Continuous-Time Casino Problems

A gambler or an investor seeks to increase his fortune by a given amount before going bankrupt. The problem can be formulated as a stochastic control problem on an interval and is called a “continuous-time casino” if certain natural conditions are satisfied. An explicit formula is given for the value function of such a problem and the collection of optimal strategies is determined. “Fair casinos” are characterized in a result analogous to one of L. E. Dubins and L. J. Savage for discrete-time problems.