| In1990, Pardoux-Peng [94] introduced the following new type of backward stochastic differential equations (BSDEs):where B is a d-dimensional Brownian motion on a standard Wiener space (Ω,F,P). Unlike the forward stochastic differential equation (SDE), the unique solution of BSDE (0.0.18) consists of two non-anticipating stochastic processes (Y,Z). In [94], they estab-lished the existence and uniqueness theorem under a standard Lipschitz condition, which generalize the linear ones of Bismut [6]. Since this pioneering work, the theory of BSDEs has gone through rapid development in many different areas of research and applications, such as partial differential equations (PDEs), mathematical finance, stochastic control and stochastic games, functional analysis, numerical analysis and stochastic computa-tions, engineering.During the last20years, there are lots of work devoted to study the the solutions’ existence and uniqueness, which extend the Pardoux-Peng’s initial result for more general formulation, e.g.. see Pardoux-Peng [96], Lepeltier-San Martin [76], Jia [69], Kobylan-ski [72], El Karoui-Kapoudjian-Pardoux-Peng-Quenez [42], Situ [132], Barles-Buckdahn-Pardoux [3], Tang-Li [141], Antonelli [1], Ma-Protter-Yong [84], Hu-Peng [64], Peng-Wu [122]. Pardoux-Tang [91], Delarue [33], Peng-Shi [120], Hamadene [50], Peng-Xu [124], Buckdahn-Djehiche-Li-Peng [9] et al. Then the BSDE theory takes shape as a distinct mathematical discipline. Moreover, Peng [106] introduced the g-expectation theory, which provides an excellent tool to the study of nonlinear stochastic analysis (more details can be founded in Peng [107,108,109,110], Chen-Tao-Matt [17], Coquet-Hu-Memin-Peng [19], Delbaen-Peng-Rosazza Gianin [28], Hu-Ma-Peng-Yao [62] et al.). Base on this view, a g-martingale is the unique solution to a semi-linear PDE in the Markovian case, which is the famous nonlinear Feynman-Kac formula. More precisely, for ξ=φ(B(T)) and f=f(t,B(t),y,z), the solution of BSDE (0.0.18) given by Y(t)=u(t,B(t)) and Z(t)=▽u(t, B(t)), where u is the unique solution of the following semi-linear PDE:(?)tu,1/2△u+f(t,x,u,▽u)=0,u(T,x)=φ(x).(0.0.19) This relation provides a probabilistic representation for a wide class of semi-linear PDEs (see Peng [100,104], Pardoux-Peng [95,97], Pardoux [92,93], Barles-Buckdahn-Pardoux [3], Buckdahn-Peng [14], Buckdahn-Hu-Peng [11], Fuhrman-Tessitore [45], Royer [130], Crisan-Delarue [24], Wu-Yu [143], Pham [126] et al.). This nonlinear Feynman-Kac for-mula also provides a nonlinear Monte-Carlo method via the BSDE to solve numerically the PDE (see Douglas-Ma-Protter [34], Zhang [149], Bouchard-Touzi [7], Peng-Xu [123], Gobet-Lemor-Warin [49], Zhao-Chen-Peng [150] et al.). The BSDE theory also provides an efficient method to deal with the problems of dynamic pricing and hedging in an in-complete financial market (see El Karoui-Peng-Quenez [40], Chen-Epstein [16], Delbaen-Peng-Rosazza Gianin [28], Duffie-Epstein [35], El Karoui-Quenez [41], Cvitanic-Karatzas [25], Jiang [70], Hu-Imkeller-Muller [60] et al.). We also refer the reader to stochastic differential maximization and games with recursive or other utilities (see Peng [105], Pham [125], Buckdahn-Li [12] and the references therein).The notion of g-expectation is an idea tool to treat the uncertainty of probability models, where all the involved uncertain probability measures are absolutely continuous with respect to the Wiener measure P (see Chen-Epstein [16]). However, for the well-known problem of volatility model uncertainty in finance, there is an uncountable number of unknown probabilities which are singular from each other. Motivated by this problem, Peng systemically established a time-consistent fully nonlinear expectation theory (see [111]). As a typical and important case, Peng [112] introduced the G-expectation theory (see also [113,114,116] and the references therein) via the following fully nonlinear PDEs (?)tu-G(Dxx2u)=0,(t,x)∈(0.T)×Rd, where G:S(d)'R is a given monotonic, bounded sublinear function and S(d) is the space of all d x d symmetric matrices. In particular, the classical linear expectation can be seen as a special case of G-expectation.Under the G-expectation framework, the notion of G-Brownian motion and the corresponding stochastic calculus of Ito’s type, were also established in an analogous way as the classical one, but in a language of "quasi-surely"(q.s.) instead of "a.s.’ (see also Peng [116], Denis-Hu-Peng [29], Hu-Peng [56,57], Denis-Martini [30], Li-Peng [77], Soner-Touzi-Zhang [135], Song [136,137,138,139] et al). On this basis, Gao [46] and Peng [116] studied the existence and uniqueness of the solution of G-SDE under a standard Lipschitz condition. In particular, the existence and uniqueness theorem for G-BSDEs was also obtained in Hu-Ji-Peng-Song [51]. The key issue is that, G-SDE and G-BSDE are connected to a larger class of fully nonlinear PDEs studied by Peng [116] and Hu-Ji-Peng-Song [52], which generalizes the ones of Pardoux-Peng [95] for the fully nonlinear case. The notion of Second Order BSDEs, proposed by Cheridito-Soner-Touzi-Victoir [18](see also Soner-Touzi-Zhang [134], and the references therein), was also motivated from the connection with fully nonlinear PDEs. For a recent account and development of the nonlinear stochastic analysis theory we refer the reader to Chen [15], Epstein-Ji [43], Hu-Ji [53], Hu-Ji-Yang [54], Bai-Lin [2], Lin [78,79], Lin [80], Gao-Hui [47], Gao [48], Dolinsky-Nutz-Soner [31], Dolinsky [32], Nutz [89], Nutz-Van Handel [90], Xu-Zhang [144], Zhang-Xu-Kannan [148] et al.From above results, we know that BSDEs with a deterministic terminal time provide probabilistic representation for solutions to semi-linear parabolic PDEs, whereas the BSDEs with a random terminal time are connected with semi-linear elliptic PDEs in the Markovian case. Thus a PDE can be regarded as a state dependent BSDE, which provides an efficient way to study the theory of PDE and stochastic analysis.In general the terminal condition of the BSDE may be a general function of Brow-nian paths. In this situation the solution (Y, Z) is regarded as a generalized "path-dependent" solution of the above BSDE (0.0.18) and it has long been discussed that this BSDE can also be viewed as a path-dependent PDE (PPDE). This problem was raised by Peng which was summarized in his ICM2010’s talk [117, Sec.1.3]. Chapter1and Chapter2will focus this topic and establish the nonlinear Feynman-Kac formula in the non-Markovian case. The key ingredient of our approach is the functional Ito formula, which introduced by Dupire [36](see also Cont-Fournie [20,21,22] for more general and systematic research). We also refer to Peng-Song [120] for the G-Sobolev-solutions of PPDEs. In order to study the fully nonlinear Feynman-Kac formula, we first study the properties of solutions to G-SDEs in the language of "quasi-surely" in Chapter3. In the linear case, Doss [33], Sussmann [140] and Huang-Xu-Hu [66](see also Huang [65] and Pardoux-Talay [98]) studied the sample solutions of SDEs, which enables us to transfer a SDE into a set of ordinary differential equations (ODEs) for each sample path. We shall study the Doss’s transform under G-expectation. We also study quasi-continuous random variables associated to the solutions of G-SDEs and state the Krylov’s estimates of G-diflusion processes,which is useful for the further development of G-Ito calculus. Then Chapter4is devoted to study fully nonlinear Feynman-Kac formula for G-SDEs in the non-Markovian case. In Chapter5,we shall explore the link between ergodic G-BSDE and the fully nonlinear ergodic elliptic PDEs and state some applications.In the linear case,Fuhrman-Hu-Tessitore[44](see also Debussche-Hu-Tessitore[26],Richou [129])introduced the notion of(markovian)ergodic BSDEs(EBSDEs). The EBSDEs provide an efficient alternative tool to study optimal control problems