We deal with the filtering problem of a general jump diffusion process, X, when the observation process, Y , is a correlated jump diffusion process having common jump times with X. In this setting, at any time t the σ -algebra F Y t provides all the available information about X t , and the central goal is to characterize the filter, π t , which is the conditional distribution of X t given observations F Y t . To this end, we prove that π t solves the Kushner-Stratonovich equation and, by applying the filtered martingale problem approach (see Kurtz and Ocone (1988)), that it is the unique weak solution to this equation. Under an additional hypothesis, we also provide a pathwise uniqueness result.

This paper is concerned with the nonlinear filtering problem for a general Markovian partially observed system (X, Y ), whose dynamics is modeled by correlated jump-diffusions having common jump times. At any time t ∈ [0, T ], the σ-algebra F Y t := σ{Y s : s ≤ t} provides all the available information about the signal X t . The central goal of stochastic filtering is to characterize the filter, π t , which is the conditional distribution of X t , given the observed data F Y t . It has been proved in [7] that π is the unique probability measure-valued process satisfying a nonlinear stochastic equation, the so-called Kushner-Stratonovich equation (KS-equation). In this paper the aim is to describe the filter π in terms of the unnormalized filter ̺, which is solution to a linear stochastic differential equation, the so-called Zakai equation. We prove equivalence between strong uniqueness for the solution to the Kushner Stratonovich equation and strong uniqueness for the solution to the Zakai one and, as a consequence, we deduce pathwise uniqueness for the solutions to the Zakai equation by applying the Filtered Martingale Problem approach ([25, 7]). To conclude, some particular cases are discussed.

Abstract. In this work, we study a dynamic portfolio optimization problem related to pairs trading, which is an investment strategy that matches a long position in one security with a short position in another security with similar characteristics. The relationship between pairs, called a spread, is modeled by a Gaussian meanreverting process whose drift rate is modulated by an unobservable continuous-time, finite-state Markov chain. Using the classical stochastic filtering theory, we reduce this problem with partial information to the one with full information and solve it for the logarithmic utility function, where the terminal wealth is penalized by the riskiness of the portfolio according to the realized volatility of the wealth process. We characterize optimal dollar-neutral strategies as well as optimal value functions under full and partial information and show that the certainty equivalence principle holds for the optimal portfolio strategy. Finally, we provide a numerical analysis for a toy example with a two-state Markov chain.

We study the optimal liquidation problem in a market model where the bid price follows a geometric pure jump process whose local characteristics are driven by an unobservable finite-state Markov chain and by the liquidation rate. This model is consistent with stylized facts of high frequency data such as the discrete nature of tick data and the clustering in the order flow. We include both temporary and permanent effects into our analysis. We use stochastic filtering to reduce the optimal liquidation problem to an equivalent optimization problem under complete information. This leads to a stochastic control problem for piecewise deterministic Markov processes (PDMPs). We carry out a detailed mathematical analysis of this problem. In particular, we derive the optimality equation for the value function, we characterize the value function as continuous viscosity solution of the associated dynamic programming equation, and we prove a novel comparison result. The paper concludes with numerical results illustrating the impact of partial information and price impact on the value function and on the optimal liquidation rate.

In this paper we investigate the local risk-minimization approach for a semimartingale financial market where there are restrictions on the available information to agents who can observe at least the asset prices. We characterize the optimal strategy in terms of suitable decompositions of a given contingent claim, with respect to a filtration representing the information level, even in presence of jumps. Finally, we discuss some practical examples in a Markovian framework and show that the computation of the optimal strategy leads to filtering problems under the real-world probability measure and under the minimal martingale measure.