A Homotopy Method for Proving Convexity in Certain Optimal Stochastic Control Problems

Abstract

Let w. be a d-dimensional Wiener process and let k:R → R⁺ be even and increasing on the half-line. It is known that the solution of the final value stochastic control problem

$$\begin{gathered} Ek(|{x_1}|) = \min ,{\text{subject to}} \hfill \\ {\text{d}}{{\text{x}}_{\text{t}}} = u(t,{x_t})dt + d{w_t}0 \leqslant t \leqslant 1 \hfill \\ |u| \leqslant 1 \hfill \\ \end{gathered} $$

is given by the control law u(t, x) = -x/|x|.