We consider the problem of how to optimize multi-stage campaigning over social networks. The dynamic programming framework is employed to balance the high present reward and large penalty on low future outcome in the presence of extensive uncertainties. In particular, we establish theoretical foundations of optimal campaigning over social networks where the user activities are modeled as a multivariate Hawkes process, and we derive a time dependent linear relation between the intensity of exogenous events and several commonly used objective functions of campaigning. We further develop a convex dynamic programming framework for determining the optimal intervention policy that prescribes the required level of external drive at each stage for the desired campaigning result. Experiments on both synthetic data and the real-world MemeTracker dataset show that our algorithm can steer the user activities for optimal campaigning much more accurately than baselines.
Existing work in stochastic optimal control includes jump diffusion stochastic differential equations (SDE) @cite_34 @cite_18 which focuses on controlling the SDEs with the jump term driven by Poisson processes not for Hawkes processes. Inspired by the opinion dynamics model proposed in @cite_1 , the authors in @cite_32 proposes a multivariate jump diffusion process framework for modeling opinion dynamics over networks and determining the control over such networks.
@cite_18 @cite_34 @cite_1 @cite_32
2221658266 2206697650 1683668173
We present a novel approach to inference in conditionally Gaussian continuous time stochastic processes, where the latent process is a Markovian jump process. We first consider the case of jump-diffusion processes, where the drift of a linear stochastic differential equation can jump at arbitrary time points. We derive partial differential equations for exact inference and present a very efficient mean field approximation. By introducing a novel lower bound on the free energy, we then generalise our approach to Gaussian processes with arbitrary covariance, such as the non-Markovian RBF covariance. We present results on both simulated and real data, showing that the approach is very accurate in capturing latent dynamics and can be useful in a number of real data modelling tasks. This paper integrates recent work on Path Integral (PI) and Kullback Leibler (KL) divergence stochastic optimal control theory with earlier work on risk sensitivity and the fundamental dualities between free energy and relative entropy. We derive the path integral optimal control framework and its iterative version based on the aforemetioned dualities. The resulting formulation of iterative path integral control is valid for general feedback policies and in contrast to previous work, it does not rely on pre-specified policy parameterizations. The derivation is based on successive applications of Girsanov's theorem and the use of Radon-Nikodým derivative as applied to diffusion processes due to the change of measure in the stochastic dynamics. We compare the PI control derived based on Dynamic Programming with PI based on the duality between free energy and relative entropy. Moreover we extend our analysis on the applicability of the relationship between free energy and relative entropy to optimal control of markov jump diffusions processes. Furthermore, we present the links between KL stochastic optimal control and the aforementioned dualities and discuss its generalizability. WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematical tools required for applications can be intimidating. Potential users often get the impression that jump and Levy processes are beyond their reach.Financial Modelling with Jump Processes shows that this is not so. It provides a self-contained overview of the theoretical, numerical, and empirical aspects involved in using jump processes in financial modelling, and it does so in terms within the grasp of nonspecialists. The introduction of new mathematical tools is motivated by their use in the modelling process, and precise mathematical statements of results are accompanied by intuitive explanations.Topics covered in this book include: jump-diffusion models, Levy processes, stochastic calculus for jump processes, pricing and hedging in incomplete markets, implied volatility smiles, time-inhomogeneous jump processes and stochastic volatility models with jumps. The authors illustrate the mathematical concepts with many numerical and empirical examples and provide the details of numerical implementation of pricing and calibration algorithms.This book demonstrates that the concepts and tools necessary for understanding and implementing models with jumps can be more intuitive that those involved in the Black Scholes and diffusion models. If you have even a basic familiarity with quantitative methods in finance, Financial Modelling with Jump Processes will give you a valuable new set of tools for modelling market fluctuations. Introduction 1 Review of Continuous Time Models 1.1 Martingales and Martingale Inequalities 1.2 Stochastic Integration 1.3 Stochastic Differential Equations: Diffusions 1.4 Reflected Diffusions 1.5 Processes with Jumps 2 Controlled Markov Chains 2.1 Recursive Equations for the Cost 2.2 Optimal Stopping Problems 2.3 Discounted Cost 2.4 Control to a Target Set and Contraction Mappings 2.5 Finite Time Control Problems 3 Dynamic Programming Equations 3.1 Functionals of Uncontrolled Processes 3.2 The Optimal Stopping Problem 3.3 Control Until a Target Set Is Reached 3.4 A Discounted Problem with a Target Set and Reflection 3.5 Average Cost Per Unit Time 4 Markov Chain Approximation Method: Introduction 4.1 Markov Chain Approximation 4.2 Continuous Time Interpolation 4.3 A Markov Chain Interpolation 4.4 A Random Walk Approximation 4.5 A Deterministic Discounted Problem 4.6 Deterministic Relaxed Controls 5 Construction of the Approximating Markov Chains 5.1 One Dimensional Examples 5.2 Numerical Simplifications 5.3 The General Finite Difference Method 5.4 A Direct Construction 5.5 Variable Grids 5.6 Jump Diffusion Processes 5.7 Reflecting Boundaries 5.8 Dynamic Programming Equations 5.9 Controlled and State Dependent Variance 6 Computational Methods for Controlled Markov Chains 6.1 The Problem Formulation 6.2 Classical Iterative Methods 6.3 Error Bounds 6.4 Accelerated Jacobi and Gauss-Seidel Methods 6.5 Domain Decomposition 6.6 Coarse Grid-Fine Grid Solutions 6.7 A Multigrid Method 6.8 Linear Programming 7 The Ergodic Cost Problem: Formulation and Algorithms 7.1 Formulation of the Control Problem 7.2 A Jacobi Type Iteration 7.3 Approximation in Policy Space 7.4 Numerical Methods 7.5 The Control Problem 7.6 The Interpolated Process 7.7 Computations 7.8 Boundary Costs and Controls 8 Heavy Traffic and Singular Control 8.1 Motivating Examples &nb