Abstract
Statement of the problem. Suppose that the state of a system at time t is described by a stochastic integral X t of the form
where X t , b ∈ R n, (σ ∈ R n×m and B t is m-dimensional Brownian motion. Here u ∈ U ⊂ R k is a parameter whose value we can choose in the given Borel set U at any instant in order to control the process X t . Thus u = u(t,ω) is a stochastic process. Since our decision at time t must be based upon what has happened up to time t, the function ω → u(t,ω) must (at least) be measurable wrt. F t , i.e. the process u t must be F t -adapted. Thus the right hand side of (11.1) is well-defined as a stochastic integral, under suitable assumptions on 6 and σ At the moment we will not specify the conditions on b and a further, but simply assume that the process X t satisfying (11.1) exists. See further comments on this in the end of this chapter.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Øksendal, B. (1992). Application to Stochastic Control. In: Stochastic Differential Equations. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02847-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-662-02847-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53335-1
Online ISBN: 978-3-662-02847-6
eBook Packages: Springer Book Archive