with ergodic cost functionals that is functionals depending only on the asymptotic behavior of the state. Moreover,by virtue of a EBSDE approach,Hu-Madec-Richou[63]study the large time behaviour of mild solutions to semi-linear PDEs(in infinite dimension).We also refer to the talk given by Pham at the "7th International Symposium on BSDEs" for the study of long time asymptotics of fully nonlinear Bellman equations.In the sequel,we list the main results in this dissertation.1.BSDE,Path-dependent PDE and Nonlinear Feynman-Kac Formula.From Dupire[36],we consider the space(Λd,d∞),where Λd=Ut∈[0,T] Λtd,Λtd is the set of all RCLL Rd-valued functions on[0,T] and for each ωt,ωt∈Λd with0≤t≤t≤T, The followingg space will be used frequently in this paper.Definition0.1.A function u is said to be in Cl,lip1,2(Λd),if the Dupire’s path-derivatives Dtu,Dxu,Dxxu exist and for φ=u,Dtu,Dxu,Dxxu,|φ(ωt)-φ(ωt)|≤C(1+||ωt||k+||ωt||k)d∞(ωt,ωt), for each ωt,ωt∈Λd, where C and k are some constants depending only on u.Now we shall briefly explain our idea.Since the solution Y of BSDE(0.0.18)is a non-anticipating stochastic process,there is a functioon u defined on Ω such that Yt(ω)=u(ωt). Assume u∈Cl,lip1,2(Λd) and d=1.Then by functional Ito formula,we obtain du(Bt)=[Dtu(Bt)+1/2Dxxu(Bt)]dt+Dxu(Bt)dB(t), which implies that(Yt,Zt)=(u(Bt),Dxu(Bt))and Dtu(Bt)+1/2Dxxu(Bt)+f(Bt,u(Bt).Dxu(Bt))=0,(0.0.20) where the derivative is the Dupire’s path derivative. In this equation, paths ωt on an interval [0,t] becomes the basic variables in the place of classical variables (t,x) and it provides a probabilistic representation for semi-linear PPDE (0.0.20). However we have assumed the function u belongs to Cl,lip1,2(Ad) and so we have to prove this claim. The rest of this part is devoted to this fact.For each ωt∈Ad, consider the following BSDE: where Bωt(u):=ωu(u)1[0,t](u)+(u(t)+B(u)-B(t))1(t,T](u). Using the Kolmogorov’s criterion as the Markovian case (see Pardoux-Peng [95]), we prove the existence of a continuous second order partial derivative of Yωtx(s) with respect to x. Theorem0.1. Assume (H1.1) and (H1.2) hold. Then for each ωt∈Λd,{Yωtx(s),s∈[0,T],x∈≧Rd} has a version which is a.e. in C0,2([0,T]×Rd).Now define u(ωt):=Yωt(t). Then by Theorem0.1, we have the following.Corollary0.1. Under assumption (H1.1) and(H1.2), the Dupire’s derivatives Dxu(ωt), Dxxu(ωt) exist and u∈Cl,lip0,2(Λd).In Pardoux-Peng [95], when the terminal of BSDE is the state-dependent case, it is shown that Z and Y are connected in the following sense under appropriate assumptions: Zωt(s)=(?)xu(s,ωt(t)+B(s)-B(t)).In the following, we shall extend this result to the path-dependent case.Theorem0.2. Under assumptions (H1.1) and (H1.2), for each fixed ωt∈Ad, the process (Zωt(s))s∈[t,T] has a continuous version given by, Dxu(Bsωt)=Zωt(s). for each s∈[t,T] a.s. Now consider the following multi-dimensional semi-linear path-dependent PDE: where u:=(u1,...,um):Λd'Rm is a function on Ad. Then we can prove the uniqueness of solution to PPDE (0.0.22). Theorem0.3. Assume that assumptions(H1.1) and (H1.2) hould and let u∈Cl,lip1,2(Λd) be a solution of the equation (0.0.22).Then we hve u(ωt)=Yωt(t) for each ωt∈Ad, where (Yωt(s),Zωt(s))t≤s≤T is the unique soulution of BSDE (0.0.21).Consequently, the path-dependent PDE(0.0.22) has at most one Cl,lip1,2-solution.Applying this theorem and the classical comparison theorem of BSDE,we have the following comparison theorem for semi-linear path-dependent PDE:Corollary0.2.We assume m=1and that f=fi,Φ=Φi,i=1.2satisfy the same assumptions as in Theorem0.3.If (ⅰ)f1(ωt,y,z)≤f2(ωt,y,z),for each (ωt,y,z)∈Λd×R×Rd,(ⅱ)Φ1(ωt)≤Φ2(ωT),for each ωTr∈ΛTd, ui∈Cl,lip1,2(≮d) is the solution of equation (0.0.22)associated with (f,Φ)=(fi,Φi), i=1,2,respectively,then for each ωt∈Λd,u1(ωt)≤u2(ωt)At the end of this chapter,we prove the existence of solution to PPDE(0.0.22)by a stochastic calculus approcah.Theorem0.4.We make assumptions (H1.1)-(H1.2).Then the