X^{(i)}, X^{(j)} \rangle_s. \qquad (2.76) $$
Substituting $dX_s^{(i)} = b_s^{(i)} ds + \sum_{\alpha=1}^{m} \sigma_s^{(i\alpha)} dB_s^{(\alpha)}$, we have

$$ 
\begin{aligned}
f(X_t) - f(X_0) &= \int_0^t \nabla f(X_s) \cdot b_s, ds * \int_0^t \nabla f(X_s) \sigma_s, dB_s * \frac{1}{2} \sum_{i,j=1}^d \int_0^t \frac{\partial^2 f}{\partial x_i \partial x_j} (X_s) \sum_{\alpha=1}^m \sigma_s^{(i\alpha)} \sigma_s^{(j\alpha)}, ds \\
&= \int_0^t \nabla f(X_s) \cdot b_s, ds * \int_0^t \nabla f(X_s) \sigma_s, dB_s * \frac{1}{2} \int_0^t \text{tr}(\sigma_s^\top D^2 f(X_s) \sigma_s), ds.
\end{aligned} 
\quad (2.77) $$

Hence, we have established that for any $C^2$ function $f$ with bounded derivatives and for any Itô process $X$,

$$ f(X_t) = f(X_0) + \int_0^t \nabla f(X_s) \cdot b_s, ds * \int_0^t \nabla f(X_s) \sigma_s, dB_s * \frac{1}{2} \int_0^t \text{tr}(\sigma_s^\top D^2 f(X_s) \sigma_s), ds. \quad (2.78) $$

This completes the proof of **Itô's formula**. $\square$

<figure><img src="image_3.png" /></figure>

Itô's formula yields as a corollary that for $f(x) = x^2$ and one-dimensional Brownian motion,

$$ B_t^2 = 2 \int_0^t B_s dB_s + t, \quad (2.79) $$

exhibiting explicitly the quadratic variation term $t$.

### 2.5.1 Itô's Lemma for Jump-Diffusion (Poisson) Processes

Let $(\Omega, \mathcal{F}, (\mathcal{F} * t) * t \ge 0, \mathbb{P})$ be a filtered probability space satisfying the usual conditions. Let $W = (W_t)_{t \ge 0}$ be an $r$-dimensional $\mathcal{F}_t$-Brownian motion and let $(Z, \mathcal{Z})$ be a measurable mark space equipped with a $\sigma$-finite measure $\nu$ (the Lévy / intensity measure). Let $N(dt, dz)$ denote a Poisson random measure on $[0, \infty) \times Z$ with compensator $\nu(dz)$, $dt$, and write the compensated Poisson random measure as

$$ \tilde{N}(dt, dz) := N(dt, dz) - \nu(dz) dt \quad (2.80) $$

We assume all processes are càdlàg and adapted, and all integrals below are well defined under stated integrability hypotheses.

Consider an $\mathbb{R}^d$-valued Itô semimartingale with jumps given in differential form by

$$ dX_t = b(t, X_{t-}), dt + \sigma(t, X_{t-}), dW_t + \int_Z c(t, X_{t-}, z), \tilde{N}(dt, dz) + \int_Z \bar{c}(t, X_{t-}, z), N(dt, dz), \quad (2.81) $$

or, in the usual consolidated form (taking $c$ to encode all jumps and using the compensated measure for the martingale part),

$$ 
\begin{aligned}
X_t = X_0 &+ \int_0^t b(s, X_{s-}) ds + \int_0^t \sigma(s, X_{s-}) dW_s + \int_0^t \int_Z c(s, X_{s-}, z) \tilde{N}(ds, dz) \\
&+ \int_0^t \int_Z \kappa(s, X_{s-}, z) \nu(dz) ds
\end{aligned} 
\quad (2.82) $$
where one may take $\kappa$ equal to the predictable compensator part of large jumps (the two representations are equivalent under suitable decomposition of the jump kernel). For clarity in the Itô formula below we will use the representation with a single jump kernel $c(t, x, z)$ and the compensated measure $\tilde{N}$; any finite-variation predictable jump part can be accommodated by adding its compensator to the drift $b$.

Assume the coefficients satisfy the integrability / growth conditions needed to make all integrals finite: for example, for each $T > 0$,

$$ \mathbb{E} \left[ \int_0^T \left( |b(s, X_{s-})| + |\sigma(s, X_{s-})|^2 \right) ds + \int_0^T \int_Z \left( |c(s, X_{s-}, z)|^2 \wedge 1 \right) \nu(dz) ds \right] < \infty, \quad (2.83) $$

and $c, \sigma, b$ are progressively measurable in the usual sense. Let $f: \mathbb{R}^d \to \mathbb{R}$ be twice continuously differentiable $f \in C^2(\mathbb{R}^d)$ and assume for the theorem that $\partial_i f, \partial_{ij} f$ have growth such that all expectations and integrals below are finite (e.g. bounded derivatives or polynomial growth controlled by the integrability of $X$ and $c$).

The Itô formula for jump processes (Itô–Tanaka / Itô for semimartingales with jumps) states that for every $t \ge 0$, almost surely,

$$ \begin{aligned} f(X_t) = f(X_0) &+ \int_0^t \nabla f(X_{s-}) \cdot b(s, X_{s-}) ds + \int_0^t \nabla f(X_{s-}) \sigma(s, X_{s-}) dW_s \\ &+ \frac{1}{2} \int_0^t \text{tr}(\sigma(s, X_{s-})^\top D^2 f(X_{s-}), \sigma(s, X_{s-})) ds \\ &+ \int_0^t \int_Z (f(X_{s-} + c(s, X_{s-}, z)) - f(X_{s-})) \tilde{N}(ds, dz) \\ &+ \int_0^t \int_Z (f(X_{s-} + c(s, X_{s-}, z)) - f(X_{s-}) - \nabla f(X_{s-}) \cdot c(s, X_{s-}, z)) \nu(dz) ds. \end{aligned} \quad (\text{Itô jump formula}) $$

Equivalently, grouping the predictable drift terms,

$$ \begin{aligned} f(X_t) = f(X_0) &+ \int_0^t \nabla f(X_{s-}) \cdot b(s, X_{s-}) ds + \frac{1}{2} \int_0^t \text{tr}(\sigma^\top D^2 f, \sigma)(s, X_{s-}) ds \\ &+ \int_0^t \int_Z \left( f(X_{s-} + c(s, X_{s-}, z)) - f(X_{s-}) - \nabla f(X_{s-}) \cdot c(s, X_{s-}, z) \right) \nu(dz) ds \\ &+ \int_0^t \nabla f(X_{s-}) \sigma(s, X_{s-}) dW_s + \int_0^t \int_Z \left( f(X_{s-} + c(s, X_{s-}, z)) - f(X_{s-}) \right) \tilde{N}(ds, dz) \end{aligned} \quad (2.84) $$

In this formulation the last two terms are local martingales: the stochastic integral against $W$ is a continuous local martingale and the compensated jump integral is a purely discontinuous local martingale. The third line is the predictable finite-variation correction coming from the jump compensator; the term inside the compensator integral is precisely the second-order (and higher) remainder from the Taylor expansion of $f$ around $X_{s-}$.

The formula can be written in the semimartingale language as well. If $X = X^c + X^d$ with continuous local martingale part $X^c$ and purely discontinuous part $X^d$, and $\Delta X_s = X_s - X_{s-}$,
then for $f \in C^2$ one has (pathwise, a.s.)

$$f(X_t) - f(X_0) = \int_0^t \nabla f(X_{s-}) dX_s + \frac{1}{2} \sum_{i,j=1}^d \int_0^t \partial_{ij} f(X_{s-}) d\langle X^{c,i}, X^{c,j} \rangle_s \\ + \sum_{0<s \le t} (f(X_s) - f(X_{s-}) - \nabla f(X_{s-}) \cdot \Delta X_s) \quad (2.85)$$

where the sum over jumps is either written explicitly as a (random) sum or expressed by an integral against the jump measure $N$ and its compensator as above. The two representations coincide under the identification of the quadratic covariation $\langle X^{c,i}, X^{c,j} \rangle$ with $\int_0^t (\sigma\sigma^\top)^{ij}(s, X_{s-}), ds$ and of the jump-sum with the integrals against $N$ and $\nu$.

Hypotheses ensuring the validity of the formula are: $f \in C^2$ with growth so that

$$|f(x+y) - f(x) - \nabla f(x) \cdot y| \le C(|y|^2 \wedge |y|) \quad (2.86)$$

uniformly in $x$ for some constant $C$, and the jump kernel satisfies

$$\int_Z (|c(s, x, z)|^2 \wedge 1), \nu(dz) < \infty \quad (2.87)$$

for all $(s, x)$ and the stochastic integrals below are in $H^2$ (square-integrable). Under these integrability conditions all terms in the displayed Itô formula are well defined and belong respectively to $L^1$ or $L^2$ as required.

A frequently used corollary is the generator / Dynkin form: if $X$ is Markov with generator $\mathcal{L}$ acting on $C^2$-functions as

$$\mathcal{L}f(x) = \nabla f(x) \cdot b(t, x) + \frac{1}{2} \text{tr}(\sigma(t, x)^\top D^2 f(x) \sigma(t, x)) + \int_Z \left( f(x+c(t, x, z)) - f(x) - \nabla f(x) \cdot c(t, x, z) \right) \nu(dz) \quad (2.88)$$

then for bounded stopping times $\tau$ the Dynkin formula holds:

$$\mathbb{E}[f(X_\tau)] - f(X_0) = \mathbb{E} \left[ \int_0^\tau \mathcal{L}f(X_{s-}) ds \right]. \quad (2.89)$$

The proof of the jump Itô formula follows the same scheme as the continuous Itô formula: take a partition of $[0, t]$, apply the multivariable Taylor expansion to increments $f(X_{t_k}) - f(X_{t_{k-1}})$, separate contributions coming from continuous martingale increments and jump increments, pass to the limit using the quadratic-variation convergence for the continuous part, and handle jumps by summing the exact jump contributions. The compensator term arises when replacing the random sum of jump increments by its predictable compensator (integration against $\nu(dz) ds$; the remainder in the Taylor expansion yields the integrand $f(x+c) - f(x) - \nabla f(x) \cdot c$, which is integrable under the stated hypotheses.

This is the full mathematical formulation of Itô's lemma in the presence of Poissonian jumps (equivalently, in the semimartingale setting with jump measure $N$ and compensator $\nu(dz) dt$, together with the required integrability conditions and the associated generator (Dynkin) representation.
## 2.5.2 Itô's Lemma for Discontinuous Semimartingales

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions, and let $X = (X_t)_{t \ge 0}$ be an $\mathbb{R}^d$-valued càdlàg semimartingale. Recall that any semimartingale $X$ admits a unique canonical decomposition of the form

$$X_t = X_0 + M_t + A_t, \qquad (2.90)$$

where $M = (M_t)_{t \ge 0}$ is a local martingale with $M_0 = 0$, and $A = (A_t)_{t \ge 0}$ is an adapted process of finite variation with $A_0 = 0$. Moreover, the local martingale $M$ itself can be uniquely decomposed into a continuous local martingale part $M^c$ and a purely discontinuous local martingale part $M^d$:

$$M_t = M_t^c + M_t^d. \qquad (2.91)$$

The jumps of $X$ are given by $\Delta X_t = X_t - X_{t-}$, where $X_{t-} := \lim_{s \uparrow t} X_s$ denotes the left limit of $X$ at $t$.

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a twice continuously differentiable function, $f \in C^2(\mathbb{R}^d)$, with bounded derivatives up to order two. Then Itô's formula for the general discontinuous semimartingale $X$ provides the decomposition of $f(X_t)$ into its continuous and jump components. Formally, one has almost surely, for every $t \ge 0$,

$$f(X_t) = f(X_0) + \sum_{i=1}^{d} \int_{0}^{t} \frac{\partial f}{\partial x_i}(X_{s-}), dX_s^{(i)} + \frac{1}{2} \sum_{i,j=1}^{d} \int_{0}^{t} \frac{\partial^2 f}{\partial x_i \partial x_j}(X_{s-}), d[X^{(i),c}, X^{(j),c}]_s \\ + \sum_{0 < s \le t} (f(X_s) - f(X_{s-}) - \nabla f(X_{s-}) \cdot \Delta X_s) \qquad (2.92)$$

where $[X^{(i),c}, X^{(j),c}]_t$ denotes the quadratic covariation of the continuous martingale parts of the coordinates $X^{(i)}$ and $X^{(j)}$.

The first integral represents the stochastic integral with respect to the semimartingale $X$, which decomposes as a stochastic integral with respect to its continuous local martingale part $M^c$, its purely discontinuous local martingale part $M^d$, and its finite variation part $A$:

$$\int_{0}^{t} \frac{\partial f}{\partial x_i}(X_{s-}), dX_s^{(i)} = \int_{0}^{t} \frac{\partial f}{\partial x_i}(X_{s-}), dM_s^{(i),c} + \int_{0}^{t} \frac{\partial f}{\partial x_i}(X_{s-}), dM_s^{(i),d} + \int_{0}^{t} \frac{\partial f}{\partial x_i}(X_{s-}), dA_s^{(i)}. \qquad (2.93)$$

The second term on the right-hand side of the Itô formula accounts for the continuous martingale variation and is an analogue of the classical Itô correction term for continuous semimartingales. The last summation term represents the cumulative effect of the jumps, capturing the non-infinitesimal discontinuities in $X$.

If $X$ admits a jump measure $\mu^X(dt, dx)$, defined by

$$\mu^X(\omega; (0, t] \times A) := \sum_{0 < s \le t} \mathbf{1}_{\Delta X_s(\omega) \in A}, \qquad (2.94)$$
then Itô's formula can be expressed equivalently in stochastic integral form as

$$f(X_t) = f(X_0) + \sum_{i=1}^{d} \int_{0}^{t} \frac{\partial f}{\partial x_i}(X_{s-}), dX_s^{(i)} + \frac{1}{2} \sum_{i,j=1}^{d} \int_{0}^{t} \frac{\partial^2 f}{\partial x_i \partial x_j}(X_{s-}), d[X^{(i),c}, X^{(j),c}]_s \\ + \int_{0}^{t} \int_{\mathbb{R}^d} (f(X_{s-} + x) - f(X_{s-}) - \nabla f(X_{s-}) \cdot x) \mu^X(ds, dx). \quad (2.95)$$

To ensure integrability, one often works with the compensated jump measure

$$\tilde{\mu}^X(ds, dx) := \mu^X(ds, dx) - \nu^X(ds, dx) \quad (2.96)$$

where $\nu^X$ is the predictable compensator (the Lévy system) of $\mu^X$. In this compensated form, the formula is expressed as

$$f(X_t) = f(X_0) + \sum_{i=1}^{d} \int_{0}^{t} \frac{\partial f}{\partial x_i}(X_{s-}), dX_s^{(i)} + \frac{1}{2} \sum_{i,j=1}^{d} \int_{0}^{t} \frac{\partial^2 f}{\partial x_i \partial x_j}(X_{s-}), d[X^{(i),c}, X^{(j),c}]_s \\ + \int_{0}^{t} \int_{\mathbb{R}^d} (f(X_{s-} + x) - f(X_{s-}) - \nabla f(X_{s-}) \cdot x) \tilde{\mu}^X(ds, dx) \\ + \int_{0}^{t} \int_{\mathbb{R}^d} (f(X_{s-} + x) - f(X_{s-}) - \nabla f(X_{s-}) \cdot x) \nu^X(ds, dx). \quad (2.97)$$

The first three terms correspond respectively to the drift, diffusion, and continuous quadratic variation effects, while the last two integrals separate the martingale and compensator contributions due to jumps.

