Stochastic Control - in Finance

In Finance

In a continuous time approach in a finance context, the state variable in the stochastic differential equation is usually wealth or net worth, and the controls are the shares placed at each time in the various assets. Given the asset allocation chosen at any time, the determinants of the change in wealth are usually the stochastic returns to assets and the interest rate on the risk-free asset. The field of stochastic control has developed greatly since the 1970s, particularly in its applications to finance. Robert Merton used stochastic control to study optimal portfolios of safe and risky assets. His work and that of Black-Scholes changed the nature of the finance literature. Major mathematical developments were by W. Fleming and R. Rishel and W. Fleming and M. Soner. These techniques were applied by J. L. Stein to the U.S. financial crisis of the decade of the 2000s.

The maximization, say of the expected logarithm of net worth at a terminal date T, is subject to stochastic processes on the components of wealth. In this case, in continuous time the Ito equation is the main tool of analysis. In the case where the maximization is an integral of a concave function of utility over an horizon (0,T), dynamic programming is used. There is no certainty equivalence as in the older literature, because the coefficients of the control variables—that is, the returns received by the chosen shares of assets—are stochastic.