A Class Of Quadratic BSDEs Driven By Gaussian Process

Backward stochastic differential equation(BSDE)is a new field in the theory of stochastic differential equation.In practical application,it can solve the prob-lem of how to set the initial state in order to achieve the expected goal,which is in the core position in physics,life science,finance,computer science,etc.Based on the theory of stochastic differential equations,this paper proposes a class of BSDE models and their solutions,which provide new research tools for dealing with many financial and stochastic control problems.Based on the work of Pardoux E,Peng S G,Kobylanski M,Bender C,this paper proposes a kind of backward stochastic differential equation model driven by Gaussian process,and proves the existence and uniqueness of the solution of the equation by combining the previous research results.The theory of this kind of equation provides a new way for the study of the theoretical problems of the solutions of partial differential equations in stochastic control and mathematical finance.In the third chapter,this paper introduces the basic definitions and theo-rems of Brownian motion and fractional Brownian motion,Gauss process,and martingale,which provides theoretical support for the derivation of the following content.In this paper,we mainly study a kind of equations.In chapter 4,we propose a class of quadratic backward stochastic differential equations driven by Gaussian process:Where X is a kind of central Gaussian process,which includes fractional Brownian motion of Hurst parameter of H?(0,1)and V(t)is the variance of Gaussian process and strictly increases,the last integral is the Wick-It? integral.To prove that the equation has a unique adaptive solution,the following auxiliary BSDE should also be introduced:Where (?) is the standard Brownian motion in the filtering probability space((?)_t?[0,V(T)],(?)).Christian Bender extended the S transformation method of Wick-It?integra-tion from fractional Brownian motion to a more general Gaussian process in their previous research,and explained the connection with Wick-Riemann sum method.Based on the previous research,we considered the case that fis quadratic with respect to y and z.By introducing the auxiliary BSDE,we first make a smooth truncation function,use the It?formula,and then use the lemma mentioned in Chapter 4 to prove the existence and uniqueness of the auxiliary BSDE solu-tion;then we use the wick-It?integral transformation theorem to prove that the original BSDE has only one adaptive solution.