scholarly journals Optimal control of ultimately bounded stochastic processes

1974 ◽  
Vol 53 ◽  
pp. 157-170
Author(s):  
Yoshio Miyahara
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Markov Process ◽  
Stochastic Differential Equation ◽  
Stochastic Processes ◽  
Wiener Process ◽  
Admissible Control ◽  
Standard Wiener Process ◽  
The Cost

We shall consider the optimal control for a system governed by a stochastic differential equationwhere u(t, x) is an admissible control and W(t) is a standard Wiener process. By an optimal control we mean a control which minimizes the cost and in addition makes the corresponding Markov process stable.

ON OPTIMAL CONTROL OF A STOCHASTIC EQUATION WITH A FRACTIONAL WIENER PROCESS

2021 ◽  
Vol 6 ◽  
pp. 5-12
Author(s):  
Pavel Knopov ◽  
◽  
Tatyana Pepelyaeva ◽  
Sergey Shpiga ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Existence Theorem ◽  
Wiener Process ◽  
Strong Dependence ◽  
Strong Mixing ◽  
Fractional Wiener Process

In recent years, a new direction of research has emerged in the theory of stochastic differential equations, namely, stochastic differential equations with a fractional Wiener process. This class of processes makes it possible to describe adequately many real phenomena of a stochastic nature in financial mathematics, hydrology, biology, and many other areas. These phenomena are not always described by stochastic systems satisfying the conditions of strong mixing, or weak dependence, but are described by systems with a strong dependence, and this strong dependence is regulated by the so-called Hurst parameter, which is a characteristic of this dependence. In this article, we consider the problem of the existence of an optimal control for a stochastic differential equation with a fractional Wiener process, in which the diffusion coefficient is present, which gives more accurate simulation results. An existence theorem is proved for an optimal control of a process that satisfies the corresponding stochastic differential equation. The main result was obtained using the Girsanov theorem for such processes and the existence theorem for a weak solution for stochastic equations with a fractional Wiener process.

A general maximum principle for mean-field forward-backward doubly stochastic differential equations with jumps processes

2019 ◽  
Vol 27 (1) ◽  
pp. 9-25 ◽  
Author(s):  
Dahbia Hafayed ◽  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Admissible Control ◽  
Mean Field ◽  
Sufficient Optimality Condition ◽  
Doubly Stochastic

Abstract In this paper, we deal with an optimal control, where the system is driven by a mean-field forward-backward doubly stochastic differential equation with jumps diffusion. We assume that the set of admissible control is convex, and we establish a necessary as well as a sufficient optimality condition for such system.

Evaluations of absorption probabilities for the Wiener process on large intervals

1980 ◽  
Vol 17 (02) ◽  
pp. 363-372 ◽  
Author(s):  
C. Park ◽  
F. J. Schuurmann
Keyword(s):  
Stochastic Processes ◽  
Wiener Process ◽  
Standard Wiener Process ◽  
Computationally Efficient ◽  
The Past ◽  
Gaussian Stochastic Processes ◽  
Absorption Probabilities

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦T W(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.

scholarly journals An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

2020 ◽  
Vol 28 (1) ◽  
pp. 1-18
Author(s):  
Dahbia Hafayed ◽  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Doubly Stochastic ◽  
Risk Sensitive ◽  
Performance Functional

AbstractIn this paper, we are concerned with an optimal control problem where the system is driven by a backward doubly stochastic differential equation with risk-sensitive performance functional. We generalized the result of Chala [A. Chala, Pontryagin’s risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.) 48 2017, 3, 399–411] to a backward doubly stochastic differential equation by using the same contribution of Djehiche, Tembine and Tempone in [B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control 60 2015, 10, 2640–2649]. We use the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of an initial control system in this type of problem, where an admissible controls set is convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. We illustrate the paper by giving two different examples for a linear quadratic system, and a numerical application as second example.

Optimal reserves growth under a Wiener process

1975 ◽  
Vol 12 (03) ◽  
pp. 457-465
Author(s):  
W. F. Foster
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Wiener Process ◽  
Order Differential Equation ◽  
Optimality Criterion ◽  
First Passage ◽  
First Order ◽  
Bounded Below ◽  
Passage Times

This paper considers a body whose funds accumulate according to a Wiener Process that has parameters which can be controlled at any stage. The process is bounded above by a level at which dividends (or savings) are set aside, and it is bounded below by a level at which a ‘rescue’ policy is invoked to avoid insolvency. Taking long-term dividend maximisation as the optimality criterion, first passage times are used to derive a general first order differential equation for the optimal control of the system at any reserves level, and this equation is solved fully for a certain class of problems. Examples are given of insurance and investment applications.

The Expected Discounted Penalty Function under Stochastic Discount Interest Force Driven by Markov Process

2012 ◽  
Vol 433-440 ◽  
pp. 5035-5039
Author(s):  
Chun Ming Zhang
Keyword(s):  
Differential Equation ◽  
Markov Process ◽  
Penalty Function ◽  
Standard Wiener Process ◽  
Exponential Distributions ◽  
Expected Discounted Penalty Function ◽  
Geometric Process ◽  
Interest Force ◽  
Discounted Penalty Function ◽  
Expected Discounted Penalty

We investigate the expected discounted penalty function in which the discount interest process is driven by markov process. We obtain the integro-differential equation satisfied by the expected discounted penalty function when interest process is perturbed by standard Wiener process and Poisson-Geometric process. A system of Laplace transforms of the expected discounted penalty function, given the initial environment state, is established from a system of integro-differential equations. One example is given with claim sizes that have exponential distributions.

scholarly journals On the machine swing dynamics: a perspective 

2021 ◽  
Author(s):  
Prashant G. Medewar ◽  
Shambhu N. Sharma
Keyword(s):  
Differential Equation ◽  
Markov Process ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Nonlinear Filtering ◽  
Applied Mathematics ◽  
Formal Approach ◽  
Augmented State ◽  
Swing Equation

Abstract A formal approach to rephrase nonlinear filtering of stochastic differential equations is the Kushner setting in applied mathematics and dynamical systems. Thanks to the ability of the Carleman linearization, the ‘nonlinear’ stochastic differential equation can be equivalently expressed as a finite system of ‘bilinear’ stochastic differential equations with the augmented state under the finite closure. Interestingly, the novelty of this paper is to embed the Carleman linearization into a stochastic evolution of the Markov process. To illustrate the Carleman linearization of the Markov process, this paper embeds the Carleman linearization into a nonlinear swing stochastic differential equation. Furthermore, we achieve the nonlinear swing equation filtering in the Carleman setting. Filtering in the Carleman setting has simplified algorithmic procedure. The concerning augmented state accounts for the nonlinearity as well as stochasticity. We show that filtering of the nonlinear stochastic swing equation in the Carleman framework is more refined as well as sharper in contrast to benchmark nonlinear EKF. This paper suggests the usefulness of the Carleman embedding into the stochastic differential equation to filter the concerning nonlinear stochastic differential system. This paper will be of interest to nonlinear stochastic dynamists exploring and unfolding linearization embedding techniques to their research.

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