# Comparison Theorem for Nonlinear Path-Dependent Partial Differential Equations

Authors: Falei Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We introduce a type of fully nonlinear path-dependent (parabolic) partial differential equation (PDE) in which the pathωt on an interval [0, t] becomes the basic variable in the place of classical variables t,x∈[0,T]×ℝd. Then we study the comparison theorem of fully nonlinear PPDE and give some of its applications.


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## Body

## 1. Introduction

Motivated by uncertainty problems, risk measures, and the superhedging in finance mathematics, Peng [1] systemically established G-expectation theory. In the G-expectation framework, the notion of G-Brownian motion and the corresponding stochastic calculus of Itô’s type were established. The key issue is that G-diffusion processes are connected to a large class of fully nonlinear PDEs in the Markov case studied by Peng [2] and Soner et al. [3].Recently, Dupire [4] and Cont and Fournié [5] introduced a new functional Itô formula which nontrivially generalized the classical one through new notion-path derivatives (see [6, 7] for more general and systematic research). It extends the Itô stochastic calculus to functionals of a given process. It provides an excellent tool for the study of path-dependence. In fact, they showed that a smooth path functional solves a linear path-dependent PDE if its composition with a Brownian motion generates a martingale, which provided a functional extension of the classical Feynman-Kac formula. Moreover, by virtue of a backward stochastic differential equation approach, we can obtain the uniqueness of the smooth solution to the semilinear path-dependent PDE (see also [8–10] and the references therein). However, these methods are mainly based on stochastic calculus.The objective of this paper is to study fully nonlinear PPDE. We refer to Krylov [11] and Wang [12] for the classical fully nonlinear PDE (see also [13–15]). Peng [16] introduced an approach of frozenness of the main course of the paths where the maximization takes place. This approach is based on techniques of PDE and can be directly applied to treat fully nonlinear path-dependent PDE. The advantage of this PDE approach is that one can treat the solution locally (path by path), whereas stochastic calculus is mainly a global approach. In this paper, we will use this method to obtain a comparison principle of fully nonlinear PPDE. In particular, some properties of the solution to fully nonlinear PPDE are also obtained. We claim that these ideas carry over to much more general frameworks, such as the case of viscosity solution to PPDE. Moreover, this method can have direct applications to stochastic analysis, for example, martingales under a fully nonlinear expectation, stochastic optimal control problems, stochastic games, nonlinear pricing and risk measuring, and backward stochastic differential equations. These more technical details are left to future work and will be presented in forthcoming papers.The paper is organized as follows. In Section2, we present some existing results in the theory of Dupire’s path derivatives that we will use in this paper. In Section 3, we obtain the comparison theorem of fully nonlinear PPDE and give some of its applications.