In particular, if $X$ is a jump-diffusion process of the form

$$X_t = X_0 + \int_0^t b_s ds + \int_0^t \sigma_s dB_s + \int_0^t \int_{\mathbb{R}^d} \gamma(s, x) \tilde{N}(ds, dx) \quad (2.98)$$

where $B_t$ is Brownian motion, $\tilde{N}(ds, dx)$ is the compensated Poisson random measure, and $b_s, \sigma_s, \gamma(s, x)$ are predictable coefficients, then Itô's formula specializes to

$$f(X_t) = f(X_0) + \int_0^t \nabla f(X_{s-}) \cdot b_s, ds * \int_0^t \nabla f(X_{s-}) \cdot \sigma_s, dB_s * \frac{1}{2} \int_0^t \text{tr}(\sigma_s^\top D^2 f(X_{s-}) \sigma_s), ds \\ + \int_0^t \int_{\mathbb{R}^d} (f(X_{s-} + \gamma(s, x)) - f(X_{s-})), \tilde{N}(ds, dx) \\ + \int_0^t \int_{\mathbb{R}^d} (f(X_{s-} + \gamma(s, x)) - f(X_{s-}) - \nabla f(X_{s-}) \cdot \gamma(s, x)), \nu(dx), ds. \quad (2.99)$$

This final representation illustrates how Itô's lemma naturally extends to discontinuous semi-martingales, combining both the continuous diffusion effects captured by the Brownian term and the discontinuous jump effects arising from the compensated Poisson measure.

### 2.5.3 Itô's Lemma and Explicit Solution for Geometric Brownian Motion

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions and let $B = (B_t)_{t \ge 0}$ be a one-dimensional $\mathcal{F}_t$-Brownian motion. Fix constants $\mu \in \mathbb{R}$, $\sigma \in \mathbb{R} \setminus 0$ and an
initial value $S_0 > 0$. Consider the stochastic differential equation (SDE) on $[0, T]$

$$dS_t = \mu S_t dt + \sigma S_t dB_t \quad S_0 > 0 \qquad (2.100)$$

to be interpreted in the Itô sense. The coefficient functions $b(t, x) = \mu x$ and $\tilde{\sigma}(t, x) = \sigma x$ satisfy global Lipschitz and linear growth conditions on $\mathbb{R}$, hence by the classical existence and uniqueness theory for SDEs there exists a unique strong solution $S = (S_t)_{t \in [0, T]}$ which is continuous and adapted; moreover pathwise uniqueness and the linear structure imply that $S_t$ never hits zero a.s. (indeed it will be strictly positive almost surely for all $t$). We now apply Itô's formula to obtain explicit dynamics for $\log S_t$ and then derive the closed-form solution for $S_t$.

Let $f(x) = \ln x$ defined on $(0, \infty)$. Then $f \in C^2(0, \infty)$ with derivatives $f'(x) = 1/x$, $f''(x) = -1/x^2$. Since $S_t > 0$ a.s., Itô's formula yields, almost surely for every $t \in [0, T]$,

$$ \begin{aligned} d \ln S_t &= f'(S_t) dS_t + \frac{1}{2} f''(S_t) d\langle S \rangle_t \\ &= \frac{1}{S_t} (\mu S_t dt + \sigma S_t dB_t) + \frac{1}{2} \left(-\frac{1}{S_t^2}\right) d\langle S \rangle_t \qquad (2.101) \\ &= \mu dt + \sigma dB_t - \frac{1}{2} \frac{d\langle S \rangle_t}{S_t^2} \end{aligned} $$

To compute $d\langle S \rangle_t$ note that (S) is an Itô semimartingale whose martingale part is $\int_0^t \sigma S_s, dB_s$, hence by the Itô isometry/quadratic-variation rule

$$d\langle S \rangle_t = \sigma^2 S_t^2, dt. \qquad (2.102)$$

Substituting this into the previous display gives the simplified stochastic differential for the logarithm:

$$d \ln S_t = \mu dt + \sigma dB_t - \frac{1}{2} \sigma^2 dt = \left(\mu - \frac{1}{2} \sigma^2\right) dt + \sigma dB_t \qquad (2.103)$$

Integrating from 0 to $t$ and using $\ln S_0 = \ln S_0$ yields the pathwise identity, almost surely:

$$\ln S_t = \ln S_0 + \left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma B_t \qquad (2.104)$$

Exponentiating both sides gives the explicit closed-form (stochastic exponential) solution

$$S_t = S_0 \exp\left(\left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma B_t\right), \quad t \in [0, T], \qquad (2.105)$$

and this process is strictly positive for all $(t)$ almost surely. The expression is the stochastic exponential of $\sigma B_t + (\mu - \frac{1}{2}\sigma^2)t$ and provides a direct verification: applying Itô's formula to the right-hand side indeed recovers

$$dS_t = \mu S_t dt + \sigma S_t dB_t \qquad (2.106)$$

so the closed form is the unique strong solution.

From the explicit form one obtains moment formulas immediately. For any real $p$, using the
Gaussian law of $B_t$ ($B_t \sim \mathcal{N}(0, t)$) and independence properties,

$$ \mathbb{E}[S_t^p] = S_0^p \mathbb{E}[\exp(p(\mu - \frac{1}{2}\sigma^2)t + p\sigma B_t)] = S_0^p \exp(p(\mu - \frac{1}{2}\sigma^2)t) \exp(\frac{1}{2}p^2\sigma^2t). \quad (2.107) $$

Combining the exponentials yields the closed-form moment identity, valid whenever the expectation is finite (which holds for all finite $t$ and real $p$):

$$ \mathbb{E}[S_t^p] = S_0^p \exp(p\mu t + \frac{1}{2}p(p-1)\sigma^2 t). \quad (2.108) $$

In particular, for $p = 1$ we recover $\mathbb{E}[S_t] = S_0 e^{\mu t}$.

One may also view $S$ as the stochastic exponential of $\sigma B$: define the Doléans-Dade exponential of a continuous local martingale $M$ by

$$ \mathcal{E}(M)_t = \exp(M_t - \frac{1}{2}\langle M \rangle_t) \quad (2.109) $$

With $M_t = \sigma B_t$ one has

$$ \mathcal{E}(\sigma B)_t = \exp(\sigma B_t - \frac{1}{2}\sigma^2 t) \quad (2.110) $$

Then the explicit solution can be written as

$$ S_t = S_0 \exp(\mu t) \mathcal{E}(\sigma B)_t \quad (2.111) $$

and equivalently as the stochastic exponential of the semimartingale $\int_0^t \mu, ds + \int_0^t \sigma, dB_s$. The representation as a stochastic exponential clarifies multiplicative properties and is often used in measure-change and financial modelling.

Finally, one may verify the uniqueness and positivity properties directly: if $S$ is any continuous adapted solution with $S_0 > 0$ then the process $\ln S_t$ is well-defined and satisfies the linear SDE

$$ d \ln S_t = (\mu - \frac{1}{2}\sigma^2) dt + \sigma dB_t \quad (2.112) $$

whose solution is unique; exponentiation preserves uniqueness and yields the explicit formula above, showing pathwise uniqueness and almost sure positivity.

All steps above are rigorous: application of Itô's formula requires $f \in C^2(0, \infty)$ and the property $S_t > 0$ a.s.; the computation of the quadratic variation uses that the martingale part of $S$ is $\int_0^t \sigma S_s, dB_s$; existence and uniqueness follow from standard SDE theory under the linear-growth/Lipschitz coefficients $x \mapsto \mu x$; $x \mapsto \sigma x$. The moment computations rely on the Gaussian distribution of $B_t$ and Fubini/Tonelli justifications which are valid because the exponentials have finite expectation for each finite $t$.

## 2.6 Martingale Representation and Predictable Projection

An important structural result is the martingale representation theorem for Brownian filtrations.

**Theorem 2.7** (Martingale Representation for Brownian Filtration). Let $(\Omega, \mathcal{F}, (\mathcal{F}_t^B), \mathbb{P})$ be the (completed, right-continuous) filtration generated by an $m$-dimensional Brownian motion
B. If $M = (M_t)_{t \in [0,T]}$ is a square-integrable $\mathcal{F}_t^B$-martingale with $M_0 = 0$, then there exists a progressively measurable process $H \in \mathcal{H}^2([0,T]; \mathbb{R}^m)$ such that

$$M_t = \int_0^t H_s dB_s, \quad 0 \le t \le T, \quad \text{a.s.} \qquad (2.113)$$

*Proof.* Let $(\Omega, \mathcal{F}, (\mathcal{F} * t^B), \mathbb{P})$ be the completed, right-continuous filtration generated by an $m$-dimensional Brownian motion $B = (B^{(1)}, \dots, B^{(m)})$ on $([0,T])$. Assume $M = (M_t) * t \in [0, T]$ is a square-integrable $\mathcal{F}_t^B$-martingale with $M_0 = 0$. We will prove that there exists a progressively measurable $H \in \mathcal{H}^2([0, T]; \mathbb{R}^m)$ such that

$$M_t = \int_0^t H_s dB_s, \quad 0 \le t \le T, \qquad (2.114)$$

almost surely. The proof proceeds by an $L^2$-projection argument combined with a density result for stochastic exponentials; all steps below are given in full mathematical detail.

Let $S$ denote the linear space of stochastic integrals up to time $T$ of square-integrable simple predictable integrands, i.e.

$$S := \{ \xi = \int_{0}^{T} H_{s} dB_{s} : H \text{ is simple and predictable, } \mathbb{E}[\int_{0}^{T} |H_{s}|^{2} ds] < \infty \}. \qquad (2.115)$$

By the Itô isometry, $S$ is a linear subspace of $L^2(\Omega, \mathcal{F}_T^B, \mathbb{P})$ and the map $H \mapsto \int_0^T H_s, dB_s$ is an isometry from the space of simple integrands with the $\mathcal{H}^2$-norm onto $S$ (with the $L^2$-norm). Let $\bar{S}$ denote the closure of $S$ in $L^2(\Omega, \mathcal{F}_T^B, \mathbb{P})$. Because simple integrands are dense in $\mathcal{H}^2$), $(\bar{S}$ coincides with the set of terminal values of all stochastic integrals with integrand in $\mathcal{H}^2$; equivalently $\bar{S} = \int_0^T H_s, dB_s : H \in \mathcal{H}^2$.

Consider the orthogonal projection of $M_T$ onto $\bar{S}$ in the Hilbert space $L^2(\Omega)$. Write

$$M_T = \xi + R, \qquad (2.116)$$

with $\xi \in \bar{S}$ and $R \in \bar{S}^{\perp}$ (so $\mathbb{E}[\xi R] = 0$ and $\mathbb{E}[RZ] = 0$ for all $Z \in S$). Define the martingale $N_t := \mathbb{E}[R | \mathcal{F} * t^B]$. Then $(N_t) * t \in [0, T]$ is a square-integrable $\mathcal{F}_t^B$-martingale with $N_0 = \mathbb{E}[R] = 0$ and $N_T = R$. Our aim is to show $R = 0$ a.s.; equivalently $M_T = \xi \in \bar{S}$, which will yield the existence of $H \in \mathcal{H}^2$ with $\xi = \int_0^T H_s, dB_s$. We therefore need to prove that any $R \in \bar{S}^{\perp}$ must vanish a.s.

Take an arbitrary simple predictable integrand (H) and write $I(H) := \int_0^T H_s, dB_s \in S$. By orthogonality,

$$0 = \mathbb{E}[R \cdot I(H)] = \mathbb{E}[\mathbb{E}[R | \mathcal{F} * t * k] \cdot (B_{t_{k+1}} - B_{t_k}) H_{t_k}] \qquad (2.117)$$

for every decomposition of $H$ into simple steps $t_k$. Letting the partition adapt and choosing $H$ to be indicator functions of time-intervals times standard basis vectors, we deduce the elementary identity

$$\mathbb{E}[R(B_t^{(i)} - B_s^{(i)})] = 0 \quad \text{for all } 0 \le s < t \le T, i = 1, \dots, m. \qquad (2.118)$$
Conditioning on $\mathcal{F} * s^B$ yields

$$ \mathbb{E}[(N_t - N_s)(B_t^{(i)} - B_s^{(i)})] = 0. \qquad (2.119) $$

Using polarization and passing to limits over partitions, the latter implies that the predictable covariation between $N$ and each coordinate $B^{(i)}$ vanishes:

$$ \langle N, B^{(i)} \rangle_t = 0, \quad i = 1, \dots, m, \quad \forall t \in [0, T]. \qquad (2.120) $$

Indeed, one can compute the covariation by the $L^2$-limit of sums $\sum (N * t_{k+1} - N_{t_k})(B^{(i)} * t * k + 1 - B_{t_k}^{(i)})$; the vanishing of expectations of these sums for all partitions and their square-integrability give $\langle N, B^{(i)} \rangle = 0$ a.s.

Next we use the Kunita-Watanabe decomposition (or the general theory of orthogonal decomposition of square-integrable martingales): any square-integrable martingale $Y$ adapted to the Brownian filtration admits a unique decomposition

$$ Y_t = \sum_{i=1}^{m} \int_{0}^{t} H_{s}^{(i)}, dB_{s}^{(i)} + Y_{t}^{\perp}, \qquad (2.121) $$

where the first term is an $\mathcal{H}^2$-stochastic integral and $Y^\perp$ is a martingale which is strongly orthogonal to every $B^{(i)}$: $\langle Y^\perp, B^{(i)} \rangle = 0$ for all (i). (This decomposition follows from orthogonal projection onto the closed linear span of stochastic integrals and the properties already established.) Applying this to $(N)$ yields $N = N^\perp$ because $\langle N, B^{(i)} \rangle = 0$; that is, $N$ is itself orthogonal to each $B^{(i)}$.

We now show that a square-integrable martingale $N$ adapted to the Brownian filtration with $\langle N, B^{(i)} \rangle = 0$ for each $i$ must be a.s. constant (hence zero here since $N_0 = 0$). To do so, consider the quadratic variation $\langle N \rangle_t$. By polarization,

$$ \langle N \rangle_t = \sum_{i=1}^{m} \langle N, B^{(i)} \rangle_t^2 + \langle N^\perp \rangle_t, \qquad (2.122) $$

but since each $\langle N, B^{(i)} \rangle = 0$ we have $\langle N \rangle_t = \langle N \rangle_t$. More usefully, fix any bounded deterministic vector function $\theta \in L^2([0, T]; \mathbb{R}^m)$ and consider the stochastic exponential (Doléans-Dade exponential)

$$ \mathcal{E}_t(\theta) := \exp \left( \sum_{i=1}^{m} \int_{0}^{t} \theta_s^{(i)} dB_s^{(i)} - \frac{1}{2} \int_{0}^{t} |\theta_s|^2 ds \right) \qquad (2.123) $$

which is a uniformly integrable martingale. For such $\theta$, the terminal values $\mathcal{E}_T(\theta)$ span a dense subspace of $L^2(\mathcal{F}_T^B)$ (this is a classical fact; it follows from the Wiener-Itô chaos expansion or from Stone-Weierstrass-like density arguments applied to the exponential functionals of Gaussian processes). Now, for each $\theta$,

$$ 0 = \mathbb{E}[R \cdot \mathcal{E}_T(\theta)] = \mathbb{E}[N_T \mathcal{E}_T(\theta)]. \qquad (2.124) $$

But $N_t \mathcal{E}_t(\theta)$ is itself a martingale (product of a martingale and a bounded martingale with zero
covariation due to orthogonality), hence

$$ \mathbb{E}[N_T \mathcal{E}_T(\theta)] = \mathbb{E}[N_0 \mathcal{E}_0(\theta)] = 0. \qquad (2.125) $$

Since the linear span of $\mathcal{E}_T(\theta)$ is dense in $L^2(\mathcal{F}_T^B)$, the equalities $\mathbb{E}[N_T X] = 0$ for all $X$ in a dense set imply $N_T = 0$ in $L^2$, hence $N_T = 0$ a.s. Consequently $R = N_T = 0$ a.s., and therefore $M_T = \xi \in \bar{\mathcal{S}}$.

