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Let μ be a Poisson random measure, let F be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let G be the initial enlargement of F with the σ -field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the σ -algebra generated by the Poisson random measure μ is equal to the expected relative entropy of the G-compensator relative to the F-compensator of the random measure μ. We then use this link to gain some insight into the changes of Doob-Meyer decompositions of stochastic processes when the filtration is enlarged from F to G. In particular, we show that if the mutual information between G and the σ -algebra generated by the Poisson random measure μ is finite, then every squareintegrable F-martingale is a G-semimartingale that belongs to the normed space S 1 relative to G.

Abstract. In the paper we propose a purely computational new method of construction of a quasi-optimal portfolio for stochastic models of a financial market. Here we present the method in the framework of a classical Black-Scholes model of a complete market (see, eg.[4], [6]), considering a well known optimal investment and consumption problem with the HARA type optimization functional. Our method is based on the idea to maximize this functional, taking into account only some subsets of possible portfolio and consumption processes. We show how to reduce the main problem to the construction of a portfolio maximizing a deterministic function of a few real valued parameters but under purely stochastic constraints. It is enough to solve several times an indicated system of stochastic differential equations (SDEs) with properly chosen parametrs. Results of computer experiments presented here were obtained with the use of the SDE-Solver software package. This is our own professional C++ application to Windows system, designed as a scientific computing tool based on Monte Carlo simulations and serving for numerical and statistical construction of solutions to a wide class of systems of SDEs, including a broad class of diffusions with jumps driven by non-Gaussian random measures (consultOur method can be easily extended to stochastic models of financial market described by systems of such SDEs. Optimal Investment and Consumption Problem for Black-Scholes Model of a Financial MarketLet us recall that an N dimensional Black-Scholes model of a financial market can be described by the following system of N + 1 SDEsS n (t) = S n (0) +