| Let us begin with the story of backward stochastic differential equations theory (BSDEs, in short). As is widely recognized now, it took a linear form as firstly appearing in [4, Bismut] in a stochastic optimal control problem, then the original work of Pardoux-Peng [92] initialized the study of genearl BSDEs Under Lipschitz condition, Pardoux-Peng established the well-posedness of such equa-tion. Nowadays scholars and related experts have revealed lots of deeply insightful connections between the theory of BSDEs and among other research fields like, partial differential equations, mathematical finance, stochastic control and stochastic differential games, stochastic numerical analysis. After the BSDEs found applications in stochastic recursive utility, a lot of work is devoted to the study of forward backward stochastic differential equations (FBSDEs in short) where the forward X and backward part Y are suitably coupled, the existence and uniqueness of solution to FBSDEs were obtained under various formulations of the prob-lem (see Antonelli [1], Delarue [26], Ma-Protter-Yong [82], Hu-Peng [56], Yong [149] etc).Since the dawn of BSDEs theory, it has prevailed in stochastic analysis especially in those areas related to the mathematic finance (see El Karoui-Peng-Quenez [41],E1 Karoui-Quenez [42], Chen-Epstein [16], Delbaen-Peng-Rosazza Gianin [35], Duffie-Epstein [29], Cvitanic-Karatzas [23] and the references therein) due to its natural backward structural appealing to financial mathematicians, i.e., lots of investment problems seek strategies that can guarantee a preset future goal ζ. The BSDEs theory in its own right provides a simple affirmative answer as long as a minimum of regularity conditions are satisfied. Part of the solution Y_ constitutes an analogy of a nonlinear martingale, according to which Peng introduced the concept of g-martingale together with a general notion of ^-expectation (g endorses the generator of BSDEs), where in particular, Doob-Meyer decomposition for g-martingale [105] holds still true. One drawback of nonlinear g-expectation is that strict restrictions upon the generator have to be put to ensure the convexity, which is considered to be one of the main features of coherent risk measures, and observing the prosper of coherent measure of risk,g-expectation is in favor of G-expectation theory.Noting the interior regularity of the following fully nonlinear PDE on [0, ∞) × Rd, with boundary condition eng [115] proposed a time-consistent sublinear expectation E, also called G-expectation, which now is one of the central topics in stochastic analysis and mathematical finance theory. The multi-fold reasons why this framework has received much attention include among others,· A type of fully nonlinear Feynman-Kac formula is established.· It allows for Knightian uncertainty or model uncertainty.· As risk measure, it is coherent that could be represented by a group of possibly non-dominated mutually singular martingale measures.Within the G-expectation framework, the pillar stones of stochastic analysis have been constructed on the canonical path space, yet all statements should be made "quasi surely" rather than "almost surely" for there exists a natural capacity associated to G-expectation. And all counterparts in G-expectation are named with a prefix G to distinguish from the classical context. Peng systematically treated the major concerns of this setting in an appropriate manner of norm extension, for example the quadrat-ic variation process of G-Brownian motion, the G-Ito’s formula, the well-posedness ofG-SDEs, the G-martingale representation. Surely there are some unsettled problems as-sociated to this framework, the interested readers are referred to [115] or Peng’s plenary lecture at ICM 2010 [116]. From then on, as the martingale representation theorem is the prototype of BSDEs, G-BSDEs are explored, in particular the existence and uniqueness results under stand Lipschitz conditions are established. For the literature, we list for instance Peng [115], Peng-Song-Zhang [120], Denis-Hu-Peng [30], Gao [47], Hu-Ji-Peng-Song [52,53], Li-Peng [75], Epstein-Ji [44]. The G-BSDEs theory could be interpreted as an extension of classical BSDEs under nonlinear expectation. Meanwhile the sec-ond order backward stochastic differential equations (2BSDEs hereafter) generalizes the classical BSDEs from a different starting point, even though they lead to similar type of Feynman-Kac formula (see Soner-Touzi-Zhang [134,135,136]). Marcel Nutz’s works [87,88] further get rid of some technical conditions to build up random G-expectation and, conditional G-expectation for Borel measurable mapping, which could essentially be seen as the interplay of G-expectation theory with 2BSDEs.In his ICM 2010 lecture, Peng proposed the concept of path dependent PDEs to allow for probabilistic interpretation for non-Markovian BSDEs, up to now there are at least three attempts towards this topic, Peng-Wang [121] utilized Dupire’s derivatives to rewrite the Feynman-Kac formula for smooth quasi-linear parabolic path dependent PDEs, Ekren-Touzi-Zhang [38,39,40] put forth path dependent viscosity solution to fully nonlinear parabolic PDEs by optimal stopping techniques, Peng-Song [119] defined in an appropriate Sobolev space the equivalence of BSDEs with path dependent quasi-linear PDEs.This dissertation deals with several problems in G-expectation theory. Noticing the work of Hu-Ji-Peng-Song [52,53], we