## 2. Preliminaries

In this section, we give an overview of the definitions concerning path derivatives. The following notations are mainly from Dupire [4] and Cont and Fournié [5].LetT>0 be fixed. For each t∈[0,T], we denote by Λt the set of right continuous ℝd-valued functions on [0,t]. For each ωT∈ΛT the value of ωT at time s∈[0,T] is denoted by ω(s). Thus ωT=ω(s)0≤s≤T is a right continuous process on [0,T] and its value at time s is ω(s). The path of ωT up to time t is denoted by ωt; that is, ωt=ω(s)0≤s≤t∈Λt. Denote Λ=⋃t∈[0,T]‍Λt. We sometimes specifically write
(1)ωt=ω(s)0≤s<t=(ω(s)0≤s<t,ω(t))
to indicate the terminal position ω(t) of ωt which often plays a special role in this framework. For each ωt∈Λ and x∈ℝd, denote by ω(s) the value of ωt at s∈[0,t] and ωtx:=(ω(s)0≤s<t,ω(t)+x) which is also an element of Λt.Now consider the functionu of path; that is, u:Λ↦ℝ. This function u=u(ωt)ωt∈Λ can be also regarded as a family of real valued functions:
(2)u(ωt)=u(t,ωt(s)0≤s≤t)=u(t,ω(s)0≤s<t,ω(t)),kkkkkkkkkkkkkkkkkkkkkkωt∈Λt,t∈[0,T].
Denote u(ωtx):=u(t,ω(s)0≤s<t,ω(t)+x), for ωt∈Λt, x∈ℝd.We introduce the distance onΛ. Let 〈·,·〉 and |·| denote the inner product and norm in ℝd. For each 0≤t≤t¯≤T and ωt,ω¯t¯∈Λ, denote
(3)∥ωt∥:=sup⁡s∈[0,t]|ωt(s)|,d∞(ωt,ω¯t¯):=sup⁡s∈[0,t∨t¯]|ω(s∧t)-ω¯(s∧t¯)|+|t-t¯|1/2.It is obvious thatΛt is a Banach space with respect to ∥·∥. Since Λ is not a linear space, d∞ is not a norm.Definition 1 (continuous).
A functionu:Λ↦ℝ is said to be Λ-continuous at ωt∈Λ, if for any ɛ>0 there exists δ>0 such that for each ω¯t¯∈Λ with d∞(ωt,ω¯t¯)<δ we have |u(ωt)-u(ω¯t¯)|<ɛ. u is said to be Λ-continuous if it is Λ-continuous at each ωt∈Λ.Remark 2.
In our framework we often regardu(ωtx) as a function of t, ωt, and x; that is, u(ωtx)=u(t,ω(s)0≤s<t,ω(t)+x). Thus, for a fixed ωt∈Λ, u(ωtx) is regarded as a function of (t,x)∈[0,T]×ℝd.Definition 3.
Givenu:Λ↦ℝ and ωt∈Λ, if there exists p∈ℝd, such that
(4)u(ωtx)=u(ωt)+〈p,x〉+o(|x|)asx⟶0,x∈ℝd,
then we say that u is (vertically) differentiable at ωt and denote Dxu(ωt)=p. u is said to be vertically differentiable in Λ if Dxu(ωt) exists for each ωt∈Λ. We can similarly define the Hessian Dxxu(ωt). It is an 𝕊(d)-valued function defined on Λ, where 𝕊(d) is the space of all d×d symmetric matrices.For eachωt∈Λ,ω¯t¯∈Λ, with t¯≥t, set
(5)ωt,t¯(r):=ω(r)1[0,t)(r)+ω(t)1[t,t¯](r),r∈[0,t¯],ωt⊗ω¯t¯(r):=ω(r)1[0,t)(r)+(ω¯(r)-ω¯(t)+ω(t))1[t,t¯](r),kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkr∈[0,t¯].
It is clear that ωt,t¯∈Λt¯ and ωt⊗ω¯t¯∈Λt¯.Definition 4.
For a givenωt∈Λ if
(6)u(ωt,s)=u(ωt)+a(s-t)+o(|s-t|)ass⟶t,s≥t,
then we say that u(ωt) is (horizontally) differentiable in t at ωt and denote Dtu(ωt)=a. u is said to be horizontally differentiable in Λ if Dtu(ωt) exists for each ωt∈Λ.Definition 5.
Defineℂj,k(Λ) as the set of functions u defined on Λ which are j times horizontally and k times vertically differentiable in Λ, such that all these derivatives are Λ-continuous.Example 6.
Ifu(ωt)=f(t,ω(t)) with f∈C1,1([0,T)×ℝ), then
(7)Dtu(ωt)=∂tf(t,ω(t)),Dxu(ωt)=∂xf(t,ω(t)),
which is the classic derivative. In general, these derivatives also satisfy the classic properties: linearity, product, and chain rule.


