Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for Backward Stochastic Differential Equations with Application

Chala, Adel

doi:10.1007/s00574-017-0031-2

Cited by 10 publications

(24 citation statements)

References 7 publications

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“…At the end, if we put in our first example A t = B t = C t = D t = 0, we get the same result as in [], and the sufficient optimality conditions are similar to those in the paper of Chala [].…”

Section: Discussionsupporting

confidence: 70%

“…The purpose of this paper, which extends to both of the results of Chala [] and Djehiche et al [], who generalized the system of FBSDE and into a fully coupled case, is to establish a necessary, as well as sufficient optimality conditions, of Pontryagin's maximum principle type, for risk‐sensitive performance functionals. We solve the problem by using the approach developed by Djehiche, Tembine and Tempone in [].…”

Section: Introductionmentioning

confidence: 79%

“…This paper contains two main results. The first one is the Theorem , which establishes the necessary optimality condition for the system of fully coupled FBSDE with risk sensitive performance, using an almost similar scheme as in Chala []. The second main result, Theorem , suggests sufficient optimality conditions of fully coupled FBSDE given in form of risk sensitive performance.…”

Section: Discussionmentioning

confidence: 99%

“…As we said, Theorem is a good SMP for the risk‐neutral of forward backward control problem. We follow the approach used in [], and suggest a transformation of the adjoint processes

((), p_{1}, q_{1})

((), p_{2}, q_{2})

and

((), p_{3}, q_{3})

in such a way to throw a way to the first component

((), p_{1}, q_{1})

in , and to obtain the SMP in terms of only the last two adjoint processes, that we denote by

((), ((), {\overset{p}{true}}_{2}, {\overset{q}{true}}_{2}), ((), {\overset{p}{true}}_{3}, {\overset{q}{true}}_{3}))

. Noting that d p 1 ( t ) = q 1 ( t ) d W t and

p_{1} false(T false) = - θ A_{T}^{θ}

, the explicit solution of this backward SDE is

p_{1} false(t false) = - θ double-struckE ([], A_{T}^{θ} false| F_{t}) = - θ V_{t}^{θ},

where

V_{t}^{θ} : = double-struckE ([], A_{T}^{θ} false| F_{t}), 0 \leq t \leq T .

…”

Section: Risk‐sensitive Stochastic Maximum Principle Of Fully Coupledmentioning

confidence: 99%

“…We also realise that necessary optimality conditions for stochastic controls, where the systems are governed by nonlinear forward stochastic differential equation with jumps, have been studied by Shi and Wu [] in the case where the set of admissible controls is convex, and [] in the general case with application to finance. Furthermore, the systems governed by a mean‐field stochastic differential equation, have been studied by Djehiche, Tembine and Tempone [], and finally the last work on this kind of problem using the backward system, with an application to linear quadratic performance functional, can be found in the paper of Chala []. The previous works have been established with risk sensitive performance functional.…”

Section: Introductionmentioning

confidence: 99%

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A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

Khallout

Chala

2019

Asian Journal of Control

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In this paper, we are interested in the problem of optimal control where the system is given by a fully coupled forward-backward stochastic differential equation with a risk-sensitive performance functional. As a preliminary step, we use the risk neutral which is an extension of the initial control system where the admissible controls are convex, and an optimal solution exists.Then, we study the necessary as well as sufficient optimality conditions for risk sensitive performance. At the end of this work, we illustrate our main result by giving an example that deals with an optimal portfolio choice problem in financial market, specifically the model of control cash flow of a firm or project where, for instance, we can set the model of pricing and managing an insurance contract. KEYWORDS cash flow, fully coupled forward-backward stochastic differential equation, logarithmic transformation, maximum principle, optimal control, risk-sensitive, variational principle Assume that the portfolio is invested in a simple Black-Scholes market model consisting of a risk-free asset (for example, a bond or a bank account) with a short interest rate r t assumed bounded and deterministic, and a risky asset evolving as a geometric Brownian motion with rate

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Section: Discussionsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 79%

Section: Discussionmentioning

confidence: 99%

“…As we said, Theorem is a good SMP for the risk‐neutral of forward backward control problem. We follow the approach used in [], and suggest a transformation of the adjoint processes

((), p_{1}, q_{1})

((), p_{2}, q_{2})

and

((), p_{3}, q_{3})

in such a way to throw a way to the first component

((), p_{1}, q_{1})

in , and to obtain the SMP in terms of only the last two adjoint processes, that we denote by

((), ((), {\overset{p}{true}}_{2}, {\overset{q}{true}}_{2}), ((), {\overset{p}{true}}_{3}, {\overset{q}{true}}_{3}))

. Noting that d p 1 ( t ) = q 1 ( t ) d W t and

p_{1} false(T false) = - θ A_{T}^{θ}

, the explicit solution of this backward SDE is

p_{1} false(t false) = - θ double-struckE ([], A_{T}^{θ} false| F_{t}) = - θ V_{t}^{θ},

where

V_{t}^{θ} : = double-struckE ([], A_{T}^{θ} false| F_{t}), 0 \leq t \leq T .

…”

Section: Risk‐sensitive Stochastic Maximum Principle Of Fully Coupledmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

Khallout

Chala

2019

Asian Journal of Control

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Stochastic maximum principle for partially observed risk‐sensitive optimal control problems of mean‐field forward‐backward stochastic differential equations

Wang

2021

Optim Control Appl Methods

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This paper deals with a partially observed risk‐sensitive optimal control problem described by mean‐field forward‐backward stochastic differential equations and the cost functional is a mean‐field exponential of integral type. By using Girsanov's theorem as well as classical convex variational techniques, we obtain two risk‐sensitive maximum principles, which are characterized in terms of the variational inequalities. Moreover, under certain concavity assumption, we give the sufficient condition for the optimality. Through the application of results, we consider the linear‐quadratic risk‐sensitive optimal control problem under partially observed information and fully observed information, respectively.

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Risk‐sensitive stochastic maximum principle for forward‐backward systems involving impulse controls

Ying

2023

Intl J Robust & Nonlinear

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In this article, we study a risk‐sensitive stochastic optimal control problem driven by forward‐backward systems in which the control variable consists of two components: the continuous control and the impulse control. The control domain is assumed to be convex. We establish the maximum principle (i.e., necessary condition) for this kind of control problem. Under some additional assumptions, the necessary optimality conditions turn out to be sufficient. To explain the theoretical results, a linear‐quadratic risk‐sensitive control problem is solved by using the maximum principle derived and the optimal control is obtained.

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Pontryagin’s Risk-Sensitive Stochastic Maximum Principle for Backward Stochastic Differential Equations with Application

Cited by 10 publications

References 7 publications

A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

Stochastic maximum principle for partially observed risk‐sensitive optimal control problems of mean‐field forward‐backward stochastic differential equations

Risk‐sensitive stochastic maximum principle for forward‐backward systems involving impulse controls

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