On Approximation of the Backward Stochastic Differential Equation

TL;DR

This paper develops an asymptotically efficient approximation method for solutions of backward stochastic differential equations with small diffusion coefficients, using a one-step maximum likelihood estimator for unknown parameters.

Contribution

It introduces a novel approximation approach for BSDE solutions in the small noise regime, leveraging one-step MLE for unknown parameters.

Findings

The approximation is asymptotically efficient in small noise asymptotics.

The method effectively estimates solutions of BSDEs with unknown parameters.

The approach improves accuracy over existing methods in the small noise setting.

Abstract

We consider the problem of approximation of the solution of the backward stochastic differential equation in the Markovian case. We suppose that the trend coefficient of the diffusion process depends on some unknown parameter and the diffusion coefficient of this equation is small. We propose an approximation of this solution based on the one-step MLE of the unknown parameter and we show that this approximation is asymptotically efficient in the asymptotics of "small noise".