## 3. Comparison Theorem for Fully Nonlinear PPDE

Now we introduce the following fully nonlinear path-dependent PDE:(8)Dtu(ωt)+G(ωt,u(ωt),Dxu(ωt),Dxx2u(ωt))=0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkωt∈Λ,u(ωT)=Φ(ωT),ωT∈ΛT,
where u:Λ↦ℝ and G:Λ×ℝ×ℝd×𝕊(d)↦ℝ are continuous functions. Moreover, G satisfies the following elliptic conditions: for each ωt∈Λ, r∈ℝ, p∈ℝd, X,Y∈𝕊(d),
(9)G(ωt,r,p,X)≥G(ωt,r,p,Y),wheneverX≥Y
and, for each r1,r2∈ℝ, p1,p2∈ℝd, X1,X2∈𝕊(d), there exists some constant C such that
(10)|G(ωt,r1,p1,X1)-G(ωt,r2,p2,X2)|≤C(|r1-r2|+|p1-p2|+|X1-X2|).Definition 7.
A functionu∈ℂ1,2(Λ) is called a ℂ1,2-solution of the path-dependent PDE (8) if, for each ωt∈Λ, t∈[0,T), equality (8) is satisfied. u is called a subsolution (resp., supersolution) of (8) if the “=” in (8) is replaced by “≥” (resp., “≤”).Remark 8.
The solution of classical PDE is a special case whenu(ωt)=u¯(t,ω(t)), u¯∈C1,2([0,T)×ℝ). Indeed, for each ωt∈Λ and t∈[0,T),
(11)∂tu¯(t,ω(t))=Dtu(ωt),∂xu¯(t,ω(t))=Dxu(ωt),∂xx2u¯(t,ω(t))=Dxx2u(ωt),
and thus u¯(t,x) is a classical solution of PDE.The following result is the so-called comparison principle or comparison theorem of path-dependent PDE.Theorem 9.
Supposeu1 is a ℂ1,2-subsolution and u2 is a ℂ1,2-supersolution. Moreover, u1 is bounded from above and u2 is bounded from below. Then the maximum principle holds: if u1(ωT)≤u2(ωT) for all ωT∈ΛT, then u1(ωt)≤u2(ωt) for each ωt∈Λ.Remark 10.
In the case whenG(ωt,r,p,X)=(1/2)tr⁡[X]+f(ωt,r,p) for some Lipschitz function f on Λ×ℝ×ℝd, the above result is the comparison theorem of semilinear path-dependent PDE, which is given by [9].In order to prove Theorem9, we will make use of the following definitions.Definition 11.
u
:
Λ
↦
R is said to be in USC*(Λ), if it satisfies the following conditions: (i)
for each fixedωt^∈Λ, u¯(t,x):=u((ωt^x)t^,t-t^) is an upper semicontinuous function of (t,x)∈[t^,T]×ℝd,(ii)
for eachωt∈Λ with ti↑t, limsup⁡i→∞⁡u(ωti)≤sup⁡x⁡u(ωtx).
We also denoteLSC*(Λ):={-u∣u∈USC*(Λ)}. A function u∈USC*(Λ) (resp., u∈LSC*(Λ)) is called a 𝒞-upper (resp., 𝒞-lower) semicontinuous function. u∈𝒞(Λ):=USC*(Λ)∩LSC*(Λ) is called a 𝒞-continuous function.Definition 12.
Define𝒞1,2(Λ) as the set of functions u∈𝒞(Λ)∪ℂ(Λ), such that, for each ωt∈Λ, Dtu(ωt), Dxu(ωt), Dxxu(ωt) exist.Definition 13.