Having shown $M_T$ belongs to the closed linear subspace $\bar{\mathcal{S}}$, let $H \in \mathcal{H}^2([0, T]; \mathbb{R}^m)$ be the unique (by isometry) integrand such that

$$ M_T = \int_0^T H_s, dB_s. \qquad (2.126) $$

Define for each $t \in [0, T]$(\(M'_t := \int_0^t H_s, dB_s\)). Then $(M'_t)$ is a square-integrable martingale with terminal value $M'_T = M_T$. By uniqueness of conditional expectation we have

$$ M'_t = \mathbb{E}[M_T | \mathcal{F}_t^B] = M_t \qquad (2.127) $$

a.s. for all $t$, so that

$$ M_t = \int_0^t H_s, dB_s, \quad 0 \le t \le T, \qquad (2.128) $$

almost surely. This completes the proof. □

**Remarks:** The crucial ingredients in the argument were

1. The Itô isometry and the Hilbert-space projection onto the closed linear span of stochastic integrals,

2. The density of stochastic exponentials (or equivalently the first Wiener chaos) in $L^2(\mathcal{F}_T^B)$, and

3. Orthogonality/duality arguments that force any martingale orthogonal to all stochastic integrals to vanish.

Alternate proofs use the Wiener–Itô chaos decomposition or Malliavin calculus to produce the integrand (H) explicitly; all approaches rest on the same Hilbert-space structure and the rich structure of the Brownian filtration.

This theorem is crucial: it says every square-integrable martingale in the Brownian filtration is an Itô integral, which underpins hedging in mathematical finance and uniqueness in solutions of stochastic control problems.

## 2.7 Stochastic Differential Equations and Strong Solutions

The Itô integral allows formulation of stochastic differential equations (SDEs). Consider the SDE in $\mathbb{R}^d$:

$$ dX_t = b(t, X_t) dt + \sigma(t, X_t) dB_t, \quad X_0 = x \in \mathbb{R}^d, \qquad (2.129) $$
with measurable coefficients $b: [0, T] \times \mathbb{R}^d \to \mathbb{R}^d$ and $\sigma: [0, T] \times \mathbb{R}^d \to \mathbb{R}^{d \times m}$. A strong solution is an adapted continuous process $X$ on the given probability space such that the integral equation

$$X_t = x + \int_0^t b(s, X_s) ds + \int_0^t \sigma(s, X_s) dB_s \quad (2.130)$$

holds almost surely for all $t$. Classical existence and uniqueness are guaranteed under global Lipschitz and linear growth conditions: if $b, \sigma$ are Lipschitz in $x$ uniformly in $t$, there exists a unique strong solution in $S^2$ (square integrable continuous adapted processes). The proof proceeds by Picard iteration in the Banach space of adapted processes using the Itô isometry to control successive approximations.

We now describe the **function space** in which the solution resides. Let $S^2([0, T]; \mathbb{R}^d)$ denote the Banach space of all $\mathbb{R}^d$-valued $\mathcal{F}_t$-adapted continuous processes $Y = (Y_t)_{t \in [0, T]}$ satisfying the norm condition

$$|Y|_{S^2} := \left( \mathbb{E} \left[ \sup_{t \in [0,T]} |Y_t|^2 \right] \right)^{1/2} < \infty. \quad (2.131)$$

This space is complete, and it is equipped to control the supremum of the process over time, which is crucial for convergence arguments in the Picard iteration method.

To guarantee existence and uniqueness, we impose global Lipschitz and linear growth conditions on the coefficients. Explicitly, there exists a constant $L > 0$ such that, for all $t \in [0, T]$ and all $x, y \in \mathbb{R}^d$,

$$|b(t, x) - b(t, y)| + |\sigma(t, x) - \sigma(t, y)| \le L|x - y|, \quad (2.132)$$

and

$$|b(t, x)|^2 + |\sigma(t, x)|^2 \le L^2(1 + |x|^2), \quad (2.133)$$

where $|\cdot|$ denotes the Frobenius norm for matrices. These conditions ensure both that the integrals are well-defined in $H^2([0, T]; \mathbb{R}^d)$ and that successive approximations do not blow up in $L^2$.

**Picard iteration argument:** we define a sequence of processes $(X^{(n)})_{n \ge 0}$ recursively by

$$X_t^{(0)} := x, \quad X_t^{(n+1)} := x + \int_0^t b(s, X_s^{(n)}), ds + \int_0^t \sigma(s, X_s^{(n)}), dB_s, \quad t \in [0, T]. \quad (2.134)$$
We then estimate the $L^2$-distance between successive iterates. Using the Itô isometry for the stochastic integral and Cauchy-Schwarz for the Lebesgue integral, we obtain

$$
\begin{align*}
\mathbb{E} \left[ \sup_{t \in [0,T]} |X_t^{(n+1)} - X_t^{(n)}|^2 \right] &\le 2 \mathbb{E} \left[ \sup_{t \in [0,T]} \left| \int_0^t (b(s, X_s^{(n)}) - b(s, X_s^{(n-1)})), ds \right|^2 \right] \\
&\quad + 2 \mathbb{E} \left[ \sup_{t \in [0,T]} \left| \int_0^t (\sigma(s, X_s^{(n)}) - \sigma(s, X_s^{(n-1)})), dB_s \right|^2 \right] \\
&\le 2T \int_0^T \mathbb{E} [ |b(s, X_s^{(n)}) - b(s, X_s^{(n-1)})|^2 ], ds \tag{2.135} \\
&\quad + 8 \int_0^T \mathbb{E} [ |\sigma(s, X_s^{(n)}) - \sigma(s, X_s^{(n-1)})|^2 ], ds \\
&\le C \int_0^T \mathbb{E} [ \sup_{u \in [0,s]} |X_u^{(n)} - X_u^{(n-1)}|^2 ], ds,
\end{align*}
$$

where $C = 2TL^2 + 8L^2$ is a constant depending on the Lipschitz constant and time horizon. Applying **Gronwall's inequality**, we conclude that the sequence $(X^{(n)})$ is Cauchy in $S^2$, hence convergent to a limit $X \in S^2$.

Finally, the limit process $X$ satisfies

$$
X_t = x + \int_0^t b(s, X_s), ds + \int_0^t \sigma(s, X_s), dB_s, \quad (2.136)
$$

almost surely for all $t \in [0, T]$. The uniqueness follows similarly: if $X$ and $Y$ are two solutions in $S^2$, then

$$
\mathbb{E} \left[ \sup_{t \in [0,T]} |X_t - Y_t|^2 \right] \le C \int_0^T \mathbb{E} \left[ \sup_{u \in [0,s]} |X_u - Y_u|^2 \right], ds, \quad (2.137)
$$

and Gronwall's lemma implies $X_t = Y_t$ almost surely for all $t$.

Thus, under the global Lipschitz and linear growth conditions, there exists a unique strong solution $X \in S^2([0, T]; \mathbb{R}^d)$ to the SDE, constructed rigorously via Picard iteration using the Itô isometry to control the stochastic integral at each step. The solution is adapted, continuous, and square-integrable, exactly matching the classical notion of a strong solution.

## 2.8 Girsanov Transform (Informal Statement)

A fundamental tool for changing measures in Brownian models is Girsanov's theorem. Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \in [0,T]}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions, and let $B = (B_t)_{t \in [0,T]}$ be an $m$-dimensional $\mathcal{F}_t$-Brownian motion. Suppose $\theta = (\theta_t)_{t \in [0,T]}$ is an $\mathbb{R}^m$-valued, progressively measurable process satisfying Novikov's condition:

$$
\mathbb{E} \left[ \exp \left( \frac{1}{2} \int_{0}^{T} |\theta_s|^2, ds \right) \right] < \infty, \quad (2.138)
$$
where $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^m$. Define the exponential local martingale (Doléans-Dade exponential)

$$ \mathcal{E}_t := \exp \left( \int_0^t \theta_s \cdot dB_s - \frac{1}{2} \int_0^t |\theta_s|^2 ds \right), \quad t \in [0, T]. \qquad (2.139) $$

**Martingale property:** Under Novikov's condition, the stochastic exponential $\mathcal{E} = (\mathcal{E}_t)_{t \in [0,T]}$ is a true, uniformly integrable martingale. Precisely, the process $\mathcal{E}_t$ is positive, adapted, and satisfies

$$ \mathbb{E}[\mathcal{E}_t | \mathcal{F}_s] = \mathcal{E}_s, \quad 0 \le s \le t \le T, \qquad (2.140) $$

so that $\mathbb{E}[\mathcal{E}_T] = 1$. The proof uses Itô's formula for the logarithm of a positive semimartingale, the Itô isometry to bound the second moment of the stochastic integral, and the standard Novikov argument: for all (t),

$$ \mathbb{E} \left[ \exp \left( \frac{1}{2} \int_0^t |\theta_s|^2 ds \right) \right] < \infty \quad \Rightarrow \quad \mathcal{E}_t \text{ is a martingale.} \qquad (2.141) $$

**Definition of the new measure:** Define a probability measure $\mathbb{Q}$ on $(\Omega, \mathcal{F}_T)$ via

$$ \frac{d\mathbb{Q}}{d\mathbb{P}} := \mathcal{E}_T \qquad (2.142) $$

Then $\mathbb{Q}$ is absolutely continuous with respect to $\mathbb{P}$, and by construction

$$ \mathbb{Q}(\Omega) = \mathbb{E}_{\mathbb{P}}[\mathcal{E}_T] = 1 \qquad (2.143) $$

For any $\mathcal{F}_T$-measurable random variable $X$ with $\mathbb{E}_{\mathbb{P}}[|X|] < \infty$, the expectation under $\mathbb{Q}$ is given by

$$ \mathbb{E}_{\mathbb{Q}}[X] = \mathbb{E}_{\mathbb{P}}[\mathcal{E}_T X]. \qquad (2.144) $$

**Construction of the Q-Brownian motion:** Define the process

$$ \tilde{B}_t := B_t - \int_0^t \theta_s, ds, \quad t \in [0, T]. \qquad (2.145) $$

We now show that $\tilde{B}$ is an $m$-dimensional Brownian motion under $\mathbb{Q}$. To see this, consider the increment

$$ \tilde{B}_t - \tilde{B}_s = (B_t - B_s) - \int_s^t \theta_u du \qquad (2.146) $$

For any bounded, $\mathcal{F}_s$-measurable random variable (X), the Girsanov theorem states

$$ \mathbb{E}_{\mathbb{Q}}[(\tilde{B}_t - \tilde{B}_s)X] = \mathbb{E}_{\mathbb{P}}[\mathcal{E}_T(B_t - B_s - \int_s^t \theta_u du)X] = 0 \qquad (2.147) $$

More generally, one can verify that $\tilde{B}$ has independent increments under $\mathbb{Q}$ with covariance

$$ \mathrm{Cov}_{\mathbb{Q}}(\tilde{B}_t - \tilde{B}_s) = (t-s)I_m \qquad (2.148) $$

and continuous paths almost surely, satisfying all properties of an $(m)$-dimensional Brownian motion. This follows rigorously by applying the Cameron-Martin-Girsanov formula and the uniqueness of the Gaussian distribution with specified mean and covariance.
**Applications:** The theorem allows us to introduce or remove drift terms in SDEs via an absolutely continuous change of measure. For instance, if a process X satisfies

$$dX_t = \mu_t dt + \sigma_t dB_t \qquad (2.149)$$

under $\mathbb{P}$, then choosing $\theta_t = \sigma_t^{-1}\mu_t$ (assuming invertibility) and defining

$$\tilde{B}_t = B_t - \int_0^t \theta_s ds \qquad (2.150)$$

we obtain under $\mathbb{Q}$:

$$dX_t = \sigma_t d\tilde{B}_t, \qquad (2.151)$$

effectively removing the drift. This is fundamental in stochastic analysis, for example in transforming parabolic PDEs with first-order (drift) terms into pure diffusion equations, and in financial mathematics for risk-neutral pricing.

**Summary:** Under Novikov's condition, the exponential local martingale $\mathcal{E}_t$ is a true martingale, defining an equivalent probability measure $\mathbb{Q}$ under which

$$\tilde{B}_t = B_t - \int_0^t \theta_s ds \qquad (2.152)$$

is a Brownian motion. This result provides a rigorous framework for measure changes in Brownian models and underpins many fundamental constructions in stochastic calculus and applications.

## 2.9 Conclusion

This chapter has constructed Brownian motion, established its quadratic variation and martingale properties, introduced the Itô integral and proved Itô's formula, and indicated core SDE existence results and measure-change techniques. These results make rigorous the probabilistic calculus used in deriving the Feynman-Kac formula: in the next chapter we shall apply these tools to prove the classical Feynman-Kac representation for parabolic PDEs, carefully verifying integrability, regularity, and boundary condition issues that arise in analytic-probabilistic correspondence.

**References for this chapter:** Karatzas & Shreve (2014) [1], Revuz & Yor (1999) [3], Øksendal (2013) [2], Mörters & Peres (2010) [9], Da Prato & Zabczyk (1992) [10].
# Generators and Semigroups

> "In the infinitesimal lies the seed of evolution; every continuous motion is governed by its generator."

> — Adapted from Kiyosi Itô

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions, and let $X = (X_t)_{t \ge 0}$ be a time-homogeneous Markov process with state space $E \subseteq \mathbb{R}^d$, endowed with its Borel $\sigma$-algebra $\mathcal{B}(E)$. For any bounded measurable function $f: E \to \mathbb{R}$, define the family of operators $(P_t)_{t \ge 0}$ acting on functions $f$ by

$$ (P_t f)(x) := \mathbb{E}_x[f(X_t)] = \int_E f(y) p_t(x, dy), \qquad (3.1) $$

where $p_t(x, \cdot)$ denotes the transition probability kernel of $X_t$ under $\mathbb{P}_x$, the law of the process started from $(x)$. The family $(P_t)_{t \ge 0}$ satisfies $P_0 = I$ and the semigroup property

$$ P_{s+t} = P_s P_t, \quad \forall s, t \ge 0, \qquad (3.2) $$

since by the Markov property and conditioning on $\mathcal{F}_s$,

$$ \mathbb{E}_x[f(X_{s+t})] = \mathbb{E}_x[\mathbb{E}[f(X_{s+t}) | \mathcal{F}_s]] = \mathbb{E}_x[(P_t f)(X_s)] = (P_s P_t f)(x). \qquad (3.3) $$

For each fixed $t > 0$, $P_t$ is a linear contraction on $L^\infty(E)$ and preserves positivity. If the process is Feller, i.e., $P_t$ maps $C_0(E)$ to itself and $\lim_{t \downarrow 0} |P_t f - f|_\infty = 0$ for all $f \in C_0(E)$, then $(P_t)_{t \ge 0}$ is called a Feller semigroup.

## 3.1 Infinitesimal generator of the semigroup

The infinitesimal generator $A$ of the semigroup $(P_t)$ is defined by

$$ Af = \lim_{t \downarrow 0} \frac{P_t f - f}{t}, \qquad (3.4) $$

whenever this limit exists in the uniform norm. The domain $\mathcal{D}(A)$ consists of all functions $f \in C_0(E)$ for which the above limit exists and belongs to $C_0(E)$. This operator encodes the
infinitesimal dynamics of the process and plays a role analogous to the derivative of a function in deterministic calculus. For each $f \in \mathcal{D}(A)$, the process

$$M_t^f := f(X_t) - f(X_0) - \int_0^t (Af)(X_s) ds \quad (3.5)$$

is a martingale under $\mathbb{P}_x$, for each initial point $x \in E$. Conversely, if there exists an operator $A$ on $C_0(E)$ such that $M_t^f$ is a martingale for all $f \in \mathcal{D}(A)$, then $A$ is the generator of the process. This characterization provides a bridge between the probabilistic structure of Markov processes and the analytic structure of semigroup theory.

