The Theory Of Weak Solution Of SDE And Properties Of Diffusions Driven By G-Brownian Motion And Their Applications

In the recent more than ten years, motivated by model uncertainty in statistics, measures of risk and superhedging in finance, many kinds of risk measures were intro-duced such as coherent measures of risk introduced by Artzner et.al. (1999), convex measures of risk by Follmer and Schied (2002), law invariant risk measures by Rosazza Gianin (2002), and comonotonic coherent risk measures and comonotonic convex risk measures by Song and Yan (2006). Especially for coherent measures of risk, there have been many significant researches and applications. In 2006, Peng introduced a notion of sublinear expectation which is an expectation with the properties of constant pre-serving, monotonicity, subadditivity and positive homogeneity. It can be corresponded to a coherent risk measure. Under this kind of sublinear expectation he introduced a notion of G-normal distribution via the following nonlinear partial differential equation where u(t, x):[0,∞]×R→R andψis a real-valued, bounded and Lipschitz continuous function defined on real line. (?) where a+=max{0, a}, a-= max{0, -a}, and (?), (?) are nonnegative constants such that (?)≤(?). This G-normal distribution is an extension of classical normal distribution in probability theory and also plays same important role as the classical normal distribution, since Peng (2007a) proved the law of large numbers and central limit theorem under sublinear expectation. Then he defined a process of G-Brownian motion (G-B.M. for short) which can be briefly said to be a continuous process with independent and stationary increments under a given sublinear expectation. This sublinear expectation is called G-expectation. In the framework of G-expectation Peng (2006) (see also Peng (2010) etc.) established the theory of Ito's typed stochastic calculus and stochastic differential equations (SDEs for short). Similar to classical SDEs theory, Peng (2006) gave a G-Ito's formula. Then Peng (2010), Gao (2009) and Li and Peng (2009) generalized this formula. For the following stochastic differential equation driven by G-Brownian motion (G-SDE for short, see Peng (2010)) where T＞0 and x are given, (Bt)t≥0 is a 1-dimensional G-Brownian motion, ((B)t)t≥0 is the corresponding quadratic variation process of (Bt)t≥0, b, h,σ:Ω×[0, T]×R→R are given functions satisfying that forψ= b, h,σ, (A1)ψ(·, x) is in a process space MG2(0, T) for eachχ∈R. (A2)ψis uniformly Lipschitz continuous in x, i.e., there exists a positive constant K such that Peng proved the existence and uniqueness theorem for solution of G-SDE (4) in MG2(0, T). Similar to the stochastic differential equation theory under mathematical expectation, existence or uniqueness for solution of G-SDE (4) with conditions on the coefficients different from conditions (Al) and (A2) above have also been studied (see Lin (2006), Jing (2006), Bai and Lin (2010)). Especially, Gao (2009) proved that the solution of equation (4) is continuous and has many good properties under assumptions (A1) and (A2) with the coefficients independent to t andω. For more about the theory of G-expectation and related theories we can refer to Hu and Peng (2009), Li and Peng (2009) and Bai (2009) etc.In this doctoral dissertation we will focus on the following issues:1. Whether does weak solution of G-SDE (4) uniquely exist under assumptions (A1) and (A2) with coefficients independent toωin the framework of G-expectation?2. Is there a relation between solution of G-SDE (4) and a class of nonlinear partial differential equation under the same conditions as the ones in issue 1?3. Do we have similar nonlinear Dykin's formula under G-expectation as the Dynkin's formula in classical stochastic differential equation theory?4. If we have the nonlinear Dykin's formula, how can we use it for calculation? This doctoral dissertation contains 3 chapters. Chapter 1 recalls basic knowledge of G-expectation, G-Brownian motion, stochastic calculus, Ito's formula, stochastic differ-ential equations, Feymann-Kac formula and