Linear filtering of systems with memory and application to finance

We study the linear filtering problem for systems driven by continuous Gaussian processes V(1) and V(2) with memory described by two parameters. The processes V(j) have the virtue that they possess stationary increments and simple semimartingale representations simultaneously. They allow for straightforward parameter estimations. After giving the semimartingale representations of V(j) by innovation theory, we derive Kalman–Bucy-type filtering equations for the systems. We apply the result to the optimal portfolio problem for an investor with partial observations. We illustrate the tractability of the filtering algorithm by numerical implementations.