Problem Of Stochastic Differential Equations Research Articles

Stochastic optimal control with random coefficients and associated stochastic Hamilton\u2013Jacobi\u2013Bellman equations

We consider the optimal control problem for stochastic differential equations (SDEs) with random coefficients under the recursive-type objective functional captured by the backward SDE (BSDE). Due to the random coefficients, the associated Hamilton–Jacobi–Bellman (HJB) equation is a class of second-order stochastic PDEs (SPDEs) driven by Brownian motion, which we call the stochastic HJB (SHJB) equation. In addition, as we adopt the recursive-type objective functional, the drift term of the SHJB equation depends on the second component of its solution. These two generalizations cause several technical intricacies, which do not appear in the existing literature. We prove the dynamic programming principle (DPP) for the value function, for which unlike the existing literature we have to use the backward semigroup associated with the recursive-type objective functional. By the DPP, we are able to show the continuity of the value function. Using the Itô–Kunita’s formula, we prove the verification theorem, which constitutes a sufficient condition for optimality and characterizes the value function, provided that the smooth (classical) solution of the SHJB equation exists. In general, the smooth solution of the SHJB equation may not exist. Hence, we study the existence and uniqueness of the solution to the SHJB equation under two different weak solution concepts. First, we show, under appropriate assumptions, the existence and uniqueness of the weak solution via the Sobolev space technique, which requires converting the SHJB equation to a class of backward stochastic evolution equations. The second result is obtained under the notion of viscosity solutions, which is an extension of the classical one to the case for SPDEs. Using the DPP and the estimates of BSDEs, we prove that the value function is the viscosity solution to the SHJB equation. For applications, we consider the linear-quadratic problem, the utility maximization problem, and the European option pricing problem. Specifically, different from the existing literature, each problem is formulated by the generalized recursive-type objective functional and is subject to random coefficients. By applying the theoretical results of this paper, we obtain the explicit optimal solution for each problem in terms of the solution of the corresponding SHJB equation.

Mean square and almost sure consensus of heterogeneous multi‐agent systems with multiplicative noise

AbstractThis paper studies the consensus problem of heterogeneous multi‐agent systems (MASs) composed of first‐order and second‐order agents. To describe the communication environment realistically, the effect of the multiplicative noise on the communication network is considered. Firstly, the consensus problem is transformed into the stability problem of stochastic differential equations with multiplicative noise. Then, a stability analysis method for such stochastic differential equations is proposed by structuring the appropriate Lyapunov function. Some sufficient conditions for both mean square (m.s.) and almost sure (a.s.) consensus are obtained for leader‐free/leader‐following heterogeneous MASs. These sufficient conditions, expressed by simple scalar inequalities, are explicitly related to control gains, noise intensities, and network topology. Furthermore, the m.s. and a.s. convergence rates of the consensus protocol are also obtained. Finally, numerical simulation shows the effectiveness of our theoretical results.

Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the SDE and the objective functional are allowed to be random, and the jump-diffusion part of the SDE depends on the state and control variables. Moreover, the cost parameters in the objective functional need not be (positive) definite matrices. Although the solution to this problem can also be obtained through the stochastic maximum principle or the dynamic programming principle, our approach is simple and direct. In particular, by using the Itô-Wentzell’s formula, together with the integro-type stochastic Riccati differential equation (ISRDE) and the backward SDE (BSDE) with jump diffusions, we obtain the equivalent objective functional that is quadratic in control u under the positive definiteness condition, where the approach is known as the completion of squares method. Then the explicit optimal solution, which is linear in state characterized by the ISRDE and the BSDE jump diffusions, and the associated optimal cost are derived by eliminating the quadratic term of u in the equivalent objective functional. We also verify the optimality of the proposed solution via the verification theorem, which requires solving the stochastic HJB equation, a class of stochastic partial differential equations with jump diffusions.

A linear–quadratic optimal control problem of stochastic differential equations with delay and partial information

This paper deals with a linear quadratic partially observed optimal control problem for the state governed by stochastic differential equations with delay. By backward separation method, variational method and filtering technique, a stochastic maximum principle for the optimal control is developed and its feedback representation is obtained. Also, a sufficient condition without the assumption of concavity is presented. At last, two examples are solved explicitly.

First-passage problem for stochastic differential equations with combined parametric Gaussian and Lévy white noises via path integral method

We study the first-passage problem for a process governed by a stochastic differential equation (SDE) driven simultaneously by both parametric Gaussian and Lévy white noises. We extend the path integral (PI) method to solve the SDE with this combined noise input and the corresponding fractional Fokker-Planck-Kolmogorov equations. Then, the PI solutions are modified to analyze the first-passage problem. Finally, numerical examples based on Monte Carlo simulations verify the extension of the PI method and the modification of the PI solutions. The detailed effects of the system parameters on the first-passage problem are analyzed.

Mean-field optimal control problem of SDDES driven by fractional Brownian Motion

We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than one half. Stochastic optimal ...

The pricing and numerical analysis of lookback options for mixed fractional Brownian motion

Abstract Using the stochastic differential equation driven by the composite Poisson process of mixed fractional Brownian motion, the price model of a mixed jump-diffusion fractional Brownian motion environment is established. Under the condition of Merton’s assumption, the Cauchy initial value problem of stochastic differential equations is iterated. The method is estimated, and the Merton formula of the European put option under the mixed jump-diffusion model is obtained, and the call-back option and the bearish option pricing formula of the mixed jump-diffusion fractional Brownian motion European floating strike price are given.

Enhancing human learning via spaced repetition optimization

Spaced repetition is a technique for efficient memorization which uses repeated review of content following a schedule determined by a spaced repetition algorithm to improve long-term retention. However, current spaced repetition algorithms are simple rule-based heuristics with a few hard-coded parameters. Here, we introduce a flexible representation of spaced repetition using the framework of marked temporal point processes and then address the design of spaced repetition algorithms with provable guarantees as an optimal control problem for stochastic differential equations with jumps. For two well-known human memory models, we show that, if the learner aims to maximize recall probability of the content to be learned subject to a cost on the reviewing frequency, the optimal reviewing schedule is given by the recall probability itself. As a result, we can then develop a simple, scalable online spaced repetition algorithm, MEMORIZE, to determine the optimal reviewing times. We perform a large-scale natural experiment using