Application of maximum principle to Stochastic Control problem

  • Sahar Fahd Alsirhani Department of Mathematics, Al Jouf University, Skaka, P.O.Box: 2014 , Saudi Arabia
Keywords: Stochastic process,Brownian motion, Markov processes, stochastic differential equation and Stochastic Control

Abstract

Suppose that the state of a system at time t is described by an Itˆo process Xt of the form
dXt = dXu t = b(t, Xt, ut)dt + σ(t, Xt, ut)dBt, t > s > 0, (1)
where Xt ∈ R n, b : R × R n × U → R n, σ : R × R
n × U → R n×m and Bt is an m-dimensional Brownian motion. Here ut ∈ U ⊂ R k is a parameter whose value we can choose in a given Borel set U at any instant t in order to control the process Xt. Thus ut = u(t, w) is a stochastic process.
Since our decision at time t must be based upon what has happened up to time t, the function w −→ u(t, w) must (at least) be measurable with respect to F (m) t, i.e. the process ut must be {F(m) t, t > 0}-adapted.

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Published
2021-06-09
How to Cite
Fahd Alsirhani, S. (2021). Application of maximum principle to Stochastic Control problem. Journal of Progressive Research in Mathematics, 18(2), 45-58. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2054
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