are curious about the wellposedness of fully coupled FBSDEs, as is done in the classical case there are other potential method, however we adopt the one by Delarue [26] for our purpose. Indeed the continuation method has turned out to be not practical in attacking this problem. Our result is only valid on small time interval, the point is that compared to the classical case, the corresponding nonlinear PDEs permit only interior regularity and G-expectation theory is up to now short of appropriate dominated convergence theorem, so the concatenation is beyond possible while extending this local result to global one following the lines of Delarue. Then we consider the property of G-normal distribution, recalling the fact the classical Gaussian distribution is invariant for OU process, we recover this property for G-normal distribution, in fact we consider more general invariant expectations for SDEs. Another obvious feature of G-expectation theory is that bounded Borel measurability does not necessarily imply integrability, indeed an extra "quasi continuous" property is required. And as simple question as whether the indicator functions of open sets are integrable remains unsolved, we will try to give an answer in third part.The first chapter of this thesis is devoted to the study of G-FBSDEs’well-posedness problem arising as a continuation of [52]. To be precise, we are interested in the existence and uniqueness of solution to the system of stochastic differential equations with standard Lipschitz and linear growth conditions upon coefficients as assumed in, (HI.3) (i) Assume for any constants (x’,y) ∈ Rm × Rn, we have So the equation allows for a unique solution Xy. Moreover, we suppose for any (y’,z’) Rn× Rn×d, the generator satisfies(ii) There exist a strictly positive constant k such that for all t ∈ [0,T],The essential result of this chapter is reported in the followingTheorem 0.1. If the Assumption (H1.3) hold, then for an arbitrary small ε> 0, there exists α constant Ck depending only on k such that, equations (0.0.4) admit a unique solution in GG2,ε(0, T) for any T≤Ck(1).Please be noted that the orders of respective spaces for X,Y, Z, K are not homoge-neous, namely the order of those for Z, K is strictly less than those for X,Y. A careful examine of the proof will clarify this point, which is to say, we construct the contraction only for X,Y, and the existence and uniqueness of Z, K in their respective spaces are found indeed due to being dominated in norm by those of X,Y. The readers to whom the statement of G-BSDE theory is familiar should not be surprised.There are a few words about our setting of the problem. We do not include Z in the forward component for there will be infinitely many solutions when only Lipschitz condition imposed on the diffusion coefficient σ as indicated in Remark 1.1. And the global results would be false in general (please refer to Example 1.2). Recalling the work of Delarue [26], we believe hopefully there will be possible global theory if the terminal functional φ is smooth enough. One essential point of his proof relies on the estimate of gradients of solutions to the corresponding partial differential equations, in sense that those estimates will hold for the whole time interval.Under further linear growth conditions, (H1.4) Apart from (H1.3), all functions b,σ,φ,g are deterministic functions and areof linear growth in all their arguments, that is, there is some constant γ such that, we get Y can be defined as a Lipschitz continuous function of state of X, even though there are too many gaps in the direction of Feynmann-Kac formula.Proposition 0.1. Suppose the assumptions (H1.4) hold true, then there is a continuous function u such that for any s ∈ [t, T] and ζ ∈ LG2(Ωt),The second chapter deals with the invariant expectations and ergodic expecta-tions of G-diffusion processes. The classical pointwise ergodic theorem of Markov process theory (see for instance, Birkhoff-Khinchin ergodic theorem) reformulates the Law of Large Numbers for this occasion, which asserts the time average of function along any trajectory converges almost surely to the space average conditioning on the invariant a-algebra. So there is a bond between the invariant measure and ergodic property, however we find these two concepts do not agree in G-framework, so it is necessary to introduce what we called "ergodic expectation" for G-diffusion process. Let us start with standing assumptions for SDEs, where b, hij:Rn'Rn, σ:Rn'Rn×d are deterministic continuous functions. In particular, denote Xx= X0,x.The notion of invariant expectations is adapted from the classical one, which more often describes the equilibrium state of the diffusion process. For a given integer p> 1, Cp,Lip(Rn) denotes all functions defined on Rn if there exists a constant Kf depending Definition 0.1. A sublinear expectation E on (Rn, C2p,LiP(Rn)) is said to be an invariant expectation for the G-diffusion process X ifThere is a weakly relatively compact probability family of probability measures that represents E on (Rn,C2p-1,Lip(Rn)) called invariant for the G-diffusion process X (see theorem 2.2), which can be understood as the uncertainty of initial distribution. Now we can state one of the main theorems of this chapter showing the existence and uniqueness of invariant expectations for diffusion process X,Theorem 0.2. Assume (H2.1) and (H2.2) hold. Then there exists a unique invariant expectation E for the