other related lemmas which will be useful in the next two chapters. We will study the former two issues in Chapter 2. Chapter 3 studies the latter two issues.Now let us give the main results of Chapter 2 and Chapter 3 in the following.Chapter 2:As is known to us, in the classical theory of stochastic differential equations (see (?)ksendal (1998),(?)ksendal (2003), Mao (1997), Ikeda and Watanabe (1989), Evans (2006), Sobczyk (2001) etc.), when Brownian motion is given in advance we can define a strong solution. And there exists a Tanaka equation (this equation under G-expectation can be referred to Lin (2009b)) which has no strong solution but another solution which is called weak solution (see (?)ksendal (1998), Example 5.3.2). Weak solution is defined when Brownian motion and probability measure are not given in advance. Similarly in this doctoral paper we will study the issues of weak solutions in the framework of G-expectation. First we can give the definitions of strong and weak solutions in the framework of G-expectation. In order to prove the existence and uniqueness of weak solution we introduce a new notion of similarity which is defined to explain the relations of random vectors and stochastic processes in different G-expectation spaces. We prove that the solutions of G-SDE (4) with coefficients independent toωare uniformly similar when G-B.M. and G-expectation are not given in advance. And we also prove the solu-tions are finite-dimension weakly identically distributed when the super variances and sub variances of G-Brownian motions at time 1 are the same respectively. In 2006 and 2010, Peng introduced backward stochastic differential equations under G-expectation (G-BSDE for short) which is the form with conditional expectation but not differential form, because general G-martingale representation theorem has not been worked out till now but the ones for special martingales (refer to Peng (2010), Soner, Touzi and Zhang (2010) and Song (2010)). And Peng proved a nonlinear Feymann-Kac formula which indicates the relation between forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs for short) and a class of nonlinear partial differen-tial equations (PDEs for short) by use of viscosity solutions. Employing the method in the proof of this formula, we can give a relation between distribution functions of weak solution of equation (2.1) and a class of nonlinear partial differential equations. We state the main results of Chapter 2 as follows. Theorem 2.4.1 Under assumptions (H1) and (H2) a weak solution Xt of stochastic differential equation (2.1) with the initial condition X0=χ∈R is weakly unique.Corollary 2.4.2 Under assumptions (H1) and (H2), ifχ∈R, and (?)≥(?)≥0 are given, then the weak solution of equation (2.1) satisfying is weakly unique in the sense that if Xt1 and Xt2 are two weak solutions satisfying our conditions, then (Xt1) and (Xt2) are finite-dimension weakly identically distributed.Theorem 2.5.1 Suppose conditions (H1) and (H2) hold, and we also assume all the coefficients of equation (2.1) are independent to t and bounded by positive constant K. Let process (?) is a weak solution of equation (2.1) with the corresponding G-Brownian motion (Bt)t≥o and G-expectation E. For eachψ∈Cb,Lip(R) we set Then u is a viscosity solution of the following partial differential equation where (?)2 and (?)2 are lower and upper variances of B1 respectively andTheorem 2.5.4 Suppose (H1) and (H2) hold. Let (Ω, LG1(Ω), (LG1(Ω))t>o, E, (Xt0,χ)t∈[0,T], (Bt)t≥0) be a weak solution of equation (2.1). We assume u,(?)2, (?)2 and G are defined the same as the ones in Theorem 2.5.1. Then u is a viscosity solution of the following partial differential equation Chapter 3:In stochastic differential equation theory, Ito's diffusions are very im-portant processes which have significant applications in stochastic controls, financial mathematics and so on (interested readers can refer to Yong and Zhou (1999) etc.). What properties do the diffusions have under G-expectation? This is an interesting problem. We can see that they are still Markovian processes under G-expectation. In this chapter we investigate the infinitesimal generator AG of a G-Ito's diffusion which is defined similar to the case under linear expectation. Under linear expectation it is very obvious from Ito's formula, but since G-expectation is not linear except for the reduced case where the G-Brownian motion has no variance uncertainty. But we can guess the representation form of generator AG at time t, since it is easily to get AG(?)