Forward-backward Estimation, With The Differential Equations

BSDEs have been studied intensely in recent years, motivated in large by applications from mathematical finance and mathematical economics; for example, large investor problems, stochastic differential utility and contingent claim replication. Such equations were introduced by Bismut for the linear case in the year 1973, by E. Pardoux and S. Peng in 1990 in the general case, where they proved an existence and uniqueness result.As to SDEs, we already have a large quantity of methods to estimate the corresponding models. The methods include parametric methods and nonparametric methods. However, there are no systematical theories to estimate a BSDE model. BSDEs and SDEs have their essential difference. We need the methods to estimate BSDE models specially. For a BSDE, the generator is the decisive factor. To estimate a BSDE model is actually to estimate the generator.In this paper, we will lay emphases on it and try to use similar methods of expected return and volatility estimation to estimate the BSDEs. We consider certain BSDEs associated with some forward classical stochastic differential equations, that is so-called FBSDEs, forward-backward stochastic differential equations:where X_t satisfyOne proposition of FBSDEs is called Generalization of the Feynman-Kac Formula. Mainly based on this proposition of FBSDEs, we can easily conduct to the approximations of g and Z:and...