G-diffusion process X. Moreover, for each f ∈ C2P,Lip(Rn), we haveThis theorem confirms why E is called "invariant". As an important example we apply the above theorem to show the G-normal expectation is the invariant expectation of G-OU processes (see lemma 2.5). Of course in this lemma 2.5, larger class of func-tions than C2p,Lip(Rn) can be considered to define the invariant expectation. It will be appealing to try to replace C2p,Lip(Rn) with the lattice of all local Lipschitz functions Cloc,Lip(Rn) for a more general result, the reason we insist using C2p,Lip(Rn) to formulate our problems is that when we are to show the existence of invariant expectations A, there should be some uniform bound on the difference of A[f] with E[f(Xt)] to generate some compactness result, but for all f∈loc,Lip(Rn) there will be troublesome. Apart from this, the reason why we use (Rn, C2p-1,Lip(Rn)) to identify the family of probabilities that represents the invariant expectation also comes from the assumption (H2.2), so that we can get lemma 2.2.Definition 0.2. A sublinear expectation E on (Mn, C2p,Lip(Rn)) is said to be an ergodic expectation for the G-diffusion process X ifSimilarly there is a group of probability measures that represents E which is called ergodic for the G-diffusion process X. It is easy to find out in broad terms this reveals there is "less" uncertainty in the ergodic expectation than the invariant one. In particular, we have example 2.4 constructed from G-OU process to show these two expectations do not always coincide.The third chapter starts with the consideration of Lusin’s theorem, in the G-framework there exists Borel measurable functions that are not quasi surely continuous ( for instance, the density process of G-Brownian motion’s quadratic process as), thus the scope of random variables under investigation are limited to LG1, of which are all quasi surely continuous on Co([0, ∞);Rd). We want to shed a little light on this problem by first showing that G-Brownian motion quasi-surely never hits any single point at fixed (H3.1) There exist two nonnegative constants C and C" such that for each (t, x), (t, x’) e [0, oo) x Rn, (H3.2) For all x ∈ Rn,|b(.,x)|,|hjk(.,x)|,|σ(.,x)|all belong to MG2(0,T), and continu-ous in t. (H3.3) There exist two constants 0<λ<(?)<∞ such that for each (t, x) ∈ [0, ∞)×Rn, (H3.4) There exists a constant L> 0 such that for each (t,x) ∈ [0, ∞) × Rn, (H3.5) There exist two constants 0< γ< Γ<∞ such that for each (t, x) ∈ [0,00) × Rn, The diffusion processes are given by, X represents the unique solution of the SDEs, a first polar set is the main concern of theorem 3.7 as recalled below,Theorem 0.3. Under suitable assumptions (H3.1-H3.3 indeed), we have for each T> 0 Furthermore, we haveThe idea for proving the theorem is to compare the amount of capacity with the behavior of viscosity solution satisfying an exponential terminal condition. Making use of this, a upper bound for the capacity of symmetric G-martingales lying in a cubic is found in corollary 3.2. In a probabilistic manner, we get polar sets out of several specific curves. All these results will make sense once we notice thatIt would immediately imply the following,We emphasize these results are essentially based on PDE arguments, and at least up to now we have no idea how to prove them in probabilistic way.For the second part of this chapter, we study the quasi continuous characterisation of MGP(0,T), which is the analogue of results obtained for LGp(ΩT)0, where p≥ 1. We adapt the concept of quasi continuous for progressively measurable processes,Definition 0.3. A progressively measurable process η:[0, T] × ΩT'R is called quasi-continuous (q.c.) if for each ε> 0, there exists a progressive open set G in [0, T] × ΩT such that c(G)<ε and η|Gc is continuous.So the suitable processes those are integrable under G-expectation are, Theorem 0.6. For each p≥1,Finally we list some necessary conditions for SDEs, so that this equation is well posed. (H3.6) There exists a constant L > 0 such that for each t≥0, (H3.8) There exists a constant (?)> 0 such that for each t> 0,The same Krylov’s estimate is established for G-Ito process, which has bounded drift coefficients and bounded nondegenerate diffusion coefficients, indeed,Theorem 0.7. Assume (H3.6) and (H3.7) hold. Let D be a bounded region in Rn and τ be a stopping time with τ≤τp, where τD is the first exit time of Xt from D. Then for each x0 ∈Rn, T≥0 and p≥n, there exists a constant N depending on p, λ, L, G,T and D such that for each t ∈ [0,T] and all Borel function f(t,x), g(x),Similar inequalities are obtained for infinite time horizon with discounting (see theo-rem 3.16) as well as truncated processes (see corollary 3.4). These estimates will enforce the quasi continuity property for the composition of Borel functions with G-Ito process X.,Theorem 0.8. Assume (H3.6)-(H3.8) hold. If φ is a Rn-valued Borel measurable func-tion of polynomial growth, then for each T> 0, we have (φ(Xt))t≤T∈MG2(0,T).Moreover we obtain the following Ito-Krylov’s formula of G-Ito processes in Sobolev spaces.In particular, we have the following dominated convergence theorem for G-random variables.Theorem 0.10.Assume(H3.6)-(H3.8)hold.Let u,uP∈wp,loc 1,2([0,T]×Rn)with p> (n+2) for P≥1.Moreover,up converges pointwise to u and (?)tup,(?)xixj2 uρ α.e converge to (?)tu,(?)xiu,(?)xixj2 u.If there exist some constants C and l such that... |
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