Stochastic Differential Equations Driven By G-Brownian Motion

Motivated by uncertainty problems, risk measures and the superhedging in finance, recently, Peng systemically established a time-consistent fully nonlinear expectation the-ory (see [69], [71] and [74]). As a typical and important case, Peng introduced the G-expectation theory (see [77] and the references therein) via the following fully nonlin-ear PDEs where G:Sd�'R is a given monotonic, bounded sublinear function and Sd is the space of all d × d symmetric matrices. In the G-expectation framework, the notion of G-Brownian motion and the corresponding stochastic calculus of Ito’s type were also established in the sense of "quasi-surely" (q.s.). On this basis, Gao [23] and Peng [76] have studied the existence and uniqueness of the solutions of stochastic differential equations driven by G-Brownian motion (G-SDEs for short) under a standard Lipschitz condition. Moreover, Lin-Bai [57] (see also Lin [53], Li-Lin-Lin [49]) obtained the existence and uniqueness of the solutions of G-SDEs under some weak conditions. Recently, Luo-Wang [59] studied the sample solutions of G-SDEs. They showed that the integration of a G-SDE in R can be reduced to the integration of an ordinary differential equation (ODE for short) pa-rameterized by a variable in (Ω,F). In particular, the existence and uniqueness theorem for backward stochastic differential equations driven by G-Brownian motion (G-BSDEs for short) was obtained in Hu-Ji-Peng-Song [34]. They established the comparison the-orem, Feynman-Kac formula and Girsanov transformation for G-BSDEs in [35]. For a recent account and development of the sublinear expectation and G-expectation theory, we refer the reader to Bai-Buckdahn [6], Denis-Martini [14], Dolinsky-Nutz-Soner [15], Dolinsky [16], Gao-Jiang [24], Gao [25], Hu-Li-Wang-Zheng [36], Hu-Peng [37,38], Hu-Wang [39], Hu-Wang-Zheng [40], Li-Peng [50], Lin [51,53,52], Nutz [64], Nutz and Van Handel [65], Peng-Song [79], Soner-Touzi-Zhang [90], Song [91,92,93,94], Xu-Zhang [98], Zhang-Xu-Kannan [101], etc.This dissertation focuses on some topics under the G-framework. We present some preliminaries in the theory of sublinear expectation and G-expectation in Chapter 1. In Chapter 2, we obtain some characterizations of G-normal distributions which com-plement the theory of G-normal distribution. In Chapter 3, we first consier explicit solutions to a class of G-SDEs and then study the generalized sample solution of G-SDE from which we establish the existence, and uniqueness of solution of G-SDE in a domain. A comparison theorem is obtained. In Chapter 4, we prove a comparison theorem for multi-dimensional G-SDEs by a probabilistic method. We also give a sufficient and nec-essary condition of comparison theorem for multi-dimensional G-SDEs through a PDE method. Motivated by the results in Chapter 4, we consider the monotonicity and order-preservation for G-diffusion processes in Chapter 5. Sufficient and necessary conditions are given respectively. In Chapter 6, we introduce the definitions of the viability proper-ty, stochastic contingent and tangent sets for G-SDEs. Equivalent criterions for viability of G-SDEs are given through stochastic contingent and tangent sets. We also study the direct and inverse image for stochastic tangent sets from which we obtain the charac-terization of viability for a family of closed sets. Chapter 7 is devoted to the study of reflected G-SDEs with nonlinear resistance. We consider an integral-Lipschitz condition of the coefficients and the increasing process also contributes to the coefficients. Exis-tence and uniqueness result is established by a Picard iteration method. Moreover, we obtain a comparison theorem. In the sequel, we list the main results in this dissertation.2. Characterizations of G-normal distribution We consider non-degenerate random variable X on a sublinear expectation space (Ω,H,E),i.e.E[X2]>(E[|X|])2.From