An asymmetric bang-bang partially observed stochastic control problem. (English) Zbl 0732.93083
The paper contains a generalization of Girsanov’s theorem for following multi-dimensional problem
\[
\min \{EL(\gamma^ Tx(t))/dx=A(t)x+B(t)[u(t)+Z]dt+C(t)dw(t)
\]
\[ x(0)=x_ 0,\quad f(t)\leq u(t)\leq g(t),\quad t\in [0,T]\} \] where \[ \dim x=\dim u=\dim z=\dim w, \] L is even continuously differentiable; \(L'(x)\geq 0\), \(x>0\), and L is of polynomial growth as \(x\to \infty\); z is a constant random vector; w(t) is a standard Wiener process. It is shown that the optimal solution has the form \[ u_ 0(t)=2^{-1}[f(t)+g(t)]+2^{-1}[g(t)-f(t)]sign \psi (t) \] where \(\psi\) (t) can be calculated.
\[ x(0)=x_ 0,\quad f(t)\leq u(t)\leq g(t),\quad t\in [0,T]\} \] where \[ \dim x=\dim u=\dim z=\dim w, \] L is even continuously differentiable; \(L'(x)\geq 0\), \(x>0\), and L is of polynomial growth as \(x\to \infty\); z is a constant random vector; w(t) is a standard Wiener process. It is shown that the optimal solution has the form \[ u_ 0(t)=2^{-1}[f(t)+g(t)]+2^{-1}[g(t)-f(t)]sign \psi (t) \] where \(\psi\) (t) can be calculated.
Reviewer: A.A.Pervozvanskij (St.Petersburg)
MSC:
93E20 | Optimal stochastic control |
References:
[1] | Benes V.E., Stochastic Differential Sytems, Stochastic Control Theory and Applications pp 1– (1988) · doi:10.1007/978-1-4613-8762-6_1 |
[2] | Weiner H.J., International Journal of Control 34 (6) pp 1215– (1981) · Zbl 0475.93081 · doi:10.1080/00207178108922594 |
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