Peng’s maximum principle for a stochastic control problem driven by a fractional and a standard Brownian motion

AbstractWe consider a stochastic control problem in the case where the set of the control domain is convex, and the system is governed by fractional Brownian motion with Hurst parameter {H\in(\frac{1}{2},1)} and standard Wiener motion. The criterion to be minimized is in the general form, with initial cost. We derive a stochastic maximum principle of optimality by using two famous approaches. The first one is the Doss–Sussmann transformation and the second one is the Malliavin derivative.

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Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem

Applied Mathematics and Computation ◽

10.1016/j.amc.2019.02.072 ◽

2019 ◽

Vol 355 ◽

pp. 282-298

Author(s):

Soukaina Douissi ◽

Jiaqiang Wen ◽

Yufeng Shi

Keyword(s):

Brownian Motion ◽

Control Problem ◽

Fractional Brownian Motion ◽

Stochastic Control ◽

Mean Field ◽

Stochastic Control Problem

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General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

Stochastic Analysis and Applications ◽

10.1080/07362994.2013.845106 ◽

2013 ◽

Vol 32 (1) ◽

pp. 88-109 ◽

Cited By ~ 12

Author(s):

Qingxin Meng

Keyword(s):

Brownian Motion ◽

Control Problem ◽

Stochastic Control ◽

Random Coefficients ◽

Optimal Stochastic Control ◽

General Linear ◽

Martingale Measure ◽

Linear Quadratic ◽

Stochastic Control Problem

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Connections Between Optimal Stopping and Stochastic Control II: Bounded-Variation Follower Problems

Advances in Applied Probability ◽

10.1017/s0001867800022072 ◽

1984 ◽

Vol 16 (1) ◽

pp. 16-16 ◽

Cited By ~ 1

Author(s):

Ioannis Karatzas ◽

Steven E. Shreve

Keyword(s):

Brownian Motion ◽

Control Problem ◽

Stochastic Control ◽

Bounded Variation ◽

Optimal Stopping ◽

Stochastic Control Problem

The stochastic control problem of tracking a Brownian motion by a process of bounded variation is reduced to a control problem with reflection at the origin, and the latter is related to a question of optimal stopping of Brownian motion absorbed at the origin. Direct probabilistic arguments can be used to show equivalences between the various problems.

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Connections between optimal stopping and stochastic control I. Monotone follower problems

Advances in Applied Probability ◽

10.1017/s0001867800022035 ◽

1984 ◽

Vol 16 (1) ◽

pp. 15-15

Author(s):

Joannis Karatzas ◽

Steven E. Shreve

Keyword(s):

Brownian Motion ◽

Control Problem ◽

Stochastic Control ◽

Optimal Stopping ◽

Optimal Solutions ◽

Stochastic Control Problem

The stochastic control problem of tracking a Brownian motion by a non-decreasing process (monotone follower) is related to a question of optimal stopping. Direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions.

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