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Using a geometric Brownian motion to control a Brownian motion and vice versa. (English) Zbl 0911.60040
Consider a one-dimensional controlled process $x(t)$ governed by the equation $$dx(t)=a(\xi (t))dt+b(\xi (t))u(\xi (t))dt+[N(\xi (t))]^{1/2}dW(t),$$ where $\xi (t):=(x(t),t)$. The aim of the homing control problem is to minimize the expectation of a functional of the form $$J(x)=\int _0^{T(x)} [\tfrac {1}{2}q(\xi (t))u^2(\xi (t))+\lambda ]dt,$$ where $q\ge 0,\ \lambda $ is real and $T(x)$ denotes the exit time from an interval $(A,B)$ for a solution starting from $x=x(0)\in (A,B)$. In the particular case $a=0,\ b=N=1$ and $q(\xi (t))=x^2(t)$ the optimal control is found by means of the mathematical expectation of a geometric Brownian motion while the optimal process is shown to be a Bessel process. Conversely, if the uncontrolled process is a geometric Brownian motion, then the optimal control is found by means of an expectation of a Brownian motion.

MSC:
60H10Stochastic ordinary differential equations
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References:
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