  We study the linear filtering problem for systems driven by continuous
Gaussian processes with memory described by two parameters. The driving
processes have the virtue that they possess stationary increments and simple
semimartingale representations simultaneously. It allows for straightforward
parameter estimations. After giving the semimartingale representations of the
processes 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.

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