Difference Methods Of Backward Stochastic Differential Equations And Their Applications In Pricing Of Contingent Claim

In recent years, the theory of Backward Stochastic Differential Equations (BSDEs) has been developed rapidly and is applied to finance extensively. At the same time, the research on numerical methods of BSDEs lags behind. Since the solutions of BSDEs can rarely be solved analytically, the numerical methods for BSDEs are indispensable and significant in theory and applications, meanwhile, there are so many scholar have already make great efforts on that.In numerous numerical methods, one of them used the discretization of filtration and the Euler difference method, gave the Scheme (1) underside:In this paper we gave a more commonly discrete version of BSDEs,then we proved the existence and uniqueness of the discrete solution, the convergence of the discrete solution, and the stabilization of the discrete solution. Furthermore the proposed numerical methods are applied to discuss financial models to validate its availability.For the discrete equation (2), ifξnand f (t , y , z )satisfy the underside conditions: measurable, and exist measurble functionψprovide n). Theorem 1: if assumption H '1 and H '2 is feasible and 1 >αhC, then discrete equation (2) has uniqueness adaptive solution , also measurable.Of course this discrete equation does not only suit for the ibid equation, it can also apply to the equation,which could transform to the ibid equation.for example,the underside equation:If the function F at a rectangular neighborhood D with P ( z 0 , y 0 , z 0) as the centre, is given by: 1) Fz ' , Fy ' ,Fz 'is continuous in neighborhood D Then function g is expressed as z = g ( z , y) is exist, in point Q ( z 0 , y 0 , z 0)'s neighborhood G . So the equation transform to:Integrable martingale z is so hard to simulate in commonly discrete BSDEs, it also brings difficulties in the prove of the convergence and stabilization of the discrete solution. So thereinafter we only discuss the instance,that f (t , y , z ) is the linear function of z , or its does not include z . Its discrete equation is:Then we obtain the theorem 2:Theorem 2: { y kn }0≤k≤n?1 , { y ' nk }0≤k≤n?1 are the discrete solution of equation (4) with ultimately conditionξn,ξ'n. Then exist a positive integer M ,which is independent of n ,and For all k ,the underside expression is legitimate:Theorem 3 when is the solution of equation (3), as the limiting case asMoreover, this methods can be extended, in equation (2), {ξin }1≤i≤n is an independent identical distribution,we can {ξin }0≤i≤n,it is not necessary,but can make the calculation easier. So we just let {ξin }1≤i≤nas independent distribution andξ0 = 0, Eξi = 0, Dξi= 1, according to the prove above,we can obtain all conclusions alike. It like use Mento-Carlo to simulate Brownian motion,but this methods could get the ultimately condition easier.If we useαk instead ofαto different k in equation (2), then equation (2) is transform to: {αk} is fixup, there is one type of discrete equation, in this case {(αk ),(ξin)} is fixup, the discrete equation is fixup, as equation (2),we can obtain the existence and uniqueness of the discrete solution, the convergence and the stabilization of the discrete solution.In the end, we discuss the application of BSDEs in finance. By pricing contingent claims in complete market, especially option pricing theory, introduce the Black-Scholes formula, at last we give an example to discuss the validity of the proposed numerical methods。...