Modern Stochastics: Theory and Applications

Given a compound mixed renewal process S under a probability measure P, we provide a characterization of all progressively equivalent martingale probability measures Q on the domain of P, that convert S into a compound mixed Poisson process. This result extends earlier works of Delbaen and Haezendonck, Lyberopoulos and Macheras, and the authors, and enables us to find a wide class of price processes satisfying the condition of no free lunch with vanishing risk. Implications to the ruin problem and to the computation of premium calculation principles in an arbitrage-free insurance market are also discussed.

In this note the maximization of the expected terminal wealth for the setup of quadratic transaction costs is considered. First, a very simple probabilistic solution to the problem is provided. Although the problem was largely studied, as far as authors know up to date this simple and probabilistic form of the solution has not appeared in the literature. Next, the general result is applied for the numerical study of the case where the risky asset is given by a fractional Brownian motion and the information flow of the investor can be diversified.