Singular Stochastic Control and Optimal Stopping with Partial Information of Itô--Lévy Processes

Øksendal, Bernt; Sulem, Agnès

doi:10.1137/100793931

Cited by 36 publications

(49 citation statements)

References 21 publications

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“…Note that our results improve those in [1,2] to the jump diffusion setting. Moreover we generalize results in [3,4], by allowing 2 International Journal of Stochastic Analysis both classical and singular controls, at least in the complete information setting. Note that in our control problem, there are two types of jumps for the state process, the inaccessible ones which come from the Poisson martingale part and the predictable ones which come from the singular control part.…”

Section: Introductionmentioning

confidence: 67%

“…Stochastic control problems of singular type have received considerable attention, due to their wide applicability in a number of different areas; see [4][5][6][7][8]. In most cases, the optimal singular control problem was studied through dynamic programming principle; see [9], where it was shown in particular that the value function is continuous and is the unique viscosity solution of the HJB variational inequality.…”

Section: Introductionmentioning

confidence: 99%

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

Chighoub

Mezerdi

2014

International Journal of Stochastic Analysis

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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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Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

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

Chighoub

Mezerdi

2014

International Journal of Stochastic Analysis

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“…Using (6.3) we know that From the construction of stopping times τ ′ δ , τ δ (the latter now defined in terms of the process X u lower ), by (6.15) and from the first step of the proof it follows that 16) for all δ > 0. Since τ δ → +∞ almost surely as δ → 0+ we conclude that X u(·) does not attain 0 for all t > 0 almost surely.…”

Section: Controlled Stochastic Logistic Equationmentioning

confidence: 99%

“…Uniqueness of the processes Y, Z then follows from (6.27). 16 To sketch the construction of the solution we first introduce for each n ∈ N the approximated equation…”

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confidence: 99%

Sufficient Stochastic Maximum Principle for Discounted Control Problem

Maslowski

Veverka

2014

Appl Math Optim

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Abstract. In this article, the sufficient Pontryagin's maximum principle for infinite horizon discounted stochastic control problem is established. The sufficiency is ensured by an additional assumption of concavity of the Hamiltonian function. Throughout the paper, it is assumed that the control domain U is a convex set and the control may enter the diffusion term of the state equation. The general results are applied to the controlled stochastic logistic equation of population dynamics.

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“…Itô-Lévy processes provide a more realistic paradigm for describing the evolution of cash reserves of insurance companies. For more details and examples on the application of jumps diffusion processes in finance and insurance, the reader is referred to Applebaum (2003), Framstad et al (2001), Øksendal and Sulem (2012) and Zou et al (2008). An Itô-Lévy model consists of a drift term, a Brownian motion component and a jump term.…”

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confidence: 99%

Optimal combined dividend and reinsurance policies under interest rate in Lévy markets

Mgobhozi

Chikodza

2017

IJMOR

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Abstract:A combined dividend and risk control problem is presented and investigated in this paper. The risk of the insurance firm is controlled by using a proportional reinsurance policy. It is considered that the evolution of the cash reserves of the firm is driven by a generalised Itô-Lévy process. The surplus cash reserves earn interest at a constant rate. The objective of the firm is to maximise the total expected discounted dividends paid out to share holders. The situation is modelled as an impulse-classical control problem. We manage to construct the value function and the optimal impulse control. The existence and uniqueness of an optimal classical control is proved.

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Singular Stochastic Control and Optimal Stopping with Partial Information of Itô--Lévy Processes

Cited by 36 publications

References 21 publications

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

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

Sufficient Stochastic Maximum Principle for Discounted Control Problem

Optimal combined dividend and reinsurance policies under interest rate in Lévy markets

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