A functionu∈𝒞1,2(Λ) is called a 𝒞1,2-solution of the path-dependent PDE (8) if for each ωt∈Λ, t∈[0,T) equality (8) is satisfied. u is called a 𝒞1,2-subsolution (resp., 𝒞1,2-supersolution) of (8) if the “=” in (8) is replaced by “≥” (resp., “≤”).Theorem 14.
Supposeu1 is a 𝒞1,2-subsolution of PPDE (8) with G1 and u2 is a 𝒞1,2-supersolution of PPDE (8) with G2, where G1(ωt,r,p,X)≤G2(ωt,r,p,X) for each (ωt,r,p,X)∈Λ×ℝ×ℝd×𝕊(d). Moreover, u1 is bounded from above and u2 is bounded from below. Then the maximum principle holds: if u1(ωT)≤u2(ωT) for all ωT∈ΛT, then u1(ωt)≤u2(ωt) for each ωt∈Λ.It is obvious thatℂ1,2(Λ)⊂𝒞1,2(Λ). Then Theorem 9 is a direct consequence of Theorem 14.For eachα>0 and ωt∈Λ, set
(12)wα(ωt)=u1(ωt)-u2(ωt)-αω2(t).In order to prove Theorem14, we need the following lemma, which is essentially from Peng [16, Lemma 6].Lemma 15.
Ifwα(ωt)>0 for some ωt∈Λ, then there exists ω¯t¯∈Λ, satisfying t¯≥t, wα(ωt)≤wα(ω¯t¯), and ω¯t¯=ωt⊗ω¯t¯, such that
(13)wα(ω¯t¯)=sup⁡γt∈Λ,t≥t¯wα(ω¯t¯⊗γt).Proof.
For eachωt∈Λ and wα(ωt)>0, we can find x^∈ℝ such that
(14)wα(ωtx^)=sup⁡xwα(ωtx),
and for each ωt∈Λ with ti↑t we get
(15)limsup⁡i→∞wα(ωti)⩽sup⁡xwα(ωtx).
Then the proof is immediate in light of Lemma 6 of Peng [16].Lemma 16.
Letu∈𝒞1,2(Λ) and ω¯t¯∈Λ be given satisfying u(ω¯t¯)≥u(ω¯t¯⊗ωt) for all ωt∈Λ, t≥t¯. Then
(16)Dtu(ω¯t¯)≤0,Dxu(ω¯t¯)=0,Dxx2u(ω¯t¯)≤0.Proof.
Sinceu(ω¯t¯)≥u(ω¯t¯,t¯+δ) for each δ, we conclude
(17)Dtu(ω¯t¯)≤0.
For each x∈ℝd,
(18)u(ω¯t¯)=sup⁡xu(ω¯t¯x),
and thus
(19)Dxu(ω¯t¯)=0,Dxx2u(ω¯t¯)≤0,
which is the desired result.Now we are going to give the proof of Theorem14.Proof of Theorem14.
We first observe that, forδ>0, the functions defined by u¯1:=u1-δ/t are a subsolution of
(20)Dtu¯1(ωt)+G¯1(Dxxu¯1(ωt)ωt,u¯1(ωt),Dxu¯1(ωt),Dxxu¯1(ωt))=δt2,
where G¯1(ωt,r,p,X):=G1(ωt,r+δ/t,p,X). It is easy to check that the function G¯1 satisfies the same conditions as G1. Since u1≤u2 follows from u¯1≤u2 in the limit δ→0, it suffices to prove Theorem 14 under the additional assumption
(21)Dtu1(ωt)+G1(ωt,u1(ωt),Dxu1(ωt),Dxxu1(ωt))≥c,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkic=δT2,lim⁡t→0u(ωt)=-∞,uniformlyon⁡[0,T).
We make the following assumption.
(A) For eachωt∈Λ, u,v∈ℝ, p∈ℝd, and Q∈𝕊(d) such that u≥v, we have
(22)G(ωt,u,p,Q)≤G(ωt,v,p,Q).
Setu¯1(ωt)=u1(ωt)e-λt, u¯2(ωt)=u2(ωt)e-λt, for some λ<-C. Then it is easy to check that u¯i(i=1,2) is a subsolution or supersolution of
