| This thesis consists of four main topics:the first one is to study scalar valued reflected stochastic differential equations driven by G-Brownian motion (RGSDE); the second one focus on the wellposedness of some types of non-Lipschitz stochastic differential equations driven by G-Brownian motion (GSDEs); in the third one, multidimension-al G-diffusions are considered; the last one contribute to the solvability of a class of second order backward stochastic differential equations (2BSDEs) with bounded terminal values and quadratic generators. In fact, the first three topics are discussed in a framework of a type of nonlinear expectation, i.e., the G-expectation introduced by Peng [58-60]. The last topic aims to generalize the theory of2BSDEs in the framework established by Soner et al.[71,72]. This theory is highly related to the G-expectation. The common point of these four topics is:stochastic differential e-quations (SDEs) are considered no longer under one probability like in the classic framework, but under a class of probabilities, which could be mutually singular and thus, such SDEs can find their applications to solve the problems with volatility un-certainty in finance.Let us be a bit more precise and explain the organization of the manuscript.In Chapter1, the notion of stochastic integrals with respect to an increasing process are established in the G-framework, and then an extension of G-Ito’s formula is given. Subsequently, a priori estimates for RGSDEs with Lipschitz coefficients are deduced and the existence and the uniqueness of solutions are proved. At the end of this chapter, the corresponding comparison theorem is presented.This chapter is mainly based on the work:Lin, Y.:Stochastic Differential Equation-s Driven by G-Brownian Motion with Reflecting Boundary Conditions. Electronic Journal of Probability18:no.9.1-23,2013. In Chapter2, as the start point of our work, some results of Li and Peng [40] are re-called, especially their idea of defining the G-Ito type integrals for the processes that are locally integrable with respect to the G-Brownian motion. In the fourth section, the notion of stochastic integrals with respect to an process of bounded variation are established and an extension of G-Ito’s formula is given. In the fifth section, the solvability of two classes of non-Lipschitz GSDEs are investigated.In Chapter3, multidimensional reflected G-diffusions are constructed by a penaliza-tion method in a convex domain and some convergence results are obtained for the case that coefficients in such equations are bounded and Lipschitz.In Chapter4, quadratic2BSDEs with bounded terminal value are discussed. In the third section of this chapter, a representation theorem and some a priori estimates are provided and thus, the uniqueness result is deduced. In the fourth section, the solutions to such2BSDEs are constructed pathwisely. In the last section, these the-oretic results are applied to solve some robust utility maximization problem with non-dominated models.This chapter is mainly based on the work:Lin, Y.:A New Result for Second Order BSDEs with Quadratic Growth and its Applications. arXiv:1301.0457, submitted. Here is an overview of the main results of this dissertation.1. Scalar valued Reflected GSDEsIn this chapter, main results are obtained on the space MGp([0,T]), for some p>1. The definition of this space can be found in Peng [58-60].In a scalar valued RGSDE, there is an increasing process appearing in the dynamic, which pushes the solution upwards (or downwards), so that the solution can always remain above (or below) the obstacle. Thus, to study such equations, we need to first establish the notion of stochastic integrals with respect to an increasing process in the G-framework. This notion are presented by the following definitions:Definition1. We denote by Mc([0,T}) the collection of all q.s. continuous processes X whose paths X.