function u is the unique Cl,lip1,2(Λd)-solution of the path-dependent PDE(0.0.22).2.BSDEs with jumps and path-dependent parabolic integro-differential equationsInspired by Chapter1,we shall study the connection between BSDEs with path-dependent parabolic integral-differential equations(PPIDEs). The difficulty in trying to follow the results in Chapter1is the problem of functional Ito formula for discon-tinuous semimartingale.The functional Ito formula for discontinuous semimartingale is introduced by Cont-Fournie[21]. However,they adopt a condition that is not usual in order to overcome some technique difficulties,and thus the applicability of this formula are limited.First we remove some additional conditions that they adopted and obtain a functional Ito formula Ito formula for discontinuous semimartingale.Theorem0.5.Let(Ω,F,(Ft)t∈[0,T],P) be a probability space.X=M+A is a semi-martingale,where M is a continuous local martingale and A is a finite variation process. If u is in Cl,lip1,2(Λd),then for any t∈[0,T[: Then consider the following decoupled FBSDE with jump, for each ωt∈Ad Now define u(ωt):=Yωt(t) for each ωt∈Λd Ad. After some calculus, it is easy to obtain the following important properties of u.Theorem0.6. Under assumptions (H2.1) and (H2.2),{Yωtx(s),s∈[0,T],x∈Rd} has a version which is a.e. in C0,22([0,T]×Rd). In particular, Dxu(ωt), Dxxu(ωt) exist and u∈Cl,lip0,2(Λd).Moreover, we have below a formula relating Z, K with Y by stochastic calculus for the non-Markovian BSDEs with jumps. We remark that the proof is much more complicated than the one in Chapter1because of the jumps.Theorem0.7. Under assumptions (H2.1) and (H2.2). the process(Zωt,Kωt) have the following a.s. left continuous version given by.Then we introduce the following semi-linear path-dependent PIDE: where u=(u1,…,un):Ad'Rn is a function on Ad and(?)ul(ωt):=Dxul(ωt)b(ω(t))+1/2tr[(σσT)(ω(t))Dxxul(ωt)] Finally we will give the main result of this part, which provides a one to one correspon-dence between BSDEs (0.0.23) and the system of path-dependent PIDEs (0.0.25)). Theorem0.8.We make assumptions(H2.1)-(H2.2).Then the function u is the unique Cl,lip1,2(Λd)-solution of the path-dependent PDE(0.0.25).3.Stochastic differential equations driven by G-Brownian motionIn section3.2,we shall establish the relations between G-SDEs and ODEs by using the extension of G-Ito formula in Lbp(Ω).We show that the integration of a G-SDE in R can be reduced to the integration of ODEs parametrized by a variable in(Ω,F).We consider the flollowing G-SDE for σ(t,x,y)∈Cb,lip1([0,T]×R2)and b(t,x,y),h(t,x, Cb,lip([0,T]×R2), where the initial condition X0∈R is a given constant.We first consider the following ODE The above ODE admits a unique solution y=φ(t,x,v)∈C1,2,1([0.T]×R×R).Denote9(t,x,v)=(/)vφ-1(t,x,v)(b(t,x,φ(t,x,v))-(?)tφ(t,x,v)).(?)vφ-1(t,x,v)(h(t,x,φ(t,x,v))-1/2((?)xσ+(?)yσσ)(t,x,φ(t,x,v)). We Call solve the following initial value problem with parameter ω: Then from the G-Ito formula and the umiqueness of solution to G-SDE(0.0.26),we obtain that X(t)=φ(t,B(t),V(t)) is the solution of SDE(0.0.26).By this result,we get a new kind of comparison theorem for G-SDEs.Moreover.a necessary and sufficient condition for comparison theorem of G-SDEs is also obtained.Theorem0.9.Consider two G-SDEs: where σi,bi,hi∈Cb,lip(R) and i∈{1,2}, then for each x1≤x2, X1,x1(t)≤X2,x2(t) if and only if b1(x)-b2(x)+2G(h1(x)-h2(x))≤0, σ1(x)=σ2(x),(?)x∈R.In section3.3, we discuss both piecewise and mean-square convergence of several ap-proximation schemes of G-SDEs. We estimate the corresponding speeds of convergence. In section3.4, we study the properties of indicator functions of G-diffusion processes and establish Krylov’s estimates for G-diffusion processes, which is important for the theory of G-expectation. In section3.5, we prove the unique viscosity solution to a nonlinear PDEs is a nonlinear expectation-martingale along the solution of the G-stochastic