(23)Dtu¯i(ωt)+G¯i(ωt,u¯i(ωt),Dxu¯i(ωt),Dxxu¯i(ωt))=0,
where, for each (ωt,u,p,Q)∈Λ×ℝ×ℝd×𝕊(d), G¯i(ωt,r,p,Q) is given by
(24)G¯i(ωt,u,p,Q)=λu+e-λtGi(ωt,eλtu,eλtp,eλtQ),
which satisfies the assumption (A). Since u¯1≤u¯2 implies u1≤u2, it suffices to prove Theorem 14 under the additional assumption (A).
Without loss of generality, assumewα≤C2. Suppose by the contrary that there exist ωt00∈Λ, t0<T, and α<c/4(C∨C2)∧1, such that
(25)m0:=wα(ωt00)>0.
Then, by Lemma15, there exists ω¯t¯∈Λ such that
(26)wα(ω¯t¯)=sup⁡γt∈Λ,t≥t¯wα(ω¯t¯⊗γt)≥m0.
Consequently,
(27)Dtu1(ω¯t¯)-Dtu2(ω¯t¯)≤0,Dxu1(ω¯t¯)-Dxu2(ω¯t¯)=2αω¯(t¯),Dxxu1(ω¯t¯)-Dxxu2(ω¯t¯)≤2α.
Since αω¯2(t¯)≤C2, we have αω¯(t¯)≤Cα. Then
(28)0<c≤Dtu1(ω¯t¯)+G1(ω¯t¯,u1(ω¯t¯),Dxu1(ω¯t¯),Dxxu1(ω¯t¯))≤Dtu2(ω¯t¯)+G2(ω¯t¯,u2(ω¯t¯),Dxu2(ω¯t¯)+2αω¯(t¯),≤Dtu2(ω¯t¯)+G2Dxxu2(ω¯t¯)+2α)≤2C(α+Cα)<c.
This induces a contradiction and the proof is completed.Corollary 17.
The path-dependent PDE (8) has at most one bounded ℂ1,2-solution.Lemma 18.
Ifu is a bounded 𝒞1,2-subsolution of PPDE (8) and Φ is a bounded function on ΛT, then, for each ωt∈Λ,
(29)u(ωt)≤sup⁡ωT∈ΛTΦ(ωT).Proof.
Sincesup⁡ωT∈ΛT⁡|Φ(ωT)| is a 𝒞1,2-supersolution of PPDE (8), applying Theorem 14, we have the desired result.Remark 19.
Ifu(ωt)=u¯(t,ω(t)) is the ℂ1,2-solution of PPDE (8) for some bounded u¯∈C1,2([0,T)×ℝ), then the above theorem is the classical comparison theorem of PDE of [17].Example 20.
Consider the following PPDE:(30)Dtu(ωt)+G(Dxx2u(ωt))=0,ωt∈Λ,u(ωT)=Φ(ωT),ω∈ΛT,
where G(a):=(1/2)(σ¯2a+-σ_2a-) for some 0<σ_2≤σ¯2<∞.
IfΦ(ω)=φ(ω(t),ω(T)) for some bounded Lipschitz function φ∈C(ℝ2) and t∈[0,T], then we can solve the PPDE (30) by the following method.
First consider the following system of fully nonlinear parabolic partial differential equations, defined on[0,T]×ℝ2 and parameterized by y∈ℝ:
(31)∂sv1(s,x,y)+G(∂xx2v1(s,x,y))=0,v1(T,x,y)=φ(x,y),
and, then, another one defined on [0,t]×ℝ:
(32)∂sv2(s,x)+G(∂xx2v2(s,x))=0,v2(t,x)=v1(t,x,x).
From Krylov [11] and Wang [18], for each y∈ℝ, v1(s,x,y)∈C1+α/2,2+α([0,T-κ)×ℝ), and v2(s,x)∈C1+α/2,2+α([0,t-κ)×ℝ), where α∈(0,1) and κ>0.
Denoteu(ωs):=v1(s,ω(s),ω(t))1t≤s+v2(s,ω(s))1s<t; we obtain u∈𝒞1,2(Λ). Applying Theorem 14, u is the unique 𝒞1,2-solution of PPDE (30). Indeed, u(ωt) is the conditional G-expectation of Φ(ωT) (see [19, 20]).



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