Approximation of BSDE with Hidden Forward Equation and Unknown Volatility

TL;DR

This paper develops a method to approximate solutions of backward stochastic differential equations (BSDEs) with hidden forward equations and unknown volatility, using estimators and PDE solutions, accounting for Gaussian noise.

Contribution

It introduces a novel approximation approach for BSDEs with hidden forward equations and unknown volatility, combining parameter estimation and PDE solutions.

Findings

Effective approximation of BSDE solutions under noise conditions

Construction of a one-step MLE-process for unknown parameters

Error analysis in various metrics

Abstract

In the present paper the problem of approximating the solution of BSDE is considered in the case where the solution of forward equation is observed in the presence of small Gaussian noise. We suppose that the volatility of the forward equation depends on an unknown parameter. This approximation is made in several steps. First we obtain a preliminary estimator of the unknown parameter, then using Kalman-Bucy filtration equations and Fisher-score device we construct an one-step MLE-process of this parameter. The solution of BSDE is approximated by means of the solution of PDE and the One-step MLE-process. The error of approximation is described in different metrics.