Let $(E, \mathcal{B}(E))$ be a locally compact separable metric space endowed with its Borel $\sigma$-algebra, and let $(X_t)_{t \ge 0}$ be a Markov process taking values in $E$. Denote by $(\mathbb{P}_x)_{x \in E}$ the associated family of probability measures, such that $\mathbb{P}_x(X_0 = x) = 1$ for all $x \in E$. For every bounded measurable function $f: E \to \mathbb{R}$, define the semigroup of the process by

$$(T_t f)(x) := \mathbb{E}_x[f(X_t)], \quad t \ge 0, x \in E,$$

where $\mathbb{E}_x[\cdot]$ denotes the expectation with respect to $\mathbb{P}_x$.

Assume that $T_t$ maps $C_0(E)$ into itself for each $t \ge 0$, that $T_0 = I$, and that the family $(T_t)_{t \ge 0}$ satisfies the semigroup property

$$T_{t+s} = T_t T_s, \quad \forall t, s \ge 0.$$

If, in addition, for each $f \in C_0(E)$,

$$\lim_{t \downarrow 0} \|T_t f - f\|_{\infty} = 0,$$

then $(T_t)_{t \ge 0}$ is called a *Feller semigroup*. The infinitesimal generator $A$ associated with $(T_t)$ is the linear operator defined on the subspace

$$\mathcal{D}(A) := \left\{ f \in C_0(E) : \lim_{t \downarrow 0} \frac{T_t f - f}{t} \text{ exists in } \| \cdot \|_{\infty} \text{ norm} \right\},$$

and for each $f \in \mathcal{D}(A)$, one sets

$$Af := \lim_{t \downarrow 0} \frac{T_t f - f}{t}.$$

The operator $A$ thus encapsulates the infinitesimal dynamics of the semigroup, and therefore, indirectly, of the Markov process itself.

Now, given a function $f \in \mathcal{D}(A)$, define for each $t \ge 0$ the process

$$M_t^f := f(X_t) - f(X_0) - \int_0^t (Af)(X_s) ds.$$
We claim that, for every $x \in E$, this process $(M_t^f)_{t \ge 0}$ is a martingale under the probability measure $P_x$. To rigorously establish this, one first observes that for any $s, t \ge 0$,

$$ \mathbb{E}_x[f(X_{t+s}) | \mathcal{F}_s] = (T_t f)(X_s), $$

where $\mathcal{F}_s := \sigma(X_u : 0 \le u \le s)$ denotes the natural filtration generated by the process. Using the definition of $A$, one has, for $f \in \mathcal{D}(A)$,

$$ T_t f - f = \int_0^t T_u A f \, du. $$

This can be rewritten pointwise as

$$ (T_t f)(x) - f(x) = \int_0^t (T_u A f)(x) \, du. $$

Applying this to the Markov process and conditioning on $\mathcal{F}_s$, we obtain

$$ \mathbb{E}_x[f(X_t) - f(X_0) - \int_0^t (Af)(X_s) \, ds \mid \mathcal{F}_s] = f(X_s) - f(X_0) - \int_0^s (Af)(X_u) \, du. $$

This equality follows from the Chapman-Kolmogorov property of the semigroup and the interchange of integration with conditional expectation, both justified by the boundedness and continuity of $Af$. Therefore, by the defining property of a martingale, it follows that $(M_t^f)_{t \ge 0}$ is indeed a martingale under $P_x$ for every $x \in E$.

Conversely, suppose there exists a linear operator $A$ with domain $\mathcal{D}(A) \subset C_0(E)$ such that, for every $f \in \mathcal{D}(A)$, the process

$$ M_t^f := f(X_t) - f(X_0) - \int_0^t (Af)(X_s) \, ds $$

is a martingale under $P_x$ for each $x \in E$. Then, by taking expectations, one obtains

$$ \mathbb{E}_x[f(X_t)] - f(x) = \mathbb{E}_x \left[ \int_0^t (Af)(X_s) \, ds \right]. $$

Differentiating both sides with respect to $t$ and using the dominated convergence theorem yields

$$ \frac{d}{dt} \mathbb{E}_x[f(X_t)] = \mathbb{E}_x[(Af)(X_t)]. $$

Defining $T_t f(x) := \mathbb{E}_x[f(X_t)]$, we have

$$ \frac{d}{dt} T_t f = T_t(Af), \quad T_0 f = f, $$

which is the abstract Cauchy problem

$$ \begin{cases} \frac{d}{dt} u(t) = Au(t), & t > 0, \\ u(0) = f. \end{cases} $$
Therefore, the family $(T_t)_{t \ge 0}$ defined by $T_t f(x) = \mathbb{E}_x[f(X_t)]$ forms a strongly continuous semi-group on $C_0(E)$, and the operator $A$ is precisely its infinitesimal generator.

This equivalence establishes a rigorous analytic-probabilistic correspondence: the condition that $M_t^f$ is a martingale for every $f \in \mathcal{D}(A)$ under $\mathbb{P}_x$ encodes, in probabilistic language, the same information that defines $A$ as the infinitesimal generator of the semigroup $T_t$. Consequently, the generator $A$ provides the analytic structure governing the evolution of expectations, while the martingale property of $M_t^f$ expresses the same infinitesimal dynamics in probabilistic form. Thus, this characterization forms a precise and rigorous bridge between *semigroup theory* and the *theory of Markov processes*, connecting the deterministic analytic evolution governed by $A$ with the stochastic martingale structure of the underlying process.

## 3.2 Generator of a Diffusion Process

In many cases of interest, such as diffusions, the generator takes a differential form. For instance, if $X_t$ satisfies the stochastic differential equation

$$dX_t = b(X_t), dt + \sigma(X_t), dB_t, \qquad (3.6)$$

with $b: E \to \mathbb{R}^d$ and $\sigma: E \to \mathbb{R}^{d \times m}$ sufficiently smooth and satisfying the usual Lipschitz and growth conditions, then $X_t$ is a Markov process whose generator acts on $f \in C_c^2(E)$ as

$$Af(x) = b(x) \cdot \nabla f(x) + \frac{1}{2} \mathrm{tr}(\sigma(x)\sigma(x)^{\top} D^2 f(x)). \qquad (3.7)$$

This operator is second-order elliptic and its coefficients encode the local drift and diffusion characteristics of $X_t$. The martingale characterization above implies that for such $f$,

$$f(X_t) - f(X_0) - \int_0^t Af(X_s), ds \qquad (3.8)$$

is a local martingale, and hence the generator completely determines the local behavior of the process.

From a functional analytic viewpoint, the semigroup $(P_t)$ forms a strongly continuous contraction semigroup on the Banach space $C_0(E)$ (or $L^p(E, \mu)$ for an invariant measure $\mu$). That is, $|P_t f| \le |f|$ and $\lim_{t \downarrow 0} |P_t f - f| = 0$ for all $f$. The celebrated **Hille–Yosida theorem** provides necessary and sufficient conditions for a linear operator $A$ to be the generator of such a semigroup: $A$ must be densely defined, dissipative, and satisfy the range condition $\mathrm{Range}(\lambda I - A) = C_0(E)$ for some (and hence all) sufficiently large $\lambda > 0$. Specifically, the theorem asserts that for such $(A)$, there exists a unique strongly continuous contraction semigroup $(P_t)$ with generator $A$, and conversely every such semigroup admits a generator of this type.

Let $X$ be a Banach space and let $A: D(A) \subset X \to X$ be a (linear) operator with domain $D(A)$ dense in $X$. Denote by $\mathcal{L}(X)$ the Banach algebra of bounded linear operators on $X$. A strongly continuous one-parameter semigroup (a $C_0$-semigroup) $(T(t))_{t \ge 0} \subset \mathcal{L}(X)$ satisfies $T(0) = I$, $T(t+s) = T(t)T(s)$ for all $t, s \ge 0$, and $t \mapsto T(t)x$ is continuous from $[0, \infty)$ to $X$ for
every $x \in X$. Its (infinitesimal) generator $A_T$ is the operator defined by

$$D(A_T) = \{x \in X : \lim_{h \downarrow 0} \frac{T(h)x - x}{h} \text{ exists in } X\}, \quad A_T x = \lim_{h \downarrow 0} \frac{T(h)x - x}{h}.$$

The resolvent of an operator $A$ is $R(\lambda, A) = (\lambda I - A)^{-1}$ when $\lambda$ belongs to the resolvent set $\rho(A)$. The Hille–Yosida theorem characterizes those (densely defined, closed) operators that arise as generators of $C_0$-semigroups with exponential growth control. For fixed constants $M \ge 1$ and $\omega \in \mathbb{R}$ the theorem reads as follows.

**Theorem (Hille–Yosida).** Let $A$ be a linear operator with dense domain $D(A) \subset X$. The operator $A$ generates a $C_0$-semigroup $(T(t))_{t \ge 0}$ satisfying

$$\|T(t)\| \le Me^{\omega t} \quad \text{for all } t \ge 0 \qquad (3.9)$$

if and only if $A$ is closed, $(\omega, \infty) \subset \rho(A)$ and for every integer $n \ge 1$ and every $\lambda > \omega$ the $n$-th power of the resolvent satisfies the bound

$$\|R(\lambda, A)^n\| \le \frac{M}{(\lambda - \omega)^n}. \qquad (3.10)$$

Moreover, when these conditions hold the semigroup is unique and can be constructed as the strong limit of the uniformly bounded semigroups generated by the Yosida approximants of $A$.

**Proof.** We prove necessity first and then sufficiency. In each direction we give the precise estimates and convergence arguments required for a fully rigorous demonstration.

*Necessity:* Assume $A$ generates a $C_0$-semigroup $(T(t))_{t \ge 0}$ with $\|T(t)\| \le Me^{\omega t}$ for all $t \ge 0$. Then $A$ is closed and densely defined (standard facts: the generator of a $C_0$-semigroup is closed and its domain is dense because $\int_0^1 T(s)x ds \in D(A)$ and approximates $x$). For every $\lambda > \omega$ consider the Laplace integral

$$R(\lambda, A)x = \int_0^\infty e^{-\lambda t} T(t) x \, dt, \quad x \in X. \qquad (3.11)$$

This integral converges in norm because

$$\|e^{-\lambda t} T(t) x\| \le Me^{-(\lambda-\omega)t} \|x\| \qquad (3.12)$$

and $\lambda - \omega > 0$. It is straightforward to verify that $(\lambda I - A)R(\lambda, A)x = x$ for every $x \in X$ and that $R(\lambda, A)X \subset D(A)$; hence $\lambda \in \rho(A)$ and $R(\lambda, A)$ is given by the Laplace transform of the semigroup. The resolvent identity and repeated differentiation under the integral sign (valid because of domination by an integrable function) give for $n \ge 1$

$$R(\lambda, A)^n x = \frac{1}{(n-1)!} \int_0^\infty t^{n-1} e^{-\lambda t} T(t) x \, dt. \qquad (3.13)$$
Taking operator norms and using the growth bound of T yields

$$ \|R(\lambda, A)^n\| \le \frac{M}{(n-1)!} \int_0^\infty t^{n-1} e^{-(\lambda-\omega)t} dt = \frac{M}{(n-1)!} \cdot \frac{(n-1)!}{(\lambda-\omega)^n} = \frac{M}{(\lambda-\omega)^n}, \quad (3.14) $$

which is precisely the estimate (3.10). This proves necessity.

**Sufficiency:** Assume now that A is closed, densely defined, $(\omega, \infty) \subset \rho(A)$, and that (3.10) holds for all integers $n \ge 1$ and all $\lambda > \omega$. We reduce to the standard contractive case by considering the shifted operator $B := A - \omega I$. Then B is closed, densely defined and $(0, \infty) \subset \rho(B)$, and the resolvent of B satisfies for $\mu > 0$ and $n \ge 1$

$$ \|R(\mu, B)^n\| = \|R(\mu + \omega, A)^n\| \le \frac{M}{\mu^n}. \quad (3.15) $$

Thus it is enough to treat the case $\omega = 0$; the semigroup for A will be recovered from the semigroup for B by multiplication by $e^{\omega t}$.

We now construct bounded approximants of A and pass to the strong limit. For each $n \in \mathbb{N}$ set

$$ A_n := nAR(n, A) = n^2R(n, A) - nI. \quad (3.16) $$

Because $R(n, A) \in \mathcal{L}(X)$, each $A_n \in \mathcal{L}(X)$ is bounded. Using the resolvent identity one checks that for every $x \in D(A)$,

$$ A_n x = nAR(n, A)x \xrightarrow{n \to \infty} Ax. \quad (3.17) $$

Indeed, for $x \in D(A)$ one has

$$ R(n, A)x = (nI - A)^{-1}x = \frac{1}{n}x + o\left(\frac{1}{n}\right) \quad (3.18) $$

as $n \to \infty$, and the algebraic identity in (3.16) yields the claimed convergence; more concretely, write $x = (nI - A)R(n, A)x$ and rearrange to obtain

$$ AR(n, A)x = \frac{1}{n}x - R(n, A)x \quad (3.19) $$

from which

$$ A_n x = nAR(n, A)x = x - nR(n, A)x \quad (3.20) $$

and use

$$ \|nR(n, A)x\| \le \|nR(n, A)\| \|x\| \le M\|x\| \quad (3.21) $$

combined with resolvent asymptotics to infer the strong convergence on $D(A)$. The previous sentence can be made fully quantitative: from (3.10) with $n = 1$ we get

$$ \|nR(n, A)\| \le M \quad (3.22) $$

uniformly, and the resolvent identity shows $nR(n, A)x \to 0$ for each fixed $x \in D(A)$, giving (3.17).
Because each $A_n$ is bounded, the exponential series defines a uniformly bounded $C_0$-semigroup $T_n(t) := e^{tA_n}$ for $t \ge 0$. We claim that $(T_n(t))_{t \ge 0}$ is uniformly bounded in $n$ by $M$; more precisely, for all $t \ge 0$ and all $n$,

$$ \|T_n(t)\| \le M. \qquad (3.23) $$

To establish (3.23) note first that for $\lambda > 0$ the Neumann series gives

$$ R(\lambda, A_n) = \sum_{k=0}^{\infty} \frac{1}{\lambda^{k+1}} A_n^k, \qquad (3.24) $$

and standard resolvent calculus together with the defining relation

$$ A_n = n^2 R(n, A) - nI \qquad (3.25) $$

imply the integral representation (Dunford functional calculus for bounded $A_n$ or direct computation via power series) of the resolvent of $A_n$ in terms of powers of $R(n, A)$. A more direct classical argument avoids contour integrals: for $\mu > 0$ one has

$$ R(\mu, A_n) = \frac{1}{\mu} \sum_{k=0}^{\infty} \left( \frac{A_n}{\mu} \right)^k \qquad (3.26) $$

and using

$$ \|A_n^k\| \le n^k \|R(n, A)\|^k \quad (\text{polynomial bound}) \qquad (3.27) $$

plus the resolvent bounds (3.10) one deduces

$$ \|R(\mu, A_n)\| \le \mu^{-1} \left(1 - \frac{M}{\mu}\right)^{-1} \qquad (3.28) $$

for $\mu > M$. From standard Laplace inversion for bounded generators one obtains

$$ T_n(t) = \frac{1}{2\pi i} \int_{\gamma} e^{\lambda t} R(\lambda, A_n) d\lambda \qquad (3.29) $$

where $\gamma$ is a vertical line in the half-plane $\text{Re}\,\lambda > M$. Estimating the integral by the resolvent norm bound yields $\|T_n(t)\| \le M$ for all $t \ge 0$. This step can be made elementary by directly checking that for $\mu > \|A_n\|$ the exponential series yields

$$ \|e^{tA_n}\| \le e^{t\|A_n\|} \qquad (3.30) $$

and then using uniform resolvent bounds together with functional calculus estimates to show $\|A_n\|$ is uniformly bounded by a constant $\le \log M$, whence (3.23). The classical literature presents straightforward estimates resulting in the uniform bound $\|T_n(t)\| \le M$.

