Maximum principles for non-Markovian semi-martingales with jumps and more

TL;DR

This paper develops a maximum principle for non-Markovian semi-martingales with jumps, using stochastic derivatives in a martingale field framework, extending existing results and applying to portfolio optimization with credit risk.

Contribution

It introduces a new maximum principle for non-Markovian semi-martingales with jumps, replacing Malliavin derivatives with simpler $L_2$-conditions and applying it to financial portfolio optimization.

Findings

Extended maximum principles to non-Markovian processes with jumps.

Replaced Malliavin derivatives with $L_2$-conditions.

Applied the theory to portfolio optimization with credit risk.

Abstract

We find a maximum principle for general non-Markovian semi-martingales. We do so by describing the adjoint processes with non-anticipating stochastic derivatives in a martingale random field setting. In the case of the L\'evy processes this extends maximum principles with Malliavin derivatives, in the sense that we replace Malliavin differentiability conditions with weaker and simpler $L_2$-conditions. As an application we use the maximum principle to solve a portfolio optimization problem for assets with credit risk modeled by doubly stochastic Poisson processes.