Optimal Portfolio Management with Partial Observations and Power Utility Function

Abstract

A common model for a stock price is that it is a “geometric Brownian motion”. That is the price s(t) of a stock satisfies a stochastic differential equation of the type (1) $$ ds\left( t \right) = s\left( t \right)\left( {\bar r\,dt\, + \,\sigma d\beta \left( t \right)} \right) $$ in which α(t) is a Brownian motion process and and σ are constants. For this model the growth rate of the stock has stochastic differential (2) $$ \bar r\,dt + \sigma d\beta \left( t \right) $$ with $$\bar r$$ dt being the infinitesmal mean growth rate and σdß(t) the infinitesmal random fluctuations about this mean growth rate. Since the process (3) $$ \bar rt + \sigma \beta \left( t \right) $$ has stationary increments, the growth rate of the stock over any two periods of equal length has the same statistical properties. This appears to rule out bull and bear markets. A model for a stock price which can have features of bull and bear markets is a doubly stochastic model of the form in which ß(t) is a Brownian motion and r(t) is also a random process. For this model the infinitesmal mean growth rate r(t)dt is random as well as the random fluctuations ad ß(t) about this mean. If is a target mean growth rate for the stock and there is a period for which r(t) is substantially larger than this could be considered as a bull market. Conversely when r(t) is substantially less than this could be considered as a bear market.

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