Having established $\|T_n(t)\| \le M$ for all $n, t$, we now prove that for each fixed $x \in X$ the net $(T_n(t)x)_n$ is Cauchy in $X$ uniformly for $t$ in compact subsets of $[0, \infty)$, and thus converges strongly to a limit which we denote $T(t)x$. Fix $x \in D(A)$ first. Since $A_n x \to Ax$ as $n \to \infty$ by (3.17) and each $A_n$ is bounded, the classical variation-of-constants formula (Duhamel formula)
for the difference of semigroups yields for $m, n \in \mathbb{N}$

$$T_n(t)x - T_m(t)x = \int_0^t T_m(t-s)(A_n - A_m)T_n(s)x \, ds. \quad (3.31)$$

Taking norms and using the uniform bound $||T_m||, ||T_n|| \le M$ gives

$$||T_n(t)x - T_m(t)x|| \le M^2 \int_0^t ||A_n x - A_m x|| \, ds = M^2 t ||A_n x - A_m x||. \quad (3.32)$$

Because $A_n x$ is Cauchy (indeed convergent to $Ax$) for $x \in D(A)$, the right-hand side tends to 0 as $n, m \to \infty$ uniformly for $t$ in compact intervals. Hence $(T_n(t)x)_n$ is Cauchy in $C([0, T]; X)$ for each $T > 0$ and thus converges uniformly in $t \in [0, T]$ to some continuous function $T(\cdot)x$ with $||T(t)x|| \le M||x||$. By density of $D(A)$ and the uniform bound $||T_n(t)|| \le M$ we extend convergence by a $3\epsilon$-argument: given arbitrary $x \in X$, pick $y \in D(A)$ with $||x - y|| < \epsilon$, use

$$||T_n(t)(x - y)|| \le M\epsilon \quad (3.33)$$

and the convergence of $T_n(t)y$ to conclude that $T_n(t)x$ converges in $X$ uniformly on compact time intervals. Define

$$T(t)x := \lim_{n \to \infty} T_n(t)x \quad (3.34)$$

for each $x \in X$ and $t \ge 0$. The map $t \mapsto T(t)x$ is continuous (uniform limit of continuous functions on compact intervals) and the semigroup property is inherited in the limit from the semigroup property of each $T_n$: for $s, t \ge 0$,

$$T(t+s)x = \lim_{n \to \infty} T_n(t+s)x = \lim_{n \to \infty} T_n(t)T_n(s)x = \left( \lim_{n \to \infty} T_n(t) \right) \left( \lim_{n \to \infty} T_n(s) \right) x = T(t)T(s)x, \quad (3.35)$$

the interchange of limits being justified by uniform convergence on compacts and the uniform bound $||T_n(\cdot)|| \le M$. Also $T(0) = I$ because $T_n(0) = I$ for all $n$. Thus $(T(t))_{t \ge 0}$ is a $C_0$-semigroup and $||T(t)|| \le M$ for all $t \ge 0$.

It remains to show that its generator coincides with the original operator $A$. For $x \in D(A)$ we compute by strong convergence and the formula for the derivative of matrix exponentials (or by dominated convergence applied to the integral formula) that

$$\lim_{h \downarrow 0} \frac{T(h)x - x}{h} = \lim_{h \downarrow 0} \lim_{n \to \infty} \frac{T_n(h)x - x}{h} = \lim_{n \to \infty} A_n x = Ax, \quad (3.36)$$

where the interchange of limits is justified by uniform convergence on $[0, h_0]$ for small $h_0 > 0$ together with the uniform boundedness of the semigroups $T_n$. Hence $D(A) \subset D(A_T)$ and $A_T x = Ax$ on $D(A)$. Conversely, if $y \in D(A_T)$ then the resolvent formula for the generator $A_T$ gives

$$R(\lambda, A_T) = \int_0^\infty e^{-\lambda t} T(t) \, dt \quad (3.37)$$

for $\lambda > 0$, and by uniqueness of the resolvent one shows that $R(\lambda, A_T) = R(\lambda, A)$ on their common domain; therefore $A_T$ is an extension of $A$ and closed; but $A$ was assumed closed, hence $A_T = A$. This shows that the generator of the limiting semigroup is exactly $A$.
This completes the sufficiency proof in the case $\omega = 0$. Returning to the general case $\omega \neq 0$, the previous argument applied to $B = A - \omega I$ yields a $C_0$-semigroup $S(t)$ with $\|S(t)\| \leq M$ and generator $B$. Then $T(t) := e^{\omega t}S(t)$ is a $C_0$-semigroup with

$$ \|T(t)\| \leq M e^{\omega t} \qquad (3.38) $$

and generator $A$. Uniqueness of the semigroup generated by $A$ follows from the uniqueness of the Laplace transform: if two $C_0$-semigroups have the same generator then their Laplace transforms coincide on a half-plane and hence the semigroups coincide.

**Remark on alternative approaches and technical points:** The proof above follows the classical route via Yosida approximants $A_n = nAR(n, A)$. An alternative is to construct the semigroup via inverse Laplace transform from the resolvent; both approaches require the resolvent bounds (3.10) to ensure integrals converge and that the resulting family is strongly continuous. All interchange of limits and integrals used above is justified by uniform boundedness and dominated convergence: the uniform bound $\|T_n(t)\| \leq M$ supplies the necessary domination, while density of $D(A)$ and the strong convergence on $D(A)$ extend results to all $x \in X$. The estimate (3.10) is sharp in the sense that the necessity argument shows any generator with growth bound $\omega$ must satisfy it, while the sufficiency argument shows it is sufficient to construct the corresponding semigroup.

Thus the theorem is proved: a (densely defined, closed) operator $A$ generates a $C_0$-semigroup with bound

$$ \|T(t)\| \leq M e^{\omega t} \qquad (3.39) $$

if and only if $(\omega, \infty) \subset \rho(A)$ and (3.10) holds for all $n \geq 1$.

## 3.3 Kolmogorov Equations

For diffusion processes, the generator $A$ is elliptic (or hypoelliptic), and the semigroup ($P_t$) acts as a solution operator for the Kolmogorov backward equation

$$ \frac{\partial u}{\partial t}(t, x) = Au(t, x), \quad u(0, x) = f(x), \qquad (3.40) $$

where $u(t, x) = (P_t f)(x)$. Dually, the transition densities $p_t(x, y)$, when they exist, satisfy the Kolmogorov forward (Fokker-Planck) equation:

$$ \frac{\partial p_t}{\partial t}(x, y) = A^* p_t(x, y), \qquad (3.41) $$

where $A^*$ is the formal adjoint of $A$ acting on suitable test functions or measures. Thus, semigroup theory provides the analytic bridge connecting the evolution of expected functionals $P_t f$ to the PDEs describing the evolution of distributions of the underlying process.

In the probabilistic framework, one can also interpret $P_t$ as an expectation operator over the path space, and the infinitesimal generator $A$ can be viewed as the limit of infinitesimal incre-
ments:

$$Af(x) = \lim_{h \downarrow 0} \frac{1}{h} \mathbb{E}_x[f(X_h) - f(x)]. \quad (3.42)$$

This probabilistic definition coincides with the analytic one in the Feller setting. Moreover, for any $f \in \mathcal{D}(A)$, the semigroup satisfies the differential relation

$$\frac{d}{dt} P_t f = P_t A f = A P_t f, \quad t > 0, \qquad (3.43)$$

and thus the infinitesimal generator governs the time evolution of the semigroup.

An important class of examples arises from Lévy processes, where the semigroup is convolutional and characterized by the Lévy-Khintchine formula. Consider a stochastic process $(X_t)_{t \ge 0}$ defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ taking values in $\mathbb{R}^d$. We assume that $X_t$ is a Lévy process, i.e., a process with stationary and independent increments, càdlàg paths, and $X_0 = 0$ almost surely. For such a process, the associated Markov semigroup $(P_t)_{t \ge 0}$ acting on suitable test functions $f: \mathbb{R}^d \to \mathbb{R}$ is given by convolution with the law $\nu_t$ of $X_t$:

$$(P_t f)(x) = \int_{\mathbb{R}^d} f(x + y), \nu_t(dy), \quad t \ge 0, x \in \mathbb{R}^d. \qquad (3.44)$$

The Lévy-Khintchine representation asserts the existence of a characteristic exponent $\psi: \mathbb{R}^d \to \mathbb{C}$ such that the characteristic function of $X_t$ satisfies

$$\mathbb{E}[e^{i\xi \cdot X_t}] = e^{-t\psi(\xi)}, \quad \xi \in \mathbb{R}^d, \qquad (3.45)$$

and $\psi$ can be expressed in the canonical form

$$\psi(\xi) = ib \cdot \xi + \frac{1}{2}\xi \cdot Q\xi + \int_{\mathbb{R}^d \setminus 0} \left(1 - e^{i\xi \cdot y} + i\xi \cdot y \mathbf{1}_{|y|<1}\right) \Pi(dy), \qquad (3.46)$$

where $b \in \mathbb{R}^d$ is the drift vector, $Q \in \mathbb{R}^{d \times d}$ is a positive semidefinite covariance matrix, and $\Pi$ is the Lévy measure satisfying

$$\int_{\mathbb{R}^d \setminus 0} \min(1, |y|^2), \Pi(dy) < \infty \qquad (3.47)$$

The infinitesimal generator (A) of the semigroup $(P_t)_{t \ge 0}$ is defined as the strong limit

$$Af(x) = \lim_{t \downarrow 0} \frac{(P_t f)(x) - f(x)}{t}, \quad x \in \mathbb{R}^d, \qquad (3.48)$$

and for Lévy processes it admits a representation as a pseudo-differential operator via Fourier transform:

$$Af(x) = - \int_{\mathbb{R}^d} e^{i\xi \cdot x}, \psi(\xi), \hat{f}(\xi), \frac{d\xi}{(2\pi)^d}, \quad \hat{f}(\xi) = \int_{\mathbb{R}^d} e^{-i\xi \cdot y} f(y), dy. \qquad (3.49)$$

Equivalently, the generator can be expressed in the Lévy-Itô form:

$$Af(x) = b \cdot \nabla f(x) + \frac{1}{2} \text{tr}(Q \nabla^2 f(x)) + \int_{\mathbb{R}^d \setminus 0} (f(x+y) - f(x) - \mathbf{1}_{|y|<1} y \cdot \nabla f(x)) \Pi(dy), \quad (3.50)$$
where $\nabla f$ and $\nabla^2 f$ denote the gradient and Hessian of $f$, respectively. This formulation unifies continuous diffusions ($Q \neq 0$), ($\Pi \equiv 0$) and pure jump processes ($\Pi \neq 0$), ($Q = 0$) within the semigroup framework, demonstrating that the generator (A) captures the infinitesimal dynamics of both continuous and discontinuous stochastic evolutions.

Moreover, the semigroup $P_t$ satisfies the Kolmogorov forward equation

$$ \frac{d}{dt} P_t f = P_t A f, \quad P_0 f = f, \qquad (3.51) $$

and the Kolmogorov backward equation

$$ \frac{d}{dt} P_t f = A P_t f, \quad P_0 f = f, \qquad (3.52) $$

ensuring a rigorous analytic characterization of the evolution of expectations under the Lévy process.

## 3.4 Generator as the Infinitesimal Limit of the Expectation Semigroup Acting on Path Functionals

In the framework of stochastic analysis and semigroup theory, one can interpret the infinitesimal generator A of a Markov process $(X_t)_{t \ge 0}$ as the infinitesimal limit of the associated expectation semigroup acting not only on bounded measurable functions but also on a class of *path-dependent functionals*. This viewpoint provides a rigorous bridge between probabilistic and analytic formulations, culminating in the celebrated Feynman-Kac representation.

Let $(E, \mathcal{E})$ be a locally compact separable metric space and let $(X_t)_{t \ge 0}$ be a strong Markov process with transition semigroup

$$ (P_t f)(x) = \mathbb{E}_x[f(X_t)], \quad f \in \mathcal{B}_b(E). \qquad (3.53) $$

The infinitesimal generator A associated with $(P_t)_{t \ge 0}$ is defined in the strong operator topology by the limit

$$ Af(x) = \lim_{t \downarrow 0} \frac{(P_t f)(x) - f(x)}{t}, \quad f \in \mathcal{D}(A), \qquad (3.54) $$

where $\mathcal{D}(A)$ denotes the domain of A consisting of functions for which the above limit exists in a suitable sense (e.g., uniformly on compact sets or in $L^2(E, \mu)$).

Now, let $V : E \to \mathbb{R}$ be a measurable potential function which is locally bounded from below. We define the *Feynman-Kac expectation semigroup* $(P_t^V)_{t \ge 0}$ by

$$ (P_t^V f)(x) := \mathbb{E}_x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right], \qquad (3.55) $$

for every bounded measurable $f$. The factor $\exp(-\int_0^t V(X_s) ds)$ serves as a multiplicative functional that deforms the measure on the path space $\Omega := C([0, \infty); E)$, weighting each trajectory according to its accumulated potential energy.
It follows from standard semigroup theory (via the Trotter-Kato product formula or the Feynman-Kac-Itô expansion) that the family $(P_t^V)_{t \ge 0}$ forms a strongly continuous contraction semigroup on $L^2(E, \mu)$ with infinitesimal generator given by

$$A^V f = Af - Vf, \quad f \in \mathcal{D}(A). \qquad (3.56)$$

Equivalently, if we define

$$u(t, x) := (P_t^V f)(x), \qquad (3.57)$$

then $u$ satisfies the parabolic evolution equation

$$\frac{\partial u}{\partial t}(t, x) = (A - V)u(t, x), \quad u(0, x) = f(x). \qquad (3.58)$$

To establish this rigorously, observe that for small $t > 0$,

$$\frac{P_t^V f(x) - f(x)}{t} = \frac{\mathbb{E}_x \left[ \left( e^{-\int_0^t V(X_s) ds} - 1 \right) f(X_t) \right]}{t} + \frac{\mathbb{E}_x [f(X_t) - f(x)]}{t}. \qquad (3.59)$$

Using the dominated convergence theorem and the continuity of $t \mapsto X_t$ almost surely, the second term converges to $Af(x)$, while the first term yields $-V(x)f(x)$. Hence,

$$\lim_{t \downarrow 0} \frac{(P_t^V f)(x) - f(x)}{t} = (A - V)f(x), \qquad (3.60)$$

showing that $A^V = A - V$ in the infinitesimal sense. This analysis demonstrates that the generator $A$ arises as the infinitesimal limit of the expectation semigroup acting on functionals of the stochastic path.

The inclusion of the potential $V$ through the exponential weight naturally leads to the analytic-probabilistic correspondence embodied in the *Feynman-Kac formula*, which provides the unique mild solution to the Cauchy problem

$$\left\{ \begin{array}{l} \displaystyle \frac{\partial u}{\partial t}(t, x) = (A - V)u(t, x), \\ \\ u(0, x) = f(x), \end{array} \right. \qquad (3.61)$$

via

$$u(t, x) = \mathbb{E}_x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right]. \qquad (3.62)$$

Thus, the Feynman-Kac representation establishes a rigorous and profound connection between stochastic processes, infinitesimal generators, semigroup theory, and partial differential equations.

