Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Optimal drift on $ [0,1]$

Author: Susan Lee
Journal: Trans. Amer. Math. Soc. 346 (1994), 159-175
MSC: Primary 60H10; Secondary 60J60, 60J65
MathSciNet review: 1254190
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Abstract: Consider one-dimensional diffusions on the interval $ [0,1]$ of the form $ d{X_t} = d{B_t} + b({X_t})dt$, with 0 a reflecting boundary, $ b(x) \geqslant 0$, and $ \int_0^1 {b(x)dx = 1} $. In this paper, we show that there is a unique drift which minimizes the expected time for $ {X_t}$ to hit $ 1$, starting from $ {X_0} = 0$. In the deterministic case $ d{X_t} = b({X_t})dt$, the optimal drift is the function which is identically equal to $ 1$. By contrast, if $ d{X_t} = d{B_t} + b({X_t})dt$, then the optimal drift is the step function which is $ 2$ on the interval $ [1/4,3/4]$ and is 0 otherwise. We also solve this problem for arbitrary starting point $ {X_0} = {x_0}$ and find that the unique optimal drift depends on the starting point, $ {x_0}$, in a curious manner.

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Keywords: Stochastic differential equations, diffusion processes, drift, hitting time, optimization
Article copyright: © Copyright 1994 American Mathematical Society