differ-ential equation. In particular, it extends the nonlinear Feynman-Kac Formula in [116] to a more general case. The difficulty in trying to follow the results in [4] is the prob-lem of optional sampling theorem of nonlinear expectation. Since there is no optional sampling theorem for G-expectation, in this paper, we give a more rigorous definition of G-martingale (resp. super-martingale, sub-martingale) which avoid this question.4. Comparison theorem for fully nonlinear PPDE and G-expectation theoryIn Chapter4, we will establish the comparison principle of fully nonlinear PPDEs. In order to study comparison theorem for fully nonlinear PPDE, we divide this chapter into two sections. In section4.1, we first give the functional G-Ito formula in LG2(Ω) for a G-Ito process, which non-trivially generalizes classical one. Theorem0.10. Suppose u∈Hγ(Ωn) for some γ∈(0,1) and αv, βvij, σvj are bounded process in MG2(0.T) for v=1,…, n and i,j=1,…, d. Then, for each t∈[0, T], in LG2(Ωt),As an application, we establish the comparison theorem for fully nonlinear PPDEs. Corollary0.3. Let uv∈Hγ(Ωn)(v=1,2) for some γ∈(0,1) be a solution of the equation (4.1.11) withGv(ωt,r,p,Q)=<p,α(ωt)>+Gv((<Qσi(ωt),σj(ωt)>+<p,βi,j(ωt)+βj,i(ωt)>)i,j), where (ωt,r,p,Q)∈Ωn×R×Rn×S(n). If (ωT)≤u2(ωT) for all ωT∈ΩTn and G1(ωt,r,p,Q)≤G2(ωt,r,p,Q), then for each ωt∈Ωn,u1(ωt)≤u2(ωt).In section4.2, we will attempt to establish the partial comparison principle of fully nonlinear PPDEs based on techniques of PDE instead of stochastic calculus. Then we obtain a partial comparison principle for the fully nonlinear path-dependent PDE through analysis method.Theorem0.11. u1is a viscosity subsolution of PPDE (4.2.1) with G1and u2is a viscosity supersolution of PPDE (4.2.1) with G2, where for each (ωt,r,p,X)∈Λd×R Rd x S(d), G1(ωt,r,p,X)≤G2(ωt,r,p,X). Suppose u1is bounded from above and u2is bounded from below. If u1or u2is in C1,2(Λd), then the maximum principle holds:if u1(ωT)≤u2(ωT) for all ωT∈ΛTd, then we also have u1(ωt)≤u2(ωt), for each ωt∈Λd.5. Ergodic BSDEs driven by G-Brownian motion and their applicationsThe aim of this paper is to study the following type of (Markovian) BSDEs dirven by G-Brownian motion with infinite horizon that we shall call G-EBSDEs:for all0≤s≤T<∞, where (B(t))t≥0is a G-Brownian motion and Xx is the solution to a forward stochastic differential equation driven by G-Brownian motion starting at x. Our aim is to find a quadruple (Y,Z,K,A), where Y. Z are adapted processes, K is a decreasing G-martingale and λ is a real number.First, we introduce a new kind of linearization method to show that the BSDEs driven by G-Brownian motion with infinite horizon have a unique solution under some certain conditions. However, the linearization methods in [10] and [52] cannot be applied to deal with this problem. Consider the following type of BSDEs driven by G-Brownian motion with infinite horizon Theorem0.12. Let assumptions (H5.1)-(H5.4) hold. Then the G-BSDE (0.0.32) has a solution (Y,Z,K) belongs to (?)G2(0,∞) such that Y is a bounded process. This solution is unique in the class of processes (Y,Z,K)∈((?)G2(0,∞) such that Y is continuous and bounded.Next, we establish the nonlinear Feynman-Kac formula for elliptic PDEs and in-troduce a new method to show the uniqueness of viscosity solution to elliptic PDEs in Rn.Theorem0.13. Function u(x) is the unique bounded continuous viscosity solution of the following PDE:Finally, we study the G-EBSDE and state some of its applications.Theorem0.14. Suppose assumptions (B5.1),(B5.2),(B5.4) and (B5.5) hold. Then for each x, the G-EBSDE (0.0.31) has a solution (Yx,Zx,Kx,λ)∈(?)G2(0,∞)×R such that|Yx(s)|≤M|Xx(s)|.For each Lipschitz function φ:Rn'R, consider the following fully nonlinear parabolic PDE:Theorem0.15. Under assumptions (B5.1),(B5.2),(B5.4) and (B5.5), there exists some constant C such that, for each T>0. In particular,... |
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