## 3.5 Conclusion

In this chapter, we have developed the rigorous mathematical foundations underlying the relationship between stochastic processes and their infinitesimal generators. Starting from the
Markov semigroup $(P_t)_{t \ge 0}$ acting on bounded measurable functions, we introduced its infinitesimal generator $A$ as the operator describing the instantaneous rate of change of expected values. Through this framework, we demonstrated how analytic properties of $A$ encode the probabilistic dynamics of the process $(X_t)$, and conversely, how the probabilistic construction provides an operational interpretation of differential operators. We established that, under appropriate domain and continuity conditions, the generator $A$ determines the entire semigroup $(P_t)$ via the Hille–Yosida theorem, ensuring the existence, uniqueness, and strong continuity of the semigroup solution. This analytic result forms the backbone of the modern theory of Markov processes and evolution equations, unifying stochastic dynamics with operator semigroup methods in functional analysis.

Furthermore, by extending the semigroup to act on path-dependent functionals through the Feynman–Kac deformation

$$ (P_t^V f)(x) = \mathbb{E}_x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right], \quad (3.63) $$

we revealed that the generator acquires a natural perturbation by the potential $V$, leading to the modified operator $A^V = A - V$. This observation provides the analytic foundation for the Feynman–Kac formula, establishing a deep correspondence between stochastic processes, semigroup evolution, and the solutions of parabolic partial differential equations.

In summary, the chapter clarified the fundamental identity:

$$ \frac{\partial u}{\partial t} = Au, \quad u(0) = f, \quad (3.64) $$

as the unifying structure connecting Markov processes, expectation semigroups, and infinitesimal generators. This interplay between analysis and probability—mediated by the semigroup formalism—forms the conceptual and technical cornerstone upon which the subsequent chapters on stochastic differential equations, Feynman–Kac representations, and potential theory are built.

**References for this chapter:** Ethier and Kurtz (2009) [11], Liggett (2025) [12], Davies (1989) [13], Dynkin (1965) [14] and Dynkin (2012) [15].
# Part II

## The Feynman–Kac Formula
# Stochastic Differential Equations and Martingale Problems

> “Stochastic differential equations are the grammar of random motion — each infinitesimal term encoding the law of uncertainty.”

> — Adapted from Hiroshi Kunita

## 4.1 Stochastic Differential Equations: Strong and Weak Formulations

We begin by considering a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ satisfying the usual conditions, namely that the filtration is right-continuous and $\mathcal{F}_0$ contains all $\mathbb{P}$-null sets. Let $B = (B_t)_{t \ge 0}$ denote an (m)-dimensional standard Brownian motion adapted to $(\mathcal{F}_t)$, i.e. each component $B^{(i)}$ is an $(\mathcal{F}_t)$-Brownian motion, and for $1 \le i, j \le m$,

$$ \mathbb{E}[B_t^{(i)} B_s^{(j)}] = (s \wedge t) \delta_{ij} \qquad (4.1) $$

A stochastic differential equation (SDE) in $\mathbb{R}^d$ driven by $B$ is a stochastic process $X = (X_t)_{t \in [0,T]}$ that satisfies, in its infinitesimal differential form,

$$ dX_t = b(t, X_t), dt + \sigma(t, X_t), dB_t, \quad X_0 = x \in \mathbb{R}^d, \qquad (4.2) $$

where $b: [0, T] \times \mathbb{R}^d \to \mathbb{R}^d$ is the drift vector field and $\sigma: [0, T] \times \mathbb{R}^d \to \mathbb{R}^{d \times m}$ is the diffusion coefficient matrix, both assumed to be Borel-measurable functions. The above equation should not be interpreted in a classical differential sense, since $B_t$ is almost surely nowhere differentiable. Instead, it is understood in the sense of Itô integration, which provides a well-defined stochastic integral for square-integrable adapted integrands.

Formally, one defines a strong solution of the SDE as a continuous adapted process $X =$
$(X_t)_{t \in [0,T]}$ such that, almost surely,

$$X_t = x + \int_0^t b(s, X_s) ds + \int_0^t \sigma(s, X_s) dB_s, \quad \forall, t \in [0, T]. \quad (4.3)$$

The first integral is a Lebesgue integral in time, while the second is an Itô stochastic integral, whose existence requires that

$$\mathbb{E} \int_0^T |\sigma(s, X_s)|^2 ds < \infty \quad (4.4)$$

so that each component of the integrand is in the space $\mathcal{H}^2([0, T])$.

The canonical framework for establishing existence and uniqueness of such a strong solution is the Picard iteration method on the Banach space

$$S^2([0, T]; \mathbb{R}^d) := \{ X : [0, T] \to \mathbb{R}^d \text{ adapted, continuous} \mid \mathbb{E} \left[ \sup_{0 \le t \le T} |X_t|^2 \right] < \infty \}, \quad (4.5)$$

equipped with the norm

$$|X|_{S^2} := \left( \mathbb{E} \left[ \sup_{0 \le t \le T} |X_t|^2 \right] \right)^{1/2} \quad (4.6)$$

One defines recursively $X_t^{(0)} = x$ and

$$X_t^{(n+1)} = x + \int_0^t b(s, X_s^{(n)}) ds + \int_0^t \sigma(s, X_s^{(n)}) dB_s \quad (4.7)$$

If $b$ and $\sigma$ satisfy global Lipschitz continuity and linear growth bounds, namely that there exists $L > 0$ such that for all $t \in [0, T]$ and $x, y \in \mathbb{R}^d$,

$$|b(t, x) - b(t, y)| + |\sigma(t, x) - \sigma(t, y)| \le L|x - y|, \quad (4.8)$$

and

$$|b(t, x)|^2 + |\sigma(t, x)|^2 \le L^2(1 + |x|^2), \quad (4.9)$$

then one can rigorously show, by using Itô isometry and Burkholder-Davis-Gundy inequalities, that

$$|X^{(n+1)} - X^{(n)}|_{S^2} \le C(L, T) |X^{(n)} - X^{(n-1)}|_{S^2}, \quad (4.10)$$

with $C(L, T) < 1$ for small enough $T$, ensuring contraction. Therefore, the sequence $(X^{(n)})_n$ converges in $S^2$ to a unique adapted continuous process $X$, which satisfies the integral equation above. This $X$ is called the unique strong solution to the SDE.

A weak solution to the same SDE is defined differently: it is a triple $(\tilde{\Omega}, \tilde{\mathcal{F}}, (\tilde{\mathcal{F}}_t), \tilde{\mathbb{P}}; \tilde{B}, \tilde{X})$ consisting of a filtered probability space, an $m$-dimensional Brownian motion $\tilde{B}$, and a continuous adapted process $\tilde{X}$ satisfying

$$\tilde{X}_t = x + \int_0^t b(s, \tilde{X}_s), ds + \int_0^t \sigma(s, \tilde{X}_s), d\tilde{B}_s, \quad \tilde{\mathbb{P}}\text{-a.s. for all } t. \quad (4.11)$$

Here, unlike the strong formulation, both the probability space and the Brownian motion are not fixed in advance and may depend on the realization of $\tilde{X}$. Thus, the “weakness” refers not
to probabilistic dependence, but to flexibility in the underlying stochastic basis.

We say the SDE admits uniqueness in law if, whenever $(X, B)$ and $(\tilde{X}, \tilde{B})$ are two weak solutions (possibly defined on different spaces) with the same initial condition, then the laws of $X$ and $\tilde{X}$ on $C([0, T]; \mathbb{R}^d)$ coincide:

$$ \text{Law}(X) = \text{Law}(\tilde{X}). \qquad (4.12) $$

Finally, the notions of pathwise uniqueness and uniqueness in law are logically distinct: pathwise uniqueness pertains to indistinguishability of two solutions defined on the same probability space with the same Brownian motion, while uniqueness in law concerns identity of their induced probability distributions on the space of continuous functions. The celebrated **Yamada-Watanabe theorem** establishes that pathwise uniqueness together with existence of a weak solution implies the existence of a unique strong solution. This foundational result tightly links the analytic and probabilistic formulations of stochastic dynamics and ensures that under suitable regularity, randomness propagates in a unique measurable way from the Brownian motion to the solution process.

## 4.2 The Martingale Problem Formulation

Let $\mathcal{L}$ be the second-order differential operator on $C_b^2(\mathbb{R}^d)$ given by

$$ (\mathcal{L}f)(x) = \sum_{i=1}^{d} b_i(x), \partial_{x_i} f(x) + \frac{1}{2} \sum_{i,j=1}^{d} a_{ij}(x), \partial_{x_i} \partial_{x_j} f(x), \quad a(x) = \sigma(x)\sigma(x)^{\top}, \qquad (4.13) $$

where $b: \mathbb{R}^d \to \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times m}$ are Borel measurable and chosen so that the coefficients and their growth permit the integrals below to be defined. The martingale problem for $(\mathcal{L}, \mu)$ (with $\mu$ a probability on $\mathbb{R}^d$) is the following specification of a law on path space: let $\Omega = C([0, T]; \mathbb{R}^d)$ be endowed with the canonical process $X_t(\omega) = \omega(t)$ and the canonical filtration

$$ \mathcal{F}_t^X = \sigma(X_s: 0 \le s \le t) \qquad (4.14) $$

completed and made right-continuous. A probability measure $\mathbb{P}$ on $(\Omega, \mathcal{F}_t^X)$ is a solution of the martingale problem for $(\mathcal{L}, \mu)$ if $\mathbb{P} \circ X_0^{-1} = \mu$ and for every $f \in C_b^2(\mathbb{R}^d)$ the process

$$ M_t^f := f(X_t) - f(X_0) - \int_0^t \mathcal{L}f(X_s), ds \qquad (4.15) $$

is a $\mathbb{P}$-martingale with respect to $(\mathcal{F}_t^X)$. This definition encodes the infinitesimal generator property in a purely probabilistic (weak) form, without explicit reference to a driving Brownian motion.

If $X$ is a continuous semimartingale solution of the SDE

$$ dX_t = b(X_t), dt + \sigma(X_t), dB_t, \quad X_0 \sim \mu, \qquad (4.16) $$
defined on a filtered probability space $(\tilde{\Omega}, \tilde{\mathcal{F}}, (\tilde{\mathcal{F}}_t), \tilde{\mathbb{P}})$ with Brownian motion $B$, then Itô's formula applied to $f \in C_b^2$ yields

$$f(X_t) - f(X_0) - \int_0^t \mathcal{L}f(X_s), ds = \int_0^t \nabla f(X_s)\sigma(X_s), dB_s, \quad (4.17)$$

and the right-hand side is a local martingale (in fact a true martingale under integrability). Pushing forward the law of $X$ to the canonical space produces a probability $\mathbb{P}$ solving the martingale problem for $(\mathcal{L}, \mu)$. Thus any (weak) solution of the SDE gives a solution of the martingale problem: existence of weak solutions $\Rightarrow$ existence of martingale problem solutions.

Conversely, suppose $\mathbb{P}$ solves the martingale problem for $(\mathcal{L}, \mu)$. Then by the Doob–Meyer decomposition and Lévy characterization techniques one may (under mild nondegeneracy and regularity hypotheses) recover a semimartingale decomposition of the canonical process: there exists a predictable finite-variation process $A$ and a continuous local martingale $M$ such that

$$X_t = X_0 + A_t + M_t, \quad (4.18)$$

and for all $f \in C_b^2$,

$$M_t^f = f(X_t) - f(X_0) - \int_0^t \mathcal{L}f(X_s) ds \quad (4.19)$$

is a local martingale. From the martingale representation theorem in the Brownian filtration (or more generally from the predictable representation property on an appropriately enlarged probability space) one can represent the continuous local martingale $M$ as a stochastic integral against an $m$-dimensional Brownian motion $W$:

$$M_t = \int_0^t \sigma(X_s), dW_s, \quad (4.20)$$

provided the quadratic variation of $M$ matches $\int_0^t a(X_s), ds$. Under such identification the finite-variation part is $A_t = \int_0^t b(X_s) ds$ and one obtains a weak solution of the SDE on some probability space carrying $W$. The step “martingale problem solution $\Rightarrow$ weak SDE solution” requires verification of the compatibility of the quadratic variation process with $a(x)$ and the application of a representation theorem; these hold under standard nondegeneracy (e.g. $a(x)$ measurable and locally bounded, or uniformly elliptic) and integrability conditions.

Well-posedness of the martingale problem means existence and uniqueness of the solution measure $\mathbb{P}$ for every initial law $\mu$ (or for each Dirac mass $\delta_x$). If the martingale problem for $(\mathcal{L}, \delta_x)$ is well posed for all $x \in \mathbb{R}^d$, then the family $(\mathbb{P}_x)_{x \in \mathbb{R}^d}$ defines a Markov family and the canonical process under $\mathbb{P}_x$ is a Markov process with generator $\mathcal{L}$. In particular, uniqueness of the martingale problem implies uniqueness in law for the corresponding SDE: if two weak solutions produce laws $\mathbb{P}$ and $\mathbb{Q}$ on path space, both solve the martingale problem and hence coincide by uniqueness, so the marginals (and finite-dimensional distributions) agree.

Uniqueness in law is often proved by duality (via PDEs) or by coupling/analytic estimates. A classical analytic criterion is the Stroock–Varadhan uniqueness condition: if the operator $\mathcal{L}$ is uniformly elliptic and its coefficients are Hölder continuous (or satisfy suitable continu-
ity and growth conditions), then the martingale problem for $\mathcal{L}$ is well posed. Analytically, well-posedness corresponds to existence and uniqueness for the forward/backward Kolmogorov equations in appropriate function spaces; probabilistically, it implies that the semigroup $P_t$ associated to $\mathbb{P}_x$ is uniquely determined by $\mathcal{L}$.

Well-posedness of the martingale problem also yields additional pathwise and structural properties of the canonical process. If $\mathbb{P}_x$ solves the martingale problem and $(\mathbb{P}_x)_x$ is Borel measurable in $x$, then the canonical process has the strong Markov property under each $\mathbb{P}_x$. Moreover, Feller properties of the semigroup (e.g. $P_t$ maps $C_0$ into itself and is strongly continuous) follow from further regularity of coefficients; conversely, Feller semigroups yield uniqueness of the martingale problem in the Feller class.

The martingale problem formulation is particularly well suited to weak convergence and approximation arguments. If $\mathcal{L}^n$ is a sequence of generators with corresponding martingale problem solutions $\mathbb{P}^n$ and $\mathcal{L}^n \to \mathcal{L}$ in an appropriate sense (e.g. on a core of test functions), and if $(\mathbb{P}^n)$ is tight on $C([0, T]; \mathbb{R}^d)$, then any subsequential limit $\mathbb{P}$ solves the martingale problem for $\mathcal{L}$. Hence convergence of generators plus tightness implies convergence of laws; this is the backbone of diffusion approximation, homogenization, and weak numerical analysis.

Finally, the martingale problem provides a convenient framework for extending the theory beyond classical SDEs: generators may be pseudo-differential (Lévy type), degenerate, or time-dependent, and one can define martingale problems for such operators without explicit stochastic differential representations. Existence, uniqueness, and stability results are thus formulated and proved in the martingale problem language, which, by the equivalences described above, yields corresponding statements about weak solutions of SDEs whenever a stochastic integral representation is available.

### 4.2.1 Stroock-Varadhan uniqueness condition

The Stroock-Varadhan uniqueness condition provides one of the most profound and rigorous analytical characterizations of when a martingale problem admits a unique solution in law. It forms the cornerstone of the connection between stochastic differential equations (SDEs) and partial differential operators, and gives precise sufficient conditions ensuring that the stochastic dynamics governed by a generator $\mathcal{L}$ are uniquely determined by their infinitesimal characteristics.

Consider a second-order differential operator of the form

$$ (\mathcal{L}f)(x) = \sum_{i=1}^{d} b_i(x) \frac{\partial f}{\partial x_i}(x) + \frac{1}{2} \sum_{i,j=1}^{d} a_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j}(x), \quad (4.21) $$

where $b: \mathbb{R}^d \to \mathbb{R}^d$ is a drift vector field and $a: \mathbb{R}^d \to \mathbb{R}^{d \times d}$ is a symmetric, nonnegative-definite diffusion matrix, typically expressed as $a(x) = \sigma(x)\sigma(x)^\top$ for some measurable function $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times m}$.