(w):t'Xt(w) are continuous in t on [0,T] outside a polar set A.Definition2. We denote by M1([0, T]) the collection of q.s. increasing processes K∈Mc([0,T]) whose paths K.(w):t'Kt(w) are increasing in t on [0.T] outside a polar set A. Then, the stochastic integrals with respect to an increasing process can be defined in the sense of Riemann-Stieltjes path by path:Definition3. We define, for a fixed X∈Mc([0, T]), the stochastic integral with respect to a given K E MI([0, T]) bywhere A is a polar set and on the complementary of which, X.(ω) is continuous and K.(ω) is continuous and increasing in t.For a process n that belongs to MGp([0, T]), where p≥1, in some special case, one can find ap’ E [1，p] such that the stochastic integral of η with respect to an increasing process is in the space LGp([0, T]), for example:Proposition1. Let K∈M1([0,T]) n MG2([0,T]), KT E LG2(ΩT) and φ: R'R be a Lipschitz function, then∫0Tφ(Kt)dKt is an element in LG1(ΩT).Proposition2. Let X be a q.s. continuous G-Ito process such that where f, h and g are elements in MGp([0, T]), p>2. Let K∈M1([0, T])∩MGq([0, T]) and KT∈LGq(ωT) where1/p+1/q=1. Then,∫0T XtdKt is an element in LG1(QT).In Peng [60], a G-Ito’s formula is given, which can be used to treat with G-Ito processes with bounded coefficients. In this chapter, a corresponding result is obtained for the process that consists of a G-Ito process and an increasing process K, which is of the following form:To obtain the desired result, we first consider a simple case, that is, the function Φ∈C2(R) and all the derivatives of Φ arc bounded and Lipschitz.Lemma1. Let Φ∈C2(R) be a real function with bounded and Lipschitz derivatives. Let f, h and g be bounded processes in MG2([0,T]) and K∈MI([0,T])∩MG2([0, T]) satisfy for each t∈[0, T], Then,Notice that in order to have such a G-Ito’s formula, some additional condition (2) on the regularity of K is adopted, this makes the G-Ito’s formula above be different from the classical one. The purpose of this condition is to ensure that A’ can be approximated in MG2([0, T]) by a sequence of simple processes constructed by K itself. Subsequently, the result above is generalized:Theorem1. Let Φ∈C2(R) be a real function such that d2Φ/dx2satisfies a, polynomi-al growth condition. Let f, h and g be bounded, processes in MG2([0, T]) and K€M1([0,T])∩MG2([0,T]) satisfies that for each t∈[0,T], and for any p>2, E[KTp]<+∞. Then. In the rest part of this chapter, we first study the reflected G-Brownian motion via the solution of a deterministic Skorokhod problem. We prove the existence and the uniqueness of such a process:Theorem2. For any p>1. there exists a unique pair of processes (X, K) in MGp([0,T])×(MI([0/T])∩MGp([0,T])), such that Xt=Bt+Kt,0≤t≤T, q.s., where (a)X is positive;(b) K0=0; and (c)∫0TXtdKt=0. q.s If the G-Brownian motion B is replaced by some G-Ito proccss in MGp([0,T]), where p>2, a similar statement holds true: Theorem3. For some p>2, consider a q.s. continuous G-Ito process Y defined in the form, of (1) whose coefficients are all elem.en.ts in MGp([0, T]). Then, there exists a unique pair of processes (X K) in MGp([0,T])×(MI([0,T])∩MGp([0, T])) such that Xt=Yt+Kt,0≤t≤T, q.s., where (a) X is positive;(b) K0=0; and (c)∫0T XtdKt=0, q.s..Then, we consider the RGSDE of the following form: where(A1) The initial condition x∈R;(A2) For some p>2, the coefficients/, h and g: Ω×[0, T]×R'R are given functions that satisfy for each x∈R,f.(x),h.(x) and g.(x) lb MGp[0, T]);(A3) The coefficients/, h and q satisfy a Lipschitz condition, i.e., for each t∈[0, T] and x, x’∈R,|ft(x)-ft(x’)|+|ht(x)-ht(x’)|+|gt(x)-gt(x’)|≤CL|x-x’|, q.s.;(A4) The obstacle S is a G-Ito process whose coefficients are all elements in MGp([0, T]), and we shall always assume that S0≤x, q.s..The solution to the RGSDE (3) is a pair of processes (X, K) that take values both in R and satisfy:(ⅰ) X∈MGp([0,T]) and Xt> St,0≤t≤T, q.s.