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Zbl 1061.93099
Lefebvre, Mario
A homing problem for diffusion processes with control-dependent variance. (English)
[J] Ann. Appl. Probab. 14, No. 2, 786-795 (2004). ISSN 1050-5164

This paper deals with optimal control problems of 1-dimensional diffusion processes defined by the stochastic differential equation $$dX(t)= f(X(t))\,dt+ (v(X(t))\vert u(t)\vert)^{{1\over 2}} dW(t),$$ with initial $X(0)= x\in[d_1, d_2]$, where $u$ is a control process and $W$ a standard Brownian motion. So, $X$ has a control-dependent infinitesimal variance. The aim is to minimize the cost criterion $$J(x)= \int^{\tau(x)}_0 (\textstyle{{1\over 2}} q(X(t)) u^2(t)+ \lambda)\,dt+ K(X(\tau(x))),$$ where $\tau(x)$ is the hitting time to the boundary points, $d_1$, $d_2$, and $\lambda$ a real constant.\par Using the dynamic programming equation, the author investigates the value function and optimal control. In particular, he obtains explicit expressions for the value function and the optimal control when the functions $f$, $v$ and $q$ are proportional to a power of $x$.
[Makiko Nisio (Kobe)]

MSC 2000:: *93E20 Optimal stochastic control (systems)
60J60 Diffusion processes

Keywords: dynamic programming equation; stochastic differential equation; hitting place; Brownian motion; diffusion processes; control-dependent variance; hitting time; value function; optimal control

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Zentralblatt MATH Berlin [Germany]