The stochastic maximum principle for a singular control problem

1994 ◽  
Vol 49 (3-4) ◽  
pp. 211-237 ◽  
Author(s):  
Abel Cadenillas ◽  
Ulrich G. Haussmann
Keyword(s):  
Maximum Principle ◽  
Control Problem ◽  
Singular Control ◽  
Stochastic Maximum Principle ◽  
Singular Control Problem

Malliavin calculus used to derive a stochastic maximum principle for system driven by fractional Brownian and standard Wiener motions with application

2020 ◽  
Vol 28 (4) ◽  
pp. 291-306
Author(s):  
Tayeb Bouaziz ◽  
Adel Chala
Keyword(s):  
Brownian Motion ◽  
Maximum Principle ◽  
Control Problem ◽  
Fractional Brownian Motion ◽  
Stochastic Control ◽  
Malliavin Calculus ◽  
Stochastic Maximum Principle ◽  
Hurst Parameter ◽  
Principle Of Optimality ◽  
Initial Cost

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.

scholarly journals The Relationship between the Stochastic Maximum Principle and the Dynamic Programming in Singular Control of Jump Diffusions

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Farid Chighoub ◽  
Brahim Mezerdi
Keyword(s):  
Dynamic Programming ◽  
Maximum Principle ◽  
Sufficient Conditions ◽  
Singular Control ◽  
Stochastic Maximum Principle ◽  
Dynamic Programming Principle ◽  
Optimal Harvesting ◽  
Spatial Gradient ◽  
Jump Diffusions ◽  
The Relationship

The main objective of this paper is to explore the relationship between the stochastic maximum principle (SMP in short) and dynamic programming principle (DPP in short), for singular control problems of jump diffusions. First, we establish necessary as well as sufficient conditions for optimality by using the stochastic calculus of jump diffusions and some properties of singular controls. Then, we give, under smoothness conditions, a useful verification theorem and we show that the solution of the adjoint equation coincides with the spatial gradient of the value function, evaluated along the optimal trajectory of the state equation. Finally, using these theoretical results, we solve explicitly an example, on optimal harvesting strategy, for a geometric Brownian motion with jumps.

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