Optimal Switching among Several Brownian Motions

Abstract

For $i = 1, \cdots ,d$, let $B_{s_i }^i $ be a one-dimensional Brownian motion on the interval $[0,a_i ]$ with absorption at the endpoints. At each instant in time, one must decide to run some subset of these d Brownian motions while holding the others fixed at their current state. The resulting process evolves in the rectangle $D = [0,a_1 ] \times \cdots \times [0,a_d ]$. If, at some instant, one decides to freeze all of the Brownian motions, then a reward is received in accordance with this final position. Two types of reward functions are considered. First, it is assumed that the reward is zero everywhere in D, except along the d edges that correspond to the coordinate axes. Along these edges, it is given by $C^3 $ strictly concave functions $\gamma _i (x_i )$, which are zero at the endpoints 0 and $a_i $ of their domains. The optimal control for this problem has a simple description. Let \[\Gamma _i \left( {x_i } \right) = - \int_0^{x_i } {u\gamma ''_i } (u)du\] and put \[M_i = \left\{ {x \in D:\Gamma _i \left( {x_i } \right) = \mathop {\max }\limits_j \Gamma _j \left( {x_i } \right)} \right\}.\] It is proved that the optimal control is: On $M_i $ run any Brownian motion except the ith and stop the first time an edge is reached. The second class of reward functions are assumed to be zero everywhere except on the facets of D that meet at the origin. On the ith such facet (i.e., where $x_i = 0$), the reward function is the product of $\gamma _j (x_j )$ for $j \ne i$. Put \[N_i = \left\{ {x \in D:\Gamma _i \left( {x_i } \right) = \mathop {\min }\limits_j \Gamma _j \left( {x_j } \right)} \right\}.\] The optimal control is: On $N_i $ run the ith Brownian motion and stop when a facet of D is reached.