Stochastic filtering of a pure jump process with predictable jumps and path-dependent local characteristics

The objective of this paper is to study the filtering problem for a system of partially observable processes $(X, Y)$, where $X$ is a non-Markovian pure-jump process representing the signal and $Y$ is a general jump-diffusion which provides observations. Our model covers the case where both processes are not necessarily quasi left-continuous, allowing them to jump at predictable stopping times. By introducing the Markovian version of the signal, we are able to compute an explicit equation for the filtering process via the innovations approach

Portfolio Optimization for a Large Investor Controlling Market Sentiment Under Partial Information

We study a portfolio optimization problem for an investor whose actions have an indirect impact on prices. We consider a market model with a risky asset price process following a pure-jump dynamics with an intensity modulated by an unobservable continuous-time finite-state Markov regime-switching process. We assume that decisions of the investor affect the generator of the regime-switching process which results in an indirect impact on the price process. Using filtering theory, we reduce this problem with partial information to one with full information and solve it for logarithmic and power utility preferences. In particular, we apply control theory for piecewise deterministic Markov processes to derive the optimality equation. Finally, we provide an example with a two-state Markov regime-switching process and discuss how an investor's ability to control the intensity of it affects optimal portfolio strategies as well as the optimal wealth under full and partial information