(ⅱ) K∈MI([0,T])∩MGp([0,T]) and K0=0, q.s.;(ⅲ)∫0T(Xt-St)dKt=0, q.s.Using the solution of a deterministic Skorokhod problem, it is easy to have the repre-sentation of the increasing process K and then to obtain the following two important a priori estimates:Proposition3. Let (X, K) be a solution to (3). Then, there exists a constant C>0such that Theorem4. Let (x1,f1,h1,g1, S1) and (x2, f2, h2, g2, S2) be two sets of coefficients thai satisfy the assumptions (Al)-(A4) and (Xi, Ki) the solution to the RGSDE cor-responding to (xi, fi,hi,gi,Si), i=1.2. DefineAx:=x1-x2, Δf:=f1-f2. Δh:=h1-h2, Ag:=g1-g2; AS:=S1-S2, AX:=X1-X2. AK:=K1-K2. Then, there exists a constant C>0such thatAs a result of the last a priori estimate, the RGSDE (3) admits at most one solution in MGp([0,T])×(M([0,T)∩MGp([0,T])).Afterwards, the existence of solution to the RGSDE (3) can be proved by Picard iteration. In summary, the following theorem of existence and uniqueness is obtained:Theorem5. Under assumptions (A1)-(A4), there exists a unique solution in MGp([0, T})×(MI([0,T))∩MGp([0,T])) to the RGSDE (3).At the end of this chapter, applying the G-Ito’s formula presented previously, the following comparison theorem is proved:Theorem6. Given two RGSDEs that satisfy the assumptions (A1)-(A4), we addi-tionally suppose in the following:(1) x1<x2and g1=g2=g;(2) for each x∈R, ft1(x)≤ft2(x) and ht1(x)≤h2(x); and St1≤St2,0≤t≤T, q.s. Let (Xi,Ki) be. a solution to the RGSDE with data (xi,fi,hi,g,Si), i=1,2. then Xt1≤Xt2,0≤t≤T. q.s..2. Localization Methods for GSDEsIn this chapter, we first recall the definition for the processes that are locally integrable with respect to the G-Brownian, and then we introduce in detail the idea of Li and Peng [40] of defining the G-Ito type integrals for those processes.Subsequently, a new space MFV([0,T];R n) is defined, i.e., the space of all the n-dimensional processes whose paths are continuous and of bounded variation q.s.. By a similar method in the last chapter, the notion of stochastic integrals with respect to a bounded variation process is given in the G-framework. Having this notion, we follow the procedure of Li and Peng [40] to consider a more general form of G-Ito’s formula, where Φ is only required to be in the class C1,2([0, T]×Rn) and X is the sum of a G-Ito’s process and a bounded variation process: We prove this result in two steps: firstly, we assume that Φ and its derivatives are all bounded and uniformly continuous, and K∈MFV([0.T];Rn) is bounded; secondly, we define a sequence of stopping times {μm}m∈N, such that Φ(X) and K can be approximated by {Φ(X^μm)}m∈N and {K^μm}m∈N, respectively, where the elements in these two sequences satisfy the conditions in the first step.The main result of this part consists of a lemma and a theorem as follows:Lemma2. Let0≤t≤T, Φ∈C1,2([0,T]×Rn) such that,(?)Φ,(?)xΦ and (?)xx2Φ are bounded and uniformly continuous and X be given in the form of (4), where fv and hvij’ are elements in M*1([0,T]), gvj is an element in M*2([0, T]),v=l,...,n, i, j=1,..., d, and K is a uniformly bounded element in MFV([0, T];Rn) satisfying that for a positive constant a and each0≤u1≤T, Then,Theorem7. Let ΦC1,2([0,T]×Rn) and X be given in the form of (4), where fv and hvij are elements in Mw1([0,T]), gvj is an element in Mw2([0,T]), v=1,...,n, i, j=1,..., d, and K is an element in MFV([0,T]);Rn) satisfying that for a positive constant a and each0<u1<T, Then,(5) holds. In particular, the G-Ito type integral in the formula above is defined as a generalized G-Ito type integral given by Li and Peng [40] and the other stochastic integrals are denned in the Lebesgue-Stieltjes sense.In the rest part of this chapter, we consider the following multidimensional GSDE:We assume that all the coefficients in the equation above are Lipschitz, then we can prove the following existence and uniqueness result:Theorem8. We assume that the following