The Convergence Property Of Solutions To Backward Doubly Stochastic Differential Equation

Backward doubly stochastic differential equation was introduced first by E.pardoux and Peng Shige .This is another most important results after Backward stochastic differential equation.The convergence theorems of Backward stochastic differential equation and Backward doubly stochastic differential equation have been proved out.But the proved convergence theorem only considered the terminal value {ξ~n} is cauchy sequence case.In my paper,I consider the case whose terminal value {ξ~n} and coefficients {f~n} , {g~n} are Cauchy sequences ai the same time.All the results are finished by author under the guidance of professor Shi Yufeng.this article has four chapters.Chapter 1:IntroductionThis Chapter introduces the development history of Backward stochastic differential equation and Backward doubly stochastic differential equation,the convergence property's development process.Chapter 2:I research the convergence property to Backward doubly stochastic differential equation in finite time interval.Let us introduce the sufficient condition:Assumed there exist constants c > 0 , 0 < α < 1 such that for any (ω, t) ∈ Ω * H(2.3)Given ,We considered the following class equations: (2.2) n= 1,2,3..... satisfying(H2.1)-(H2.3).Theorem 2.3.1As n tends to .There exists such that . Furthermore (y_t,Z_t) is solution of the following equation.I prove the theorem by three steps,that is,three propositions.Proposition 2.3.2 Given any n,under the conditions (H2.1)-(h2.3),The solution pair (y_t~n,z_t~n) to Eq.(2.2) has the following uniformly estimation.Where K is a constant that has nothing with n.Proposition 2.3.3 Under the conditions of theorem 2.3.1, (y_t~i,z_t~i) is Cauchy sequences in S~2 * M~2 for each other.Proposition 2.3.4 Under the conditions of theorem 2.3.1 and Eq.(3.2).we have the following resultsChapter 3:I consider the convergence property to Backward doubly stochastic differential equation in infinite time interval.Let T = oo ,1 borrow the sufficient condition from Han and Shi[3]:coefficient g is independent of variable z.(H2.3) is transformed into : (H3.2)There exist non-negative and integrable function v(t),u(t) such that (￡,?/*, Zi) 6 Rk*Rk*d*R+,i = 1,2. < v{t)\yi - y2\ + u(t)\\Zl - z2\\\9(t,yu)-g(t,y2,)\ oo,with fn -f f,gn -> ^^n -?￡2.Then we have yt ￡ S2, zt ￡ M2 such that y" —> t/t,2" —> z(.moreover (yt,zt) is thesolution pair to(3.1) yt = <￡ + /t°° /(s, y8, zs)ds + /t°° 5(5, ys)d5s - /t°° z,dW^8,0 < t < oc By using the continuous dependence theorem ,1 prove the theorem.Chapter 4:I consider the convergence property to reflected backward doubly stochastic differential equation.Reflected backward doubly stochastic differential equation has a reflecting barrier {St}.As soon as process {yt} come into contact with {St},another continuous and increasing process {Kt} can let {yt} back.With regard to Reflected backward doubly stochastic differential equation ,1 get the existence and uniqueness theorem and the representation to solution triple independently,but it's a great pity that my tutor and his copartner had obtained it before me.Let's see the conditions: (i) ￡ € V .(ii) V(y, z) G R x Rd, /(-, y, z) G M2, g(-, y, z) G M2.(iii) There exists constants K>0, 0 < a < 1 , Vy, y' e R, z, z' e Rd. a.s.,a.e.And the last one is an "obstacle" {St, 0 < t < T}, which is a continuous progressively measurable real-valued process, St is Tt measurable, satisfying(iv) ￡{supo St, 0St,O