Viscosity Solutions To PDEs On Riemannian Manifolds, Viability Property And Their Applications In SDEs And BSDEs With Jumps

The theory of viscosity solutions to nonlinear PDEs on Rn was introduced by M. G. Crandall and P. L. Lions [23] in the early 1980's. The primary virtues of this theory are that it allows merely continuous functions to be solutions of fully nonlinear equations of second order, that it provides very general existence and uniqueness theorems. More recently there have been various approaches to extend the theory of viscosity solutions to the setting of Riemannian manifolds. This is a natural thing to do, because many functions arising from geometrical problems are not differentiable.In Chapter 2, we consider the Cauchy-Dirichlet problem on Riemannian manifolds. With a key property of the Hessian of the function d2(x,y) introduced in [9], we get several uniqueness results with some restrictions on curvature. Then we apply Perron's method to get the existence result.In stochastic control theory, there exists a kind of cost functional which is described by the solution of a BSDE. For example, in the utility theory, economists use the solutions to describe recursive utilities. Peng [62] studied this kind of problem. In fact, as the solution to the corresponding SDE, the wealth process of the investor may be constrained. For example, it should be nonnegative. In particular, for some special need, it may be a process in some curving spaces. So in Chapter 3, we consider one kind of recursive optimal control problem involved with Riemannian manifolds, i.e. the corresponding SDE is defined on Riemannian manifolds. We get the corresponding dynamic programming principle and prove that the value function is the unique viscosity solution of the H-J-B equation on manifolds. This work is spirited by Peng [62].Viability properties for stochastic differential equations and inclusions had been introduced and studied by Aubin and Da Prato in [4], [5] and [6]. Buckdahn, Quincam- poix and Rascanu [19] got the viability property for backward stochastic differential equations. The theory of viability property is widely applied in stochastic control with constrained state, mathematical finance, partial differential equation and so on. For example, as we know, the comparison theorems for SDEs and BSDEs are very impor-tant and useful. But with traditional methods, we can only get results for the case of one-dimensional. Moreover, they are just sufficient or necessary. If we apply viability property, we can get the comparison theorems for multidimensional SDEs and BSDEs and the results are both sufficient and necessary. The brilliant idea is from Peng [50].We consider some problems on viability properties as follows.In Chapter 2, we consider the viability property of SDEs defined on Riemannian manifolds which keeps the state within closed subsets of the manifolds. In Chapter 4, we consider the viability property for SDEs with jumps which can make the state within a closed submanifold of Rm.Being spirited by [19], in Chapter 5, we generalize the backward stochastic viability property (BSVP in short) to the case with jumps with the same method as [19]. This with the stochastic viability property (SVP in short) with jumps obtained by Peng and Zhu [51], we study the comparison theorems for multidimensional SDEs with jumps (in Chapter 3) and BSDEs with jumps (in Chapter 5).This thesis consists of five chapters. In the following, we list the main results of this thesis.Chapter 1:We introduce problems studied from Chapter 2 to Chapter 5.Chapter 2:We consider the following Cauchy-Dirichlet problem on Riemannian manifolds We study the existence and uniqueness of the viscosity solution to the above PDE. At the end of this chapter, we consider the viability property of SDEs defined on Rieman-nian manifolds which keeps the state within closed subsets of Riemannian manifolds. The main results are the following Theorem 2.2.1, Theorem 2.2.8 and Theorem 2.4.4.Theorem 2.2.1. Let Q be a bounded open subset of a complete finite-dimensional Riemannian manifold M, and for each fixed t∈(0, T), F∈C(χ, R) be continuous, proper and satisfy:there exists a functionω:[0,+∞]→[0,+∞] withω(0+)= 0 and such that for all fixed t∈(0, T) and for all x,y∈Ω, r∈R,P∈Ls2(TMx), Q∈Ls2(TMy) with where Aa is the second derivative of the functionφα(x,y)=(?)d2(x,y)(α> 0) at the point (x, y)∈M×M, and the points x,y are assumed to be close enough to each other so that d(x,y)< min{iM(x),iM(y)}.Let u∈USC([0, T)×Ω) be a subsolution andυ∈LSC([0, T)×Ω) a supersolution of (2.1). Then u≤υon [0, T)×Ω.In particular the Cauchy-Dirichlet problem (2.1) has at most one viscosity solution.Theorem 2.2.8. Let comparison hold for (2.1), i.e., ifωis a subsolution of (2.1) and v is a supersolution of (2.1), thenω≤υ. Suppose also that there exists a subsolution u and a supersolution u of (2.1) that satisfy u*(0, x)= u*(0, x)=ψ(x) for x∈Ωand u*(t,x)= u*(t,x)= h(t,x) for (t,x)∈[0,T)×(?)