## Abstract

For i = 1, ... ,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}] x ... x [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 γ_{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 Γ_{i}(x_{i}) = - ∫_{0}^{x(i)} uγ_{i}″(u)du and put M_{i} = {x ε D:Γ_{i}(x_{i}) = max _{j} Γ_{j}(x_{j})}. 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 γ_{j}(x_{j}) for j ≠ i. Put N_{i} = {x ε D:Γ_{i}(x_{i}) = max _{j} Γ_{j}(x_{j})}. The optimal control is: On N_{i} run the ith Brownian motion and stop when a facet of D is reached.

Original language | English (US) |
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Pages (from-to) | 1150-1162 |

Number of pages | 13 |

Journal | SIAM Journal on Control and Optimization |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - 1992 |

## All Science Journal Classification (ASJC) codes

- Control and Optimization
- Applied Mathematics