Optimal portfolios with bounded capital at risk. (Q2770980)

This paper, which is basically a research announcement, analyses the work of Bielecki and Pliska on dynamic asset management.NEWLINENEWLINE Suppose that you have decided to invest in a predetermined collection of assets. Your objective is to specify an investing self-financing strategy that will maximize the utility of your wealth at a certain time horizon.NEWLINENEWLINE It is assumed that the evolution of the assets follows a certain joint stochastic evolution. It is further assumed that the local rates of return of the assets themselves follow a certain stochastic process. This second evolution tries to capture the uncertainty on the general economy, like economic cycles, and so on.NEWLINENEWLINE In a well known and much acclaimed paper Bielecki and Pliska have been able to obtain the optimal strategy for a certain utility function.NEWLINENEWLINE In this paper the authors explain how to transform this problem of selecting an optimal investing strategy into a stochastic differential game (where, in a certain sense, the investor plays a game against nature). This approach appears to have originated with the work of Dai Pra, Meneghini and Runggaldier. With this approach, the difficulty posed by the exponential utility function is somehow linearized.NEWLINENEWLINE The prescription of Bielecki and Pliska for the evolution of the rates of return is a linear Itô process and so, could reach negative values. To circumvent this difficulty (?) the authors force reflection at \(0\) of the rates of returns and show how to handle this situation under the stochastic differential game approach.