Optimal switching among several Brownian motions

Robert J. Vanderbei

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1 Scopus citations


For i = 1, ... ,d, let Bs(i)i be a one-dimensional Brownian motion on the interval [0, ai] 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, a1] x ... x [0, ad]. 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 C3 strictly concave functions γi(xi), which are zero at the endpoints 0 and ai of their domains. The optimal control for this problem has a simple description. Let Γi(xi) = - ∫0x(i)i″(u)du and put Mi = {x ε D:Γi(xi) = max j Γj(xj)}. It is proved that the optimal control is: On Mi 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 xi = 0), the reward function is the product of γj(xj) for j ≠ i. Put Ni = {x ε D:Γi(xi) = max j Γj(xj)}. The optimal control is: On Ni run the ith Brownian motion and stop when a facet of D is reached.

Original languageEnglish (US)
Pages (from-to)1150-1162
Number of pages13
JournalSIAM Journal on Control and Optimization
Issue number5
StatePublished - 1992

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Applied Mathematics


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