William D. Sudderth

Continuous-Time Red and Black: How to Control a Diffusion to a Goal

A player starts at x in (0, 1) and tries to reach 1. The process ( X t , t (ge) 0) of his positions moves according to a diffusion process (or, more generally, an Ito process) whose infinitesimal parameters (mu), (sigma) are chosen by the player at each instant of time from a set depending on his current position. To maximize the probability of reaching 1, the player should choose the parameters so as to maximize (mu)/(sigma) 2 , at least when the maximum is achieved by bounded, measurable functions. This implies that bold (timid) play is optimal for subfair (superfair), continuous-time red-and-black. Furthermore, in superfair red-and-black, the strategy which maximizes the drift coefficient of {log X t } minimizes the expected time to reach 1.

Minimizing or maximizing the expected time to reach zero

We treat the following control problems: the process

Construction of stationary Markov equilibria in a strategic market game

X_1 (t)

Finitely additive stochastic games with Borel measurable payoffs

with Values in the interval

An operator solution of stochastic games

( { - \infty ,0} ]

Finitely additive and measurable stochastic games

(or

Subgame-Perfect Equilibria for Stochastic Games

[ {0,\infty } )

Controlling a Process to a Goal in Finite Time

) is given by the stochastic differential equation \[dX_1 (t) = \mu (t)dt + \sigma (t)dW_t ,\quad X_1 (0) = x_1 \] where the nonanticipative controls

Reaching zero rapidly

\mu

Perfect Information Games with Upper Semicontinuous Payoffs

and