Recent Advances in Nonlinear Filtering with a Financial Application to Derivatives Hedging under Incomplete Information

Abstract

In this chapter, we present some recent results about nonlinear filtering for jump diffusion signal and observation driven by correlated Brownian motions having common jump times. We provide the Kushner-Stratonovich and the Zakai equation for the normalized and the unnormalized filter, respectively. Moreover, we give conditions under which pathwise uniqueness for the solutions of both equations holds. Finally, we study an application of nonlinear filtering to the financial problem of derivatives hedging in an incomplete market with partial observation. Precisely, we consider the risk-minimizing hedging approach. In this framework, we compute the optimal hedging strategy for an informed investor and a partially informed one and compare the total expected squared costs of the strategies.

Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems

“…Another prominent situation where SPDEs such as the Kushner equation (cf., for example, Kushner [75]) and the Zakai equation (cf., for example, Zakai [105]) appear is in the case of nonlinear filtering problems where SPDEs describe the density of the state space of the considered system. In particular, we refer, e.g., to [16,21,25,29,36,37] for filtering problems in financial engineering, we refer, e.g., to [18,24,94,95,98,103] for filtering problems in chemical engineering, and we refer, e.g., to [28,19,20,23,33,89] for filtering problems in weather forecasting. SPDEs arising in nonlinear filtering problems are usually high-dimensional as the number of dimensions corresponds to the state space of the considered filtering problem.…”