(χ) for each real valued, bounded and Lipschitz continuous function fdefined on real line and any real number x from equation (2.11). We will give the representation of gener-ator AG under the conditions that (?)and the coefficients of the G-SDE which a G-Ito process Xt satisfies are bounded. Then we can extend this theorem to other general cases. Then we will give a special kind of G-martingale and present a property of process f(Xt). And from Theorem 3.3.1 it is easy to get a nonlinear formula which is different from Dynkin's formula under linear expectation and we also give its corollary. Then by use of this corollary we give an estimating interval of IE[(Bt+χ)2n+1] where n is a positive integer and x, t are given real numbers such that t≥0. We state the main results of Chapter 3 as follows.Theorem 3.2.1 f DG and for any given real number x, we haveTheorem 3.2.2 If we remove the bounded conditions on a, b and c and suppose f∈C2(R) satisfying (?) is bounded and Lipschitz continuous instead of(?), then Theorem 3.2.1 still holds.Theorem 3.2.3 We suppose (Xt)t≥0 is a G-Ito's diffusion with boundedness and uniform Lipschitz continuity conditions on the coefficients a, b and c of G-SDE (3.1) which this diffusion satisfies. Then the generator of Xt can be presented the same as before for all functions f∈C2(R) satisfying (?) is polynomially increasing, i.e., there exist a constant C and a positive integer n such that for any real number x,Theorem 3.2.4 Suppose Xt is a G-Ito's diffusion. And we assume f∈C2(R) such that (?) is locally Lipschitz continuous, i.e., for each m≥1 there exists a constant Lm such that for any real numbers x, y with|x|≤m and|y|≤m, and forψ= a,b, c, c2 andψ= (?),processψ(X)ψ(X) is uniformly mean-square continuous, i.e., Then AGf(Xt) has the same representation as before in (?).Theorem 3.3.2 Suppose real functions f∈C2(R) and a G-Ito's diffusion (Xt)t≥0 satisfy any one of the following conditions:(i) (?) is polynomially growing and the coefficients of of G-SDE which Xt satisfies are bounded.(ii) (?). Then the process (Mt)t≥0 defined by is a G-Martingale with super-mean f(x). Further more, we have the following decomposition where Mt is a non-positive martingale in (?) with zero super-mean and Mt is a martingale with zero sub and super means.Theorem 3.4.4 Suppose function f and processes Xt are defined the same as the ones in Theorem 3.3.2, then we have the following results:(1) f(Xt) is a G-martingale (resp. G-supermartingale, G-submartingale) if and only if (2) f(Xt) is a G-martingale with no mean uncertainty if and only if one of the following conditions holdsCorollary 3.4.5 Suppose function f and process (Xt)t≥0 satisfy the same conditions as the ones in Theorem 3.3.2, then process Mt defined as in Theorem 3.3.2 is a G-martingale with no mean uncertainty is equivalent to any one of the following conditions(1) For any t≥0, Mt=0, in (?).(2)σ= 1 or for any t≥0, (?).Theorem 3.5.1 (nonlinear Dynkin's formula) Suppose real functions f∈C2(R) and a G-Ito diffusion (Xt)t≥o with initial condition X0=χ∈R satisfy any one of the following conditions:(i) (?) is polynomially growing and the coefficients of G-SDE that Xt satisfies are bounded.(ii)(?). Then for any t≥0, we have where is a non-positive G-martingale with zero super-mean.Corollary 3.5.3 Process Xt and functions f are the same as the ones in Theorem 3.5.1, then we have for any t≥0, where the equalities of both sides hold whenσ=1 and the right equality holds when Mt defined the same as the ones in Theorem 3.5.1 has no mean uncertainty.Theorem 3.5.4 For any given integer n≥1 and any nonnegative real number t, we have the following estimating inequalities...