Application of maximum principle to Stochastic Control problem

Sahar Fahd Alsirhani

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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References

[1] M. Adams and V. Guillemin, Theory and probability, 1st edition, Brikh¨auser, Boston, 1996.
[2] A. Al-Hussein, Backward stochastic differential equations in infinite dimensions and application, Arab
J. Math. Sc., 1-18, 2004.
[3] I. Alzughaibi, Topics in stochastic analysis and applications to finance, MSc dissertation, Qassim University, Saudi Arabia, 2017.
[4] K. B. Athreya and S. N. Lahiri, Measure theory and probability theory, Springer Texts in Statistics,
New York, 2006.
[5] R. M. Dudley, Wiener functionals as Itˆo integrals. Ann. Probability 5, 140-141, 1977.
[6] K Ito and H. P. Jr. McKean, Diffusion processes and their sample paths, Springer-Verlag, Berlin/New
York, 1974.
[7] H.-H. Kuo, Introduction to stochastic integration, Universitext, Springer, New York, 2006.
[8] B. Øksendal, Stochastic differential equations: An introduction with applications, 6th edition, Springer,
Berlin, 2003.
[9] E. Pardoux and S. G. Peng, Adapted solution of backward stochastic differential equations, System
Control Lett, no. 14, 1, 55-61, 1990.
[10] D. Revuz and M. Yor, Continuous martingales and Brownian motion, Springer, 1991.
[11] J. L. Speyer and W. H. Chung, Stochastic processes, estimation, and control, University of California,
Los Angeles, 2008.
[12] D. Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University
Press, Cambridge, 1992.
[13] H. Ying, Backward stochastic differential equations and applications in finance, Lecture notes, August
10, 2013, http://hkqf.se.cuhk.edu.hk/short_course_201308.
[14] J. Yong and X.Y Zhou, Stochastic controls, Hamiltonian systems and HJB equations, Springer, New
York, 1999.