Homeomorphism Flows For Stochastic Differential Equations And Applications

After the fundamental works of Ito and Gihman in 1940s, the theory of stochastic differential equations (SDEs) has been studied extensively. The flow property of solutions of SDEs was studied around 1980 by Elworthy, Malliavin, Bismut, Ikeda-Watanabe. Kunita, Meyer etc. It was proved that under Lipschitz coefficients, the solutions of SDEs driven by Brownian motion or a continuous semimartingale form a stochastic homeomorphism flow. Further, when the coefficients are smooth, the solutions form a stochastic diffeomorphism flow.In this thesis, we mainly study the homeomorphism flow properties of solutions for SDEs with jumps and backward stochastic differential equations (BSDE), and also give some applications.First consider SDEs with jumps. Let (U,(?),v) be aσ-fmite measure space. We fix U₀∈(?) such that v(U - U₀) <∞. Assume that the continuity modulus of the drift coefficient b(t, x) and the diffusion coefficientσ(t, x) in x are respectively |x| log(|x|^-1+e) and |x| log^1/2 (|x|^-1 + e). Moreover, we also suppose that for some q > (2d)∨4 and any p∈[2, q], the continuity modulus for the coefficient of small jumps f(t, x, u) with respect to x in L^p(U₀,(?),v) is |x| log^1/p(|x|^-1 + e) and the norm of f(t,x,u) in L^p(U₀,(?), v) is dominated by a linear function in x; the coefficient of large jumps g(t, x, u) is continuous in x. We first prove some moment estimates for the solutions of SDEs by Bihari's inequality, and then apply Kolmogorov's criterion to obtain the continuity of solutions with respect to the initial data. On the other hand, we further assume that f(t,x,u) is Lipschitz continuous in x with the Lipschitz constant L(u) and f(t,0,u) is dominated by L(u), which belongs to L²(U₀,(?), v) and is bounded on U₀ by 0 <δ< 1 satisfying that for some q > Adx(?)x+g(t,x,u) is a homeomorphism mapping. Using Gronwall's inequality, we first prove some moment estimates with the negative index, and then apply Kolmogorov's criterionto obtain the homeomorphism of solutions. We also apply our result to stochastic differential equations driven by L(?)vy processes, and compare it with the well-known results such as Fujiwara-Kunita and Protter. We find that our result is more general.To BSDEs, if the terminal conditionξis replaced by a family of random variablesξ(x,ω), which depends on a parameter x∈R such that x(?)ξ(x,ω) is a homeomorphism a.s., we prove that the solutions of BSDEs x (?) Y_t^ξ(x) form a homeomorphism. When the generator f(s,ω, y, z) is Lipschitz continuous in y, z, by the monotonicity of x (?)ξ(x,ω) and a generalized comparison theorem of BSDEs we show that x (?) Y_t^ξ(x) is continuous and one-to-one. If for some R₀ > 0 andε> 0.where y(x) is a real continuous function on R satisfying (?) g(x) =±∞(or(?)g(x)=(?)∞). The proof of onto property for the first theorem is based on thecomparison theorem, which is used to compare from above and below the solutions of BSDEs with a backward ordinary differential equation. And if for someα< (?)∧0,the proof of onto property for the second theorem is completed by supremum and infimumlimits. Next, we apply the above results to a backward stochastic differential equation coupled with a forward stochastic differential equation, and obtain the homeomorphismof its solution. Since it is related with a quasilinear deterministic parabolic partial differential equation, we also get that the solution for the latter equation is a homeomorphism.