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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(ξ(t))dt+b(ξ(t))u(ξ(t))dt+[N(ξ(t))] 1/2 dW(t),

where ξ(t):=(x(t),t). The aim of the homing control problem is to minimize the expectation of a functional of the form

J(x)= 0 T(x) [1 2q(ξ(t))u 2 (ξ(t))+λ]dt,

where q0,λ is real and T(x) denotes the exit time from an interval (A,B) for a solution starting from x=x(0)(A,B). In the particular case a=0,b=N=1 and q(ξ(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.

60H10Stochastic ordinary differential equations