Ω. Then is a solution of (2.1).Theorem 2.4.4. Under the assumptions (H1)-(H3), the following conditions are equiv-alent:(i) K enjoys viability with respect to (2.7);(ii)dK2(x) satisfies partial differential inequality (2.9).Chapter 3:We study one kind of recursive optimal control problem involved with Riemannian manifolds, i.e. the corresponding SDE is defined on Riemannian manifolds. In details, we consider the following control system where M is a compact Riemannian manifold without boundary. And the associated cost functional is where (Y.t,x;v., Z.t,x;v.) is the solution to the BSDE: We will minimize the cost functional and define the value function Using the idea and framework from Peng [62] (Chapter 2), with the help of the com-pactness of the Riemannian manifold, we get the continuity of value function u and the general dynamic programming principle (DPP).Lemma 3.2.2. (Continuity on x) u(t,x) is bounded and uniformly continuous in x, uniformly in t.Theorem 3.2.5. (DPP) (?)(t,x)∈[0,T] x M,(?)δ∈[0,T-t], we haveProposition 3.2.6. (Continuity on t) The value function u(t, x) is continuous in t,t∈[0,T].Finally, we prove that u(t, x) is the unique viscosity solution of the following non-linear parabolic PDE on M which is called H-J-B equation:Chapter 4:Applying the characterization of viability property of multidimen-sional SDEs with jumps in [51], we study two questions. One is for the viability prop-erty for SDEs with jumps which can make the state within a closed submanifold of Rm. Another is for the comparison theorem of multidimensional SDEs with jumps. We get the following main results: Theorem 4.2.2. Under the assumptions (A1) and (A2), SDE (4.2) enjoys SVP in a closed submanifold K if and only if:(?)t∈[0, T], x∈K, where m(x) is any normal vector of K at x andTheorem 4.3.1. suppose that (bi,σi,γi),i= 1,2 satisfy (A1) and (A2). Then the following are equivalent:(i) For any t∈[0, T], x1,x2∈Rm such that x1≥x2, the unique adapted solutions X1 and X2 in S[t,T]2 to the SDE (4.12) over time interval [t, T] satisfy:(ii)σ1≡σ2, and for any t∈[0,T],k= 1,2,...,m,Chapter 5:With the same method as [19], We study the viability property for BSDEs with jumps which can make the solution with a convex closed subset of Rm. As an application, we consider the comparison theorem for multidimensional BSDEs with jumps. The following are the main results in Chapter 5:Theorem 5.2.1. (BSVP with jumps) Suppose that f:Ω×[0,T]×Rm×Rm×d×L2(E,BE,n;Rm)→Rm is a measurable function which satisfies (H1)-(H3). Let K be a nonempty closed set. The set K enjoys the BSVP for BSDE (5.1) if and only if:(?)(t,z,u)∈[0,T]×Rmxd×L2(E,BE,n;Rm) and for all y×Rm such that dK2(·) is twice differentiable at y, where C*> 0 is a constant which does not depend on (t, y, z, u).Theorem 5.3.1. Suppose that f1 and f2 satisfy (H1)-(H3). Then the following are equivalent:(i) For any s∈[0,T],(?)ξ1,ξ2∈L2(Ω,Fs,P;Rm) such thatξ1≥ξ2, the unique solutions (Y1,Z1,U1) and (Y2, Z2, U2) inΒ[0,s]2 to the BSDE (5.23) over interval [0, s] satisfy:(ii)(?)t∈[0,T], (?)(y,y'),(?)(z,z'), (?)(u,u'), where C is a constant which dose not depend on t, (y,y'), (z, z'), (u, u').Theorem 5.3.2. Let m= 1. Suppose that f1 and f2 satisfy (H1)-(H3). Then the following are equivalent:(i) For any s∈[0,T], (?)ξ1,ξ2∈L2(Ω,Fs, P;R) such thatξ1≥ξ2, the unique solutions (Y1,Z1,U1) and (Y2, Z2, U2) inΒ[0,s]2 to the BSDE (5.23) over interval [0, s] satisfy:(ii) (?)(t,y',z)∈[0,T]×R×R1×d,Corollary 5.3.8. Suppose that f satisfies (H1)-(H3) and fk depends only on uk. Then the following are equivalent:(i) For any s∈[0,T], (?)ξ1,ξ2∈L2(Ω,Fs, P;Rm) such thatξ1≥ξ2, the unique solutions (Y1,Z1,U1) and (Y2, Z2,U2) inΒ[0,s]2 to the BSDE (5.23) over interval [0, s] satisfy:(ii) For any k = 1,2,..., m,...