the definition of G-normal distribution, an equivalent characterization of G-normal distribution is that, for any a, b> 0 where Y is an independent copy of X. Denote then We are interested in the case that is replaced by f(λ) which is a nonnegative function of λ. We obtain the following characterizations of G-normal distribution.Theorem 0.1. Let f be a nonnegative function defined on some interval of R, which contains 0 as an interior point and X be a non-degenerate random variable on a sublinear expectation space (Ω, H,E, for all λ such that f (λ) is non-negative, λX+f(λ)Y=dX where Y is an independent copy of X, then:(i) X is G-normal distributed;Theorem 0.2. Let X, Y be two non-degenerate random variables on a sublinear expec-tation space (Ω,H,E) and f be a given non-negative function defined on some interval of R, which contains 0 as an interior point. Assuming that Y is independent with X, and λX+f(λ)Y is a non-degenerate random variable whose distribution does not depend on λ for all λ such that f(λ) is non-negative, then:(i) for some a, b> 0.(ii) X and Y are G-normal distributed with3. Stochastic differential equations driven by G-Brownian motionConsider the following stochastic differential equation driven by 1-dimensional G-Brownian motion: where X0 ∈ R,b, h, σ are R-valued functions defined on R and for some constant C. When σ ∈C1 (R) satisfies when b satisfies some conditions, by constructing the solution to a family of PDEs, we give the explicit solution of (0.0.14), where f is R-valued continuous function defined on R, φ satisfies some differentiability. When σ ∈ C1(R) satisfies when b satisfies some conditions, we give the explicit solution φ(t,x,<B>t,∫0tf(s)dBs) of (0.0.14), where f is R-valued continuous function defined on R, φ satisfies some differentiability.Now we consider generalized sample solution, suppose F is a domain of R+× R2 and σ(t,x,y) ∈ Cb,lip2(F) and b(t,x,y), h(t,x,y) ∈ Cb,lip(F). We consider the following G-SDE where (0,0,X0) ∈F.We first consider the following deterministic initial value problem: where t serves as a parameter. Moreover, the above ODE admits a unique solution g=φ(t,x,v) ∈ C2(F), where F is some domain in R+× R2. Define: We also have: then we solve the initial value problem with parameter ω of the following ODE: Note that <B>t is a continuous finite variation process, the ODE (0.0.17) has a unique solution V=Vt(ω),0≤t≤τ(ω) and τ is the "explosion time". We get the existence and uniqueness of solution of G-SDE in a domain.Theorem 0.3. Suppose F is a domain of R+×R2 and σ(t,x,y) ∈ Cb,lip2(F) and b(t,x,y), h(t,x,y) ∈Cb,lip(F). Then G-SDE (0.0.15) admits a unique solution where φ and V are given by equation (0.0.16) and (0.0.17) respectively, τ is the "explo-sion time" for Xt.From this result, we obtain the comparison theorem for G-SDEs in a domain.Theorem 0.4. Let and be given. If there exist three functions σ,f and g satisfying the Caratheodory conditions and the inequality Then for the unique solution Xt of G-SDE (0.0.15) holds for q.s. ω and every t in the common interval where both sides are defined. Here φ and V are the maximal solutions to the problems and with X0≤X0, respectively.4. Comparison theorem for multi-dimensional G-SDEsWe consider the following SDEs driven by a d-dimensional G-Brownian motion: and where the initial conditions X0, Y0,Y0 ∈Rn are given constants together with X0≤Y0.We obtain the following comparison theorem for multidimensional G-SDEs by virtue of a stochastic calculus approach.Theorem 0.5. Suppose that the following two conditions hold.(B1) For any t ∈ [0, T], and i=1,..., n, the inequality are fulfilled, whenever Xi=yi and Xi≤yj for all j≠i.