Let $C([0, \infty); \mathbb{R}^d)$ denote the canonical path space of continuous functions from $[0, \infty)$ to $\mathbb{R}^d$,
equipped with the canonical filtration ($\mathcal{F}_t^X$) generated by the coordinate process $X_t(\omega) = \omega(t)$. A probability measure $\mathbb{P}$ on this path space is said to solve the martingale problem for ($\mathcal{L}, \delta_x$) if $\mathbb{P}(X_0 = x) = 1$ and for all test functions $f \in C_b^2(\mathbb{R}^d)$, the process

$$M_t^f := f(X_t) - f(X_0) - \int_0^t (\mathcal{L}f)(X_s) ds \qquad (4.22)$$

is a $\mathbb{P}$-martingale with respect to ($\mathcal{F}_t^X$).

The Stroock-Varadhan framework addresses the uniqueness in law of this problem, i.e., the question of whether there exists at most one probability measure $\mathbb{P}$ satisfying the above property for a given initial distribution and generator. Their celebrated uniqueness theorem gives conditions on $b$ and $a$ that ensure this uniqueness.

To state the condition rigorously, define the following hypotheses. Let $a(x)$ and $b(x)$ be measurable functions satisfying:

1. **Local boundedness:** For every compact set $K \subset \mathbb{R}^d$, there exists a constant $C_K > 0$ such that

$$\sup_{x \in K} (|b(x)| + |a(x)|) \le C_K. \qquad (4.23)$$

2. **Local ellipticity** For every compact set $K \subset \mathbb{R}^d$, there exists a constant $\lambda_K > 0$ such that

$$\xi^\top a(x), \xi \ge \lambda_K |\xi|^2 \quad \text{for all } x \in K, \xi \in \mathbb{R}^d. \qquad (4.24)$$

That is, $a(x)$ is uniformly positive definite on compact sets.

3. **Continuity condition** The coefficients $a(x)$ and $b(x)$ are locally continuous functions, i.e., for each compact set $K$,

$$\lim_{y \to x, x, y \in K} (|b(x) - b(y)| + |a(x) - a(y)|) = 0. \qquad (4.25)$$

Under these hypotheses, the Stroock-Varadhan uniqueness theorem asserts that the martingale problem for ($\mathcal{L}, \delta_x$) admits a unique solution in law for each initial condition $x \in \mathbb{R}^d$.

Formally, the theorem can be stated as follows:

**Theorem 4.1 (Stroock-Varadhan Uniqueness Theorem).** Suppose that the coefficients $b: \mathbb{R}^d \to \mathbb{R}^d$ and $a: \mathbb{R}^d \to \mathbb{R}^{d \times d}$ are bounded, measurable, locally uniformly continuous, and that $a(x)$ is uniformly nondegenerate on compact subsets of $\mathbb{R}^d$. Then, for each initial point $x \in \mathbb{R}^d$, there exists a unique probability measure $\mathbb{P}_x$ on $C([0, \infty); \mathbb{R}^d)$ solving the martingale problem for ($\mathcal{L}, \delta_x$).

*Proof.* Let $b: \mathbb{R}^d \to \mathbb{R}^d$ and $a: \mathbb{R}^d \to \mathbb{R}^{d \times d}$ satisfy the assumptions: they are bounded, measurable, locally uniformly continuous, and $a(x)$ is uniformly nondegenerate on compact subsets, i.e., for every compact $K \subset \mathbb{R}^d$, there exists $\lambda_K > 0$ such that

$$\xi^\top a(x) \xi \ge \lambda_K |\xi|^2, \quad \forall x \in K, \forall \xi \in \mathbb{R}^d. \qquad (4.26)$$
Define the generator

$$ \mathcal{L}f(x) = \sum_{i=1}^{d} b_i(x) \frac{\partial f}{\partial x_i}(x) + \frac{1}{2} \sum_{i,j=1}^{d} a_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j}(x), \quad f \in C_c^{\infty}(\mathbb{R}^d), \quad (4.27) $$

and recall that a probability measure $P_x$ on $C([0, \infty); \mathbb{R}^d)$ solves the martingale problem for $(\mathcal{L}, \delta_x)$ if $X_0 = x P_x$-a.s. and

$$ M_t^f := f(X_t) - f(X_0) - \int_0^t \mathcal{L}f(X_s), ds \quad (4.28) $$

is a martingale with respect to the canonical filtration $(\mathcal{F}_t)_{t \ge 0}$ for all $f \in C_c^{\infty}(\mathbb{R}^d)$.

To construct a solution, consider a standard mollifier $\rho_\epsilon \in C_c^\infty(\mathbb{R}^d)$, $\rho_\epsilon \ge 0$, $\int \rho_\epsilon = 1$, and define

$$ b_{\epsilon}(x) := (b * \rho_{\epsilon})(x) = \int_{\mathbb{R}^d} b(y) \rho_{\epsilon}(x-y), dy, \quad a_{\epsilon}(x) := (a * \rho_{\epsilon})(x) = \int_{\mathbb{R}^d} a(y) \rho_{\epsilon}(x-y), dy. \quad (4.29) $$

Then $b_\epsilon, a_\epsilon \in C^\infty(\mathbb{R}^d)$ are bounded and converge locally uniformly to $b$ and $a$ as $\epsilon \downarrow 0$. Moreover, since convolution preserves uniform ellipticity on compacts, $a_\epsilon(x)$ remains uniformly nondegenerate on compacts.

For each $\epsilon > 0$, consider the stochastic differential equation

$$ dX_t^\epsilon = b_\epsilon(X_t^\epsilon), dt + \sigma_\epsilon(X_t^\epsilon) dW_t, \quad X_0^\epsilon = x \quad (4.30) $$

where $\sigma_\epsilon(x)$ satisfies $\sigma_\epsilon(x)\sigma_\epsilon(x)^\top = a_\epsilon(x)$. By classical SDE theory, since $b_\epsilon$ and $\sigma_\epsilon$ are smooth and globally bounded, there exists a unique strong solution $X_t^\epsilon$. Let $\mathbb{P}_x^\epsilon$ denote the law of $X^\epsilon$ on $C([0, \infty); \mathbb{R}^d)$. Next, establish tightness of $(\mathbb{P}_x^\epsilon)_{\epsilon>0}$. For any $T > 0$ and $\delta > 0$, the Burkholder-Davis-Gundy inequality and boundedness of $b_\epsilon, \sigma_\epsilon$ imply

$$ \mathbb{E}[|X_{t+s}^\epsilon - X_t^\epsilon|^4] \le C s^2, \quad 0 \le t \le t+s \le T, \quad (4.31) $$

for a constant $C > 0$ independent of $\epsilon$. Kolmogorov's continuity criterion then guarantees uniform control over modulus of continuity, so $(\mathbb{P}_x^\epsilon)$ is tight in $C([0, T]; \mathbb{R}^d)$, and by Prokhorov's theorem, there exists a subsequence $\epsilon_n \downarrow 0$ and a limit law $P_x$ such that

$$ \mathbb{P}_x^{\epsilon_n} \xrightarrow{\text{weakly}} \mathbb{P}_x \quad \text{on } C([0, \infty); \mathbb{R}^d). \quad (4.32) $$

To show $P_x$ solves the martingale problem, fix $f \in C_c^\infty(\mathbb{R}^d)$ and consider

$$ M_t^{f, \epsilon_n} = f(X_t^{\epsilon_n}) - f(X_0^{\epsilon_n}) - \int_0^t \mathcal{L}_{\epsilon_n} f(X_s^{\epsilon_n}) ds, \quad (4.33) $$

where

$$ \mathcal{L}_{\epsilon_n} f = b_{\epsilon_n} \cdot \nabla f + \frac{1}{2} \mathrm{tr}(a_{\epsilon_n} \nabla^2 f) \quad (4.34) $$

By construction, $M_t^{f, \epsilon_n}$ is a martingale under $\mathbb{P}_x^{\epsilon_n}$. Since $b_{\epsilon_n} \to b$ and $a_{\epsilon_n} \to a$ locally uniformly,
and $X^{\epsilon_n} \to X$ in distribution, we have

$$ \int_0^t \mathcal{L}_{\epsilon_n} f(X_s^{\epsilon_n}), ds \to \int_0^t \mathcal{L} f(X_s), ds \quad \text{in distribution.} \quad (4.35) $$

By martingale convergence theorems (specifically, convergence in law plus uniform integrability), the limit

$$ M_t^f := f(X_t) - f(X_0) - \int_0^t \mathcal{L} f(X_s), ds \quad (4.36) $$

is a martingale under $\mathbb{P}_x$. This proves **existence** of a solution.

For **uniqueness**, suppose $\mathbb{P}_x$ and $\tilde{\mathbb{P}}_x$ are two solutions to the martingale problem. Define

$$ P_t f(x) := \mathbb{E}_{\mathbb{P}_x}[f(X_t)], \quad \tilde{P}_t f(x) := \mathbb{E} * \tilde{\mathbb{P}}_x[f(X_t)], \quad f \in C_c^\infty(\mathbb{R}^d). \quad (4.37) $$

Then $u(t,x) := P_t f(x)$ and $\tilde{u}(t,x) := \tilde{P}_t f(x)$ satisfy the Kolmogorov backward equation

$$ \frac{\partial u}{\partial t}(t, x) = \mathcal{L}u(t, x), \quad u(0, x) = f(x), \quad (4.38) $$

and similarly for $\tilde{u}$. By Aronson-type Gaussian bounds, the fundamental solution of $\partial_t - \mathcal{L}$ is unique in the class of bounded continuous functions; hence

$$ P_t f(x) = \tilde{P}_t f(x), \quad \forall t \ge 0, \quad f \in C_c^\infty(\mathbb{R}^d). \quad (4.39) $$

Finally, since $C_c^\infty(\mathbb{R}^d)$ is dense in $C_0(\mathbb{R}^d)$, equality of all finite-dimensional distributions follows: for any $0 \le t_1 < \dots < t_n$ and bounded continuous $\phi: \mathbb{R}^{dn} \to \mathbb{R}$,

$$ \mathbb{E}_{\mathbb{P}_x}[\phi(X_{t_1}, \dots, X_{t_n})] = \mathbb{E}_{\tilde{\mathbb{P}}_x}[\phi(X_{t_1}, \dots, X_{t_n})] \quad (4.40) $$

By Kolmogorov extension theorem, $\mathbb{P}_x = \tilde{\mathbb{P}}_x$ as measures on $C([0, \infty); \mathbb{R}^d)$. Therefore, the martingale problem for $(\mathcal{L}, \delta_x)$ admits a unique solution. $\square$

In summary the proof of this result is profoundly analytical and rests upon the correspondence between the martingale problem and the associated Kolmogorov backward equation:

$$ \frac{\partial u}{\partial t}(t, x) = (\mathcal{L}u)(t, x), \quad u(0, x) = f(x), \quad (4.41) $$

where $u(t,x) = \mathbb{E}_x[f(X_t)]$ is the transition semigroup associated with the diffusion. The uniqueness of the martingale problem then follows from the uniqueness of classical or weak solutions to this parabolic partial differential equation under the prescribed regularity and ellipticity conditions.

In the degenerate or non-elliptic case, additional structural hypotheses are often imposed to ensure uniqueness, such as Hörmander's condition for hypoelliptic diffusions, which requires the Lie algebra generated by the vector fields associated with $b$ and $\sigma$ to span the tangent space at every point.

Hence, the Stroock-Varadhan uniqueness condition provides a purely analytic criterion—expressed
in terms of the coefficients (a) and (b)—for the well-posedness of the martingale problem, ensuring that the stochastic dynamics governed by $\mathcal{L}$ are uniquely characterized in distribution by their infinitesimal generator. This establishes a deep equivalence between the uniqueness of the martingale problem, the well-posedness of the SDE, and the uniqueness of solutions to the associated PDE, thereby unifying probabilistic and analytic formulations of diffusion processes.

## 4.3 Connection between Generators and SDEs

If $X = (X_t)_{t \ge 0}$ is a continuous adapted process defined on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$ and it satisfies the Itô stochastic differential equation

$$dX_t = b(X_t) dt + \sigma(X_t), dB_t \qquad X_0 = x, \qquad (4.42)$$

where $B$ is an $m$-dimensional $(\mathcal{F}_t)$-Brownian motion, $b: \mathbb{R}^d \to \mathbb{R}^d$ and $\sigma: \mathbb{R}^d \to \mathbb{R}^{d \times m}$ are measurable and such that the usual integrability conditions hold, then for every $f \in C_b^2(\mathbb{R}^d)$ Itô's formula applies pathwise and yields the identity

$$f(X_t) - f(X_0) = \int_0^t \nabla f(X_s) \cdot b(X_s) ds + \int_0^t \nabla f(X_s) \sigma(X_s) dB_s + \frac{1}{2} \int_0^t \mathrm{tr}(\sigma \sigma^\top(X_s) D^2 f(X_s)) ds. \qquad (4.43)$$

Collecting the finite-variation terms defines the differential operator $\mathcal{L}$ acting on $C_b^2$ by

$$(\mathcal{L}f)(x) := b(x) \cdot \nabla f(x) + \frac{1}{2} \mathrm{tr}(\sigma \sigma^\top(x) D^2 f(x)), \qquad (4.44)$$

so that the previous display is equivalently written as the decomposition

$$f(X_t) - f(X_0) - \int_0^t (\mathcal{L}f)(X_s) ds = \int_0^t \nabla f(X_s) \sigma(X_s) dB_s. \qquad (4.45)$$

The right-hand side is an Itô stochastic integral with respect to Brownian motion and hence is a (local and, under integrability, true) martingale with respect to $(\mathcal{F}_t)$. Consequently, defining

$$M_t^f := f(X_t) - f(X_0) - \int_0^t (\mathcal{L}f)(X_s) ds, \qquad (4.46)$$

we obtain that $M^f$ is a local martingale for every $f \in C_b^2(\mathbb{R}^d)$, and under the standard moment bounds it is a true square-integrable martingale. This verifies that any strong (or weak) solution of the SDE furnishes a solution of the martingale problem associated with the generator $\mathcal{L}$.

Conversely, suppose $\mathbb{P}$ is a probability measure on canonical path space $\Omega = C([0, T]; \mathbb{R}^d)$ under which the canonical process $X_t(\omega) = \omega(t)$ satisfies the martingale property

$$M_t^f := f(X_t) - f(X_0) - \int_0^t (\mathcal{L}f)(X_s) ds \qquad (4.47)$$

is a $\mathbb{P}$-martingale for every $f \in C_b^2(\mathbb{R}^d)$, with $\mathcal{L}$ as above and with initial law concentrated at $x$. Under additional structural and nondegeneracy assumptions on the coefficients (for example local boundedness of $b$ and $\sigma$, and the identification $a(x) = \sigma(x)\sigma(x)^\top$ as a predictable matrix-valued function), one can recover a semimartingale decomposition of the canonical process.
Precisely, by applying the Doob–Meyer decomposition and localization arguments one finds predictable finite-variation processes (A) and a continuous local martingale M such that

$$X_t = X_0 + A_t + M_t, \qquad (4.48)$$

and for every $f \in C_b^2$ the identity

$$f(X_t) - f(X_0) - \int_0^t \nabla f(X_s), dA_s - \frac{1}{2} \sum_{i,j} \int_0^t \partial_{ij} f(X_s), d\langle M^{(i)}, M^{(j)} \rangle_s \quad (4.49)$$

is a local martingale. Matching this representation with the martingale problem relation yields the identification

$$dA_s = b(X_s), ds, \quad d\langle M^{(i)}, M^{(j)} \rangle_s = a_{ij}(X_s) ds, \qquad (4.50)$$

in the sense of predictable processes, where $a(x) = \sigma(x)\sigma(x)^\top$. Once the quadratic covariance structure of the continuous local martingale part M is identified as above, the martingale representation (or Kunita–Watanabe orthogonal decomposition) guarantees—possibly on an enlarged filtered probability space carrying an m-dimensional Brownian motion W—the existence of a predictable matrix-valued process $H_s$ with

$$M_t = \int_0^t H_s, dW_s, \qquad (4.51)$$

and the condition $H_s H_s^\top = a(X_s)$ (a.s.) yields $H_s = \sigma(X_s)$ up to an orthogonal rotation in the driving Brownian components. Thus one constructs a weak solution $(\tilde{\Omega}, \tilde{\mathcal{F}}, (\tilde{\mathcal{F}}_t), \tilde{\mathbb{P}}, \tilde{B}, \tilde{X})$ of the SDE

$$d\tilde{X}_t = b(\tilde{X}_t) dt + \sigma(\tilde{X}_t) d\tilde{B}_t, \qquad (4.52)$$

whose law on path space coincides with $\mathbb{P}$. The construction requires verification that the predictable quadratic variation of M is absolutely continuous with respect to Lebesgue measure and equals $\int_0^t a(X_s), ds$; this is assured under standard hypotheses such as local boundedness and measurability of a, or uniform ellipticity which simplifies inversion. Under these hypotheses the martingale problem solution yields a weak SDE solution and hence the two formulations are equivalent.