conditions hold:(H1) For some p>2and each x∈R, fv(·,x), hijv(·,x),.gjv(·,x) G M*p([0,T]), v=1....,n, i,j=1,...,d;(H2) The. coefficients f,h, and g are uniformly Lipschitz in x, i.e., for each t G [0. T] andx,x’∈Rn,where||·||is the Hilbert-Schmidt norm of a matrix.Then, there is a unique process X G M*p([0, T]; Rn) that has t-continuous paths and satisfies (6). For two initial values x, y∈Rn, let Xx and Xy be two solutions of (6) respectively with the initial values x and y, then there exists a constant C>0that depends only on p, n, T and CL, such thatThanks to Li and Peng [40] who open the door for localization methods in the G-frarnework, we can consider some GSDE with weaker assumptions, for example, the GSDE with non-Lipschitz coefficients, whose solution may be constructed via ap-proximation by solutions of Lipschitz GSDEs. In this chapter, we discuss two types of non-Lipschitz GSDEs: in equations of the first type, Lipschitz constants of their coefficients depend on t, while equations of the second type are time-homogenous and their coefficients are locally Lipschitz with respect to x and satisfy a condition of the Lyapunov type.Theorem9. We assume that the following conditions hold:(H1’) For some p>2and each x∈R,fv(·.x), hijv(·,x), gjv(·,x)∈Mwp([0,T]) with respect to a common sequence of stopping times {σm}m∈N, v=1,..., n, i, j=1...., d: (H2’) Outside a polar set A, the coefficients f, h and g are locally Lipschitz in x, i.e., for each t e [0, T] and x, x’∈Rn,|f(t,x)(w)-f(t,x’)(w)|+||h(t,x)(w)-h(t,x’)(w)||+||g(t,x)(w)-g(t,x’)(w)||≤Ct(w)|x-x’|,where C is a positive process whose paths C.(w) are continuous on [0, T] outside the polar set A.Then, there exists a unique process in X∈Mwp([0, T]; Rn) that has t-continuous paths on [0,T] satisfying (6).Theorem10. We assume that the following conditions hold:(H2") The coefficients fv, hijv and gjv: Rn'R are deterministic functions, v=1,..., n, i,j=1,..., d, such that f, h and g are locally Lipschitz in x, i.e., for each x, x’∈{x:|x|<R}, there exists a positive constant CR that only depends on R, such that|f(x)-f(x’)|+||h(x)-h(x’)||+||g(x)-g(x’)||<CR|x-x’|;(H3") There exists a deterministic Lyapunov function V E C1,2([0. T]×Rn) satisfying V≥1, such that and there exists a constant CLY≥0, such that for all (t,x)∈[0, T]×Rn,(?)V(t,x)≤CLYV(t,x), where (?) is a differential operator defined by(?)V(t, x):=(?)tV(t, x)+(?)xvV(t, x)fv(x) in which Then, there exists a unique process X∈Mwp([0, T];Rn) that has t-continuous paths on [0,T], and the following estimate holds: E[V(t,Xtx)]<eCLYTV(0,x). 3. Reflected Multidimensional GSDEsIn this chapter, the main result are obtained on the space M*2([0, T]). The definition of this space can be found in Li and Peng [40].We consider the following reflected multidimensional GSDEs in a given open and convex domain (?):where K is a bounded variation process, which ensures that X stays in the closure of (?).Let p=2and the coefficients f, h. and g of (7) satisfy (H1) and (H2) in the last chapter. We say that a couple of processes (X, K) solves the reflected GSDE (7) if(i) X and A’ are M*2([0,1];Rn) processes whose paths are continuous on [0,1] out-side a polar set A;(ii) For ω∈Ac, X.(ω) takes values in (?), K.(ω) is of bounded variation on [0,1] and K0(ω)=0;(iii) Z is a process satisfying that for ω∈Ac,Z(ω) takes values in (?) and is contin-uous, then for any t∈[0.1],Beside (H1) and (H2), we assume moreover (H3) in this chapter:(H3) For all (t,x)∈[0,1]×Rn,fv(·,x), hijv(·,x) and gjv(·,x), v=1,...,n,i,j=1,...,d, are uniformly bounded.Proceeding similarly the procedures in Menaldi [50], we construct the following se-quence of G-diffusions, the element of which is a solution of a Lipschitz GSDE, which comprises a term of penalization:We prove the following two convergence results and obtain the existence and the uniqueness of the solution to the n-dimensional RGSDE (7): for any p≥1. Theorem11. Assume that (H1)-(H3) hold, then (7) admits a unique solution (X, K)∈M*2([0,1];Rn×(MFV([0. l];Rn)∩M*2([0, l];Rn)).4. Quadratic Second Order BSDEsThe main results of this chapter are established in the framework of quasi-sure s-tochastic analysis introduced by Soner et al.