(B2) b, hij, σi and b, hij, σi satisfy (H2) and(σi)k depends only on xk, for each k=1,...,n, i,j=1,...,d, i.e., for all t ∈ [0,T], x,y ∈ Rn.Then for all ∈ [0,T], We now introduce the definition of the viability property of G-SDE:Definition 0.1. Given a closed set K C (?) Rn. K is said to be viable for the equation (0.0.20) if starting at any time t ∈ [0, T] and from any point x in K, the solution (Xst,x)t≤s≤T to G-SDE (0.0.20) satisfies for each s ∈ [t,T],Then we define the following real valued function u: where C is a constant, dK(x) denotes the distance function of K:dK(x)=inf{|x-x’|: x’ ∈ K}.We know that u is a continuous function on [0, T] ×Rn with quadratic growth in x and K is viable for the G-SDE (0.0.20) if and only ifMoreover, from Theorem 3.7 in Peng [76], function u(t, x) is the unique viscosity solution of the following equation: where for φ∈C1,2([0,T]×Rn),The following theorem states the equivalent relation between the viability property of G-SDE and the square of the distance to the constraint set is a viscosity supersolution of the associated PDE.Theorem 0.6. Assume that (H2’) holds. Then the following conditions are equivalent:(1) K is viable for G-SDE (0.0.20).(2) dK2(·) is a viscosity supersolution of PDE (0.0.22).From the above theorem, we get a sufficient and necessary condition for comparison theorem of G-SDEs.For each v ∈{1,2} and t ∈ [0,T], s ∈ [t, T], consider the following G-SDE: where x1, x2 ∈ Rn.Theorem 0.7. If bv,hij and σiv satisfy assumption (H2’) for each v ∈{1,2}, then the following conditions are equivalent:(1) For any t e [0, T] and x1≤x2,(2) σ1=σ2 and for any t ∈ [0, T], k ∈{1,...,n}, where is a d x d symmetric matrix.5. On monotonicity and order-preservation for multidimensional G-diffusion processesWe suppose (Xt)0≤t≤T to be n-dimensional G-Ito diffusion process where (Bt)0≤t≤T is a d-dimensional G-Brownian motion and b, h, σ are Lipschitz con-tinuous functions on Rn. The Markov semigroup εt is defined by εtf(x)=E[f(Xt0,x)], where X0,x represents the G-Ito process with initial condition x at initial time t=0 and f is a function defined on Rn. The infinitesimal generator L of the Markov semigroup, which satisfies for f appropriately taken such that the above limit exists, is of the following form: where <(?)xf,h>+((?)xx2fσ,σ) is a d×d symmetric matrix, defined by:Similar to that in Herbst-Pitt [31] and Chen-Wang [11], we introduce the following definitions. Let "≤" denote the usual semi-order in Rn.(1) A measurable function f is called monotone if f(x)≤f(x) for all x≤x.Denote by M the set of all bounded Lipschitz continuous monotone functions.(2) For two semigroups{εt}0≤t≤T and{εt}0≤t≤T, we write εt≥εt, if for all f ∈M, for all x≥x and 0≤t≤T, εtf{x)≥εtf(x).If in addition, εt=εt, we call εt monotone. Let and let {εt}0≤t≤T, {εt}0≤s≤T and {εt}0≤t≤T be the semigroups generated by L,L and L’ respectively. And we always assume that b, hij,σi and 6, hij, σi satisfy (H2) for each i,j=1,... ,d.We have the following results concerning the monotonicity and order-preservation property of G-diffusion processes.Theorem 0.8. Suppose the following conditions hold:(Cl) for all i,j,σliσkj depends only on xi and xj,l,k =1,...,d.(C2) for all i, whenever x≤y with xj = yj. then εt is monotone.Theorem 0.9. If εt is monotone, then the following conditions hold:(Cl) for all i,j, σliσkj depends only on xi and xj,l,k=1,...,d.(C2’) for all i, whenever x≥y with xi=yi.Theorem 0.10. If εt≥εt then the following two conditions hold:(D1) for all i,j, σilσjk≡σilσjk depends only on xi and xj,l,k=1,...,d.(D2) for all i, whenever x≥y with xi=yi.Theorem 0.11. Assume (H3) holds and assume that σσ* (or resp. σσ*) is uniformly positive definite, i.e., there exists a constant β>0, such that for all y ∈ Rn, x ∈ Rn, y*σ(x)σ*(x)y≥β|y|2. If one of εt and εt is monotone, if the following hold:(D1) for all i,j, σilσjk≡σilσjk and σilσjk depends only on xi and xj, l,k=1,...,d(D5) for all x,K ∈ Rn, K≥0, then εt≥εt.6. Viability property for stochastic differential equations driven by G-Brownian motionWe consider the universally augmented filtration which allows us to use the measur-able selection argument. Thanks to the works of Peng [76] and Li-Peng [50], we define the G-Ito integral under