Non-linear filtering and optimal investment under partial information for stochastic volatility models

TL;DR

This paper develops a framework for filtering and optimizing investment strategies under partial information in stochastic volatility models, transforming the problem into a full information setting using change of measure and stochastic PDEs.

Contribution

It introduces a novel approach to filter estimation and optimal investment in stochastic volatility models with partial information, utilizing stochastic PDEs and martingale duality.

Findings

Filters depend on prior models for trend and volatility

Optimal portfolio characterized via semilinear PDEs

Framework applicable to popular stochastic volatility models

Abstract

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by a stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filters processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priorimodels for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations". Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programmingapproach, in terms of the solution to a semilinear partial differential equation. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.