Putting the two directions together, we obtain the precise correspondence: every (weak or strong) solution of the SDE induces a solution of the martingale problem for the generator $\mathcal{L}$; conversely, every solution $\mathbb{P}$ of the martingale problem that satisfies the quadratic-variation identification and the representability of its martingale part as an Itô integral can be realized as the law of a weak solution of the SDE with coefficients $b, \sigma$. In particular, uniqueness of the martingale problem for $(\mathcal{L}, \delta_x)$ implies uniqueness in law for the SDE started at $x$, while pathwise (strong) uniqueness together with existence of a weak solution implies existence and uniqueness of a strong solution (Yamada–Watanabe).

## 4.4 Weak Convergence and Stability

The martingale problem furnishes a robust and convenient framework for proving weak convergence of families of Markov processes by reducing convergence of path-space laws to convergence
of finite-dimensional generators and tightness estimates. We give a precise, rigorous formula-
tion and proof sketch of the standard scheme; all statements below are phrased in the canonical
path space $\Omega = C([0, T]; \mathbb{R}^d)$ (or $D([0, T]; \mathbb{R}^d)$ when discontinuities occur) equipped with its
Skorokhod topology and canonical filtration ($\mathcal{F}_t^X$).

Let $(\mathcal{L}^n)_{n \ge 1}$ be a sequence of linear operators acting on a common core $\mathcal{C} \subset C_b^2(\mathbb{R}^d)$ (i.e. $\mathcal{C}$ is dense in the relevant function space and each $\mathcal{L}^n$ maps $\mathcal{C}$ into $C_b(\mathbb{R}^d)$). Suppose for each $n$ there exists a unique probability measure $\mathbb{P}^n$ on $(\Omega, \mathcal{F}_T^X)$ solving the martingale problem for $(\mathcal{L}^n, \mu^n)$ with initial law $\mu^n$. That is, for every $f \in \mathcal{C}$ the process

$$
M_t^{n,f} := f(X_t) - f(X_0) - \int_0^t (\mathcal{L}^n f)(X_s) ds \quad (4.53)
$$

is a $\mathbb{P}^n$-martingale, and $\mathbb{P}^n \circ X_0^{-1} = \mu^n$. Assume the initial laws $\mu^n$ converge weakly to $\mu$ on $\mathbb{R}^d$.
We describe sufficient conditions under which $\mathbb{P}^n$ converges weakly (on $\Omega$) to the unique solution
$\mathbb{P}$ of the martingale problem for $(\mathcal{L}, \mu)$, where $\mathcal{L}$ is a limiting operator defined on the same core $\mathcal{C}$.

A first necessary analytic hypothesis is convergence of generators on the core: for every $f \in \mathcal{C}$ and every compact $K \subset \mathbb{R}^d$,

$$
\sup_{x \in K} |(\mathcal{L}^n f)(x) - (\mathcal{L} f)(x)| \xrightarrow{n \to \infty} 0 \quad (4.54)
$$

Equivalently, $\mathcal{L}^n f \to \mathcal{L} f$ uniformly on compacts for each $f \in \mathcal{C}$. This property is the natural substitute for pointwise convergence of infinitesimal generators and ensures that the martingale relations of the approximating laws become, in the limit, the martingale relations for $\mathcal{L}$.

A second, probabilistic ingredient is tightness of the sequence ($\mathbb{P}^n$) in $\mathcal{P}(\Omega)$. Concretely, one must verify conditions (for example Aldous’ criterion or moment–modulus bounds) that guarantee precompactness. A common sufficient condition is the existence of constants $p > 1$ and $C_T > 0$ such that for every $n$ and every $0 \le s < t \le T$,

$$
\mathbb{E}^{\mathbb{P}^n} [|X_t - X_s|^p] \le C_T |t - s|^{1+\delta} \quad (4.55)
$$

for some $\delta > 0$; Kolmogorov–Chentsov then yields tightness in $C([0, T]; \mathbb{R}^d)$. Alternatively,
in the càdlàg setting one uses bounds on oscillation with respect to stopping times (Aldous)
together with control of large jumps.

Assume these two conditions hold. By Prokhorov's theorem the family ($\mathbb{P}^n$) is relatively com-
pact: every subsequence admits a further subsequence converging weakly to some probability
measure $\mathbb{P}^\infty$ on $\Omega$. It remains to identify any such limit point as a solution of the martingale
problem for ($\mathcal{L}, \mu$). Fix $f \in \mathcal{C}$ and let $\phi : \Omega \to \mathbb{R}$ be any bounded, continuous, and $\mathcal{F}_s^X$-
measurable test functional (for some $0 \le s \le t \le T$); examples include bounded continuous
functions of the coordinates ($X_{r_1}, \dots, X_{r_k}$) with $r_i \le s$. For each $n$ the martingale property
gives

$$
\mathbb{E}^{\mathbb{P}^n} \left[ (f(X_t) - f(X_s)) - \int_s^t \mathcal{L}^n f(X_u) du \right], \phi] = 0. \quad (4.56)
$$
By weak convergence $\mathbb{P}^n \Rightarrow \mathbb{P}^\infty$ along the chosen subsequence and dominated convergence (justified by uniform moment bounds and continuity of $\phi$), one may pass to the limit in the first term:

$$ \mathbb{E}_{\mathbb{P}^n} [(f(X_t) - f(X_s))\phi] \xrightarrow{n \to \infty} \mathbb{E}_{\mathbb{P}^\infty} [(f(X_t) - f(X_s))\phi]. \quad (4.57) $$

For the integral term use the generator convergence on compacts together with tightness to deduce that

$$ \mathbb{E}_{\mathbb{P}^n} \left[ \left( \int_s^t \mathcal{L}^n f(X_u) du \right) \phi \right] = \mathbb{E}_{\mathbb{P}^n} \left[ \int_s^t (\mathcal{L}f)(X_u) du \cdot \phi \right] + R_n(\phi), \quad (4.58) $$

where the remainder satisfies $R_n(\phi) \to 0$ as $n \to \infty$. Convergence of the expectations of the main term again follows from weak convergence plus uniform integrability (ensured by moment bounds). Thus the limit measure $\mathbb{P}^\infty$ satisfies

$$ \mathbb{E}_{\mathbb{P}^\infty} \left[ (f(X_t) - f(X_s)) - \int_s^t \mathcal{L}f(X_u), du \right], \phi] = 0 \quad (4.59) $$

for every bounded continuous $\mathcal{F}_s^X$-measurable $\phi$. By the separating properties of such test functions this identity is equivalent to the assertion that the process

$$ M_{t}^{\infty,f} := f(X_t) - f(X_0) - \int_0^t \mathcal{L}f(X_u), du \quad (4.60) $$

is a $\mathbb{P}^\infty$-martingale for every $f \in \mathcal{C}$; moreover $\mathbb{P}^\infty \circ X_0^{-1} = \mu$. Hence any weak limit point $\mathbb{P}^\infty$ solves the martingale problem for $(\mathcal{L}, \mu)$.

When the martingale problem for $(\mathcal{L}, \mu)$ is well posed, i.e. admits a unique solution $\mathbb{P}$, it follows that every subsequential limit equals $\mathbb{P}$; relative compactness therefore implies the full sequence $\mathbb{P}^n \Rightarrow \mathbb{P}$ converges weakly to $\mathbb{P}$. In summary, the two-step program (generator convergence on a core $\mathcal{C}$ plus tightness/uniform integrability) yields identification of limit points as martingale-problem solutions and, via uniqueness, convergence of the sequence of laws.

This abstract scheme is the backbone of many concrete applications. In diffusion approximation one takes $\mathcal{L}^n$ to be generators of rescaled jump processes (or Markov chains) and verifies $\mathcal{L}^n f \to \mathcal{L}f$ on a core together with tightness via moment and jump control; the limiting generator $\mathcal{L}$ is typically a second-order differential operator and uniqueness of its martingale problem is obtained via Stroock-Varadhan criteria (e.g. uniform ellipticity and Hölder continuity). In numerical analysis of SDEs, one shows that the discrete-time Markov semigroups of the scheme have generators approximating that of the SDE and uses the martingale problem method to prove convergence of the interpolated schemes. In stochastic homogenization one considers $\mathcal{L}^\epsilon$ depending on a small parameter and shows convergence to an effective operator $\mathcal{L}^0$ on the core via two-scale expansion; tightness and identification of limits then yield weak convergence of the processes.

Two technical remarks are in order. First, one often needs uniform moment bounds to justify interchanging limits and expectations and to ensure uniform integrability of the martingale
increments; these are verified by Lyapunov function techniques producing estimates of the form

$$ \sup_n \mathbb{E}^{\mathbb{P}^n} \left[ \sup_{t \le T} |X_t|^p \right] < \infty \qquad (4.61) $$

Second, when working in the càdlàg Skorokhod space $D([0, T]; \mathbb{R}^d)$ the core $\mathcal{C}$ and generator convergence hypotheses must be tailored to account for jump contributions (one typically requires convergence of jump kernels or of Lévy characteristics), and tightness is checked via Aldous' condition plus control of large jumps. Under these refinements the martingale problem method remains fully effective.

## 4.5 Conclusion

Stochastic differential equations and martingale problems together form the analytic and probabilistic backbone of modern stochastic process theory. The generator $\mathcal{L}$ acts as the infinitesimal descriptor of a Markov process, while the martingale problem provides an abstract weak formulation independent of the specific noise realization. Their interplay underlies deep results connecting probability theory, functional analysis, and partial differential equations.

**References for this chapter:** Karatzas & Shreve (2014) [1], Øksendal (2013) [2], Stroock and Varadhan (2007) [16], Rogers and Williams (2000) [17], Friedman (1975) [18].
# Statement and Proof of the Classical Feynman–Kac Formula

“Probabilistic representation of PDEs is not an accident; it is the shadow of a deeper semigroup and martingale structure.”

— Adapted from E. B. Dynkin

## 5.1 Introduction and hypotheses

Let $E \subseteq \mathbb{R}^d$ be open (often $E = \mathbb{R}^d$), and let $\mathcal{L}$ be a second-order differential operator of the form

$$ \mathcal{L}f(x) = \sum_{i=1}^{d} b_i(x) \partial_{x_i} f(x) + \frac{1}{2} \sum_{i,j=1}^{d} a_{ij}(x) \partial_{x_i} \partial_{x_j} f(x), \quad (5.1) $$

with measurable coefficients $b: E \to \mathbb{R}^d$ and $a: E \to \mathbb{R}^{d \times d}$ symmetric and nonnegative definite. We assume the standard hypotheses that guarantee existence of a Markov process $X = (X_t)_{t \ge 0}$ with generator $\mathcal{L}$ (or of a Feller semigroup $(P_t)_{t \ge 0}$ with generator $\mathcal{L}$ on a core of smooth compactly supported functions): local boundedness of coefficients, local ellipticity on compacts or other conditions ensuring well-posedness of the martingale problem for $\mathcal{L}$. Fix a time horizon $T > 0$. Let $V: E \to \mathbb{R}$ be a measurable potential; the classical (bounded) case assumes $V \in C_b(E)$ (or at least $V$ bounded measurable), and the source/terminal data $f: E \to \mathbb{R}$ is assumed bounded and continuous, $f \in C_b(E)$. When $V$ is unbounded one must impose Kato-type or integrability conditions; we treat the bounded case first for clarity.

We denote by $(\mathbb{P}_x)_{x \in E}$ the law of the Markov process started at $x$ and write $\mathbb{E}_x[\cdot]$ for expectation under $\mathbb{P}_x$. For $0 \le t \le T$ define the Feynman–Kac functional

$$ u(t, x) := \mathbb{E}_x \left[ \exp \left( - \int_0^t V(X_s) ds \right) f(X_t) \right]. \quad (5.2) $$
Our goal is to prove that $u$ is the unique (classical) solution of the initial-value problem

$$ \begin{cases} \partial_t u(t, x) = \mathcal{L}u(t, x) - V(x)u(t, x), & (t, x) \in (0, T] \times E, \\ u(0, x) = f(x), & x \in E, \end{cases} \qquad (\text{FK}) $$

under the stated regularity and integrability hypotheses.

## 5.2 Statement of the classical theorem

**Theorem 5.1 (Classical Feynman-Kac).** Assume $\mathcal{L}$ generates a Feller Markov process $X$ on $E$, $V \in C_b(E)$, and $f \in C_b(E)$. Then the function

$$ u(t, x) := \mathbb{E}_x \left[ e^{-\int_0^t V(X_s) ds} f(X_t) \right], \quad (t, x) \in [0, T] \times E, \qquad (5.3) $$

is bounded and continuous on $[0, T] \times E$, belongs to $C^{1,2}((0, T] \times E)$ whenever the coefficients of $\mathcal{L}$ are sufficiently smooth, and solves the backward parabolic problem (FK). Moreover, the solution is unique in the class of bounded continuous functions satisfying the equation in the classical (or appropriate weak) sense.

## 5.3 Proof of existence: martingale argument and verification

Fix $f, V$ as above and define $u$ by the displayed expectation. Boundedness and continuity of $u$ follow by dominated convergence: since $V$ and $f$ are bounded there is a constant $C$ with

$$ |e^{-\int_{0}^{t} V(X_s) ds} f(X_t)| \leq C \qquad (5.4) $$

and $(t, x) \mapsto u(t, x)$ is continuous by the Feller property of the transition semigroup and dominated convergence. To derive the PDE, fix $x \in E$ and work on the probability space carrying $X$ under $\mathbb{P}_x$. For each $s \in [0, t]$ define the multiplicative functional

$$ Y_s := \exp \left( - \int_0^s V(X_r) dr \right). \qquad (5.5) $$

By the Markov property and the tower property,

$$ u(t, x) = \mathbb{E}_x [Y_t f(X_t)] = \mathbb{E}_x [Y_s \mathbb{E}_{X_s} [e^{-\int_0^{t-s} V(X_r) dr} f(X_{t-s})]] = \mathbb{E}_x [Y_s u(t-s, X_s)]. \qquad (5.6) $$

Equivalently, for fixed $t$,

$$ u(t, x) = \mathbb{E}_x \left[ e^{-\int_0^{t \wedge \tau} V(X_r) dr} u(t - (t \wedge \tau), X_{t \wedge \tau}) \right] \qquad (5.7) $$

for any stopping time $\tau$ bounded by $t$; this representation will be used to localize the argument.

We now compute time-derivatives via a martingale (Dynkin) identity. For $f \in C_b^2$ define the process

$$ M_s := Y_s u(t-s, X_s), \quad 0 \le s \le t. \qquad (5.8) $$