this new setting in a similar way and extend the space of suit-able integrants. Under a standard Lipschitz assumption on the coefficients, the unique solution of a G-SDE falls in a new space M2(0, T). Then, we introduce the definition of the viability property for G-SDE:Definition 0.2. Let κ be a family of closed subsets of Rd, κ is said to be viable for G-SDE (0.0.24) if starting at any time t ∈ [0,T] and from any random variable ζ ∈ L2(Ft) in κt, the solution (Xst,ζ)t≤s≤T to (0.0.24) satisfies for each s ∈ [t,T], Xst,ε ∈ κs q.s.Consider a random variable ζ ∈ L2(Ft) in κt, we introduce the concept of contingent set and tangent set.Definition 0.3. The stochastic contingent set Cκ(t,ζ) to κ at ζ is the set of all bounded triples (u,v,w) of Ft-random variables, such that for any ε> 0, there exists δ’>0 such that for each δ ∈ (0,δ’], we can find three Ft+δ-random variables as, bs and cδ so that, and satisfyDefinition 0.4. The stochastic tangent set Tκ(t,ζ) to κ at ζ is the set of all bound-ed triples (u,v,w) of Ft-random variables, such that there exist three bounded adapted stochastic processes ab,bs,cs converging to 0 when s�'t such that for some δ’> 0, where the stochastic process d=a,b,c satisfies:for anyp> 0 there exists some constant Cp depending on p and T such that,We obtain the following equivalent criterions of viability of G-SDEs through stochas-tic contingent and tangent sets.Theorem 0.12. Let κ be a family of closed subsets of Rd, then the following conditions are equivalent:(1) κ is viable for G-SDE (0.0.24).(2) For any ζ ∈ L2(Ft) in κt,(3) For any ζ ∈ L2(Ft) in κt,We establish the characterization of viability of κ through the study of direct and inverse image for stochastic tangent sets.Theorem 0.13. Let K:=(κt)0≤t≤T be a family of closed subsets of Rd and If thenTheorem 0.14. Let κ:=(κt)0≤t≤T be a family of closed subsets of Rd and If the matrix φ’(x) has a right inverse denoted by φ’(x)+, which is a bounded Lipschitz function, then if and only if7. Reflected stochastic differential equations driven by G-Brownian mo-tion with nonlinear resistanceWe consider the following scalar valued reflected G-SDE with nonlinear resistance, i.e., the increasing process also contributes to the coefficients, where(A1) The initial condition x ∈R;(A2) For some p> 2, the coefficients f, h and g:Ω×[0,T]×R×R�'R are given functions satisfying for each x, y ∈ R, f.(x, y), h.(x, y), and where β1 ∈ MGP([0,T]) and β2 ∈ R+;(A3) The coefficients f, h and g satisfying an integral-Lipschitz condition, i.e., for each t ∈ [0,T] and where β:[0, T]�'R+ is integrable, and ρ:(0,+∞)�'(0,+∞) is continuous increasing and concave function that vanishes at 0+ and satisfies(A4) The obstacle is a G-Ito process whose coefficients are all elements in MGp([0,T]), and we shall always assume that S0≤x, q.s..We establish the existence and unique result.Theorem 0.15. Let the assumptions (A1)-(A4) hold true, then the reflected G-SDE (0.0.26) admits a unique solution in MGp([0,T]).In order to give the comparison principle, we consider the following scalar valued reflected G-SDE: and we assume that:(A2’) For some p> 2, the coefficients f, h:Ω×[0,T]×R×R�'R and g:Ω× [0,T]×R�'R are given functions satisfying for each x,y ∈ R,f.(x,y),h.(x, y), and g(x) ∈ MGp([0,T])and where β1 ∈ MGp([0,T]) and β2 ∈ R+;(A3’) The coefficients f, h and g satisfy an integral-Lipschitz condition, i.e., for each t ∈ [0, T] and where ρ:(0,+∞)�'(0,+∞) is continuous increasing and concave function that vanishes at 0+ and satisfiesWe obtain the following comparison result.Theorem 0.16. Suppose that for i=1,2 fi,hi,gi satisfy the conditions (A1),(A2’),(A3’) and (A4), and we assume in addition the following:(1) x1≤x2 and g1=g2=g;(2) ft1(x,0)<ft2(x,0) and ht1(x,0)< ht2(x,0),for x ∈ R,f1, h1 are decreasing in y, and f2, h2 are increasing in y, and st1≤St2,0≤t≤T,q.s..If (X1,K1) and (X2,K2) are the solutions to reflected G-SDEs above resepectively, then, Xt1≤Xt2,0≤t≤T,q.s.