[70,71].2BSDEs are first investigated by Cheridito et al.[7] and by Soner et al.[71,72], afterwards, Possamai and Zhou [G3] have studied the wellposedness of these equations when the coefficients have a quadratic growth. However, in order to overcome some technique difficulties, they adopt a condition that is not usual and thus, the applicability of such equations are limited. The objective of this chapter is to remove some additional conditions that they adopted and to obtain a more general result on this topic.In this chapter, we consider the following equation with a class pH of probabilities that are mutually singular:where PH-q.s. means that the equation above holds P-a.s. for each P∈PH.Besides, paths of K are non-decreasing, P-a.s., for each PH-q.s., and K satisfies a minimal condition.For the wellposedness of the2BSDE (8), we assume the following assumptions on the generator F:Ω×[0,l]×R×Rd×DFt'R:(A1) DFt(y,z)=DFt is independent of (w,y, z);(A2) F is F-progressively measurable and uniformly continuous in ω;(A3) F is continuous in (y,z) and has a quadratic growth, i.e., there exists a triple (α, β,γ)∈R+×R+×R+, such that for all (w, t, y, z,a)∈Ω×[0,1]×R×Rd×DFt,(A4) F is uniformly Lipschitz in y, i.e., there exists α μ.>0, such that for all (ω, t, y,y’, z, a)∈Ω×[0,1]×R×R×Rd×DFt,|Ft(ω,y,z,a)-Ft(w,y’, z, a)|≤μ|y-y’| (A5) F is local Lipschitz in z, i.e., for each (ω, t, y, z, z’, a)∈Ω×[0,1]×R×Rd×Rd×DFt,|Ft(ω,y,z,a)-Ft(ω,y, z’,a)|C(1+|a1/2z|+|a1/2z’|)|a1/2(z-z’)|.We notice that, in the classical framework, we need some similar conditions to obtain the wellposedness of quadratic BSDEs (cf. Kobylanski[35] and Morlais[51]), and these assumptions that we adopted are weaker than those in Possamai and Zhou [63].Similarly to the results in Possamai and Zhou [63], assuming that the terminal value ξ is uniformly bounded, we deduce a representation theorem for solutions of the2BSDE (8) with respect to solutions of classical BSDEs. Also, some a priori estimates are established and thus, the uniqueness is obtained.For the existence of a solution to the2BSDE (8), we first prove the following lemma:Lemma3. Assume (A1)-(A5) hold. For a given ξ∈LH∞and a fixed P∈PH, we have, for each t∈[0,1] and P-a.s. ω∈Ω, ytP(1,ξ)(ω)=ytPt,w(1,ξ).Both sides of (9) in the lemma above are solutions of classical BSDEs. Under a Lipschitz condition on the generators, this lemma can be easily proved by Picard iteration. When generators have a quadratic growth, we can pathwisely apply the monotone stability theorem for classical quadratic BSDEs to prove this lemma.Then, we first suppose the terminal value ξ is uniformly continuous in ω and uniformly bounded, and we pathwisely define the following process: for each (ω, t)∈Ω×[0,1],Following the procedures in Soner et al.[72], we verify that defines a solution of the2BSDE (8).Secondly, we make use of a priori estimates to generalize the existence result and have the main result of this chapter:Theorem12. Under (A1)-(A5) and for a given ξ∈LH∞, the2BSDE (8) has a unique solution (Y，Z)∈DH∞×HH2. At the end of this chapter, we re-consider the financial problems in Matoussi et al.[49] by using the theoretic results of quadratic2BSDEs obtained previously, and we re-solve some robust utility maximization problems with non-dominated models. Since the wellposedness result of quadratic2BSDEs is generalized, i.e., the existence and the uniqueness of solutions no longer depend on some conditions that are difficult to verify, thus the financial problems that we deal with is more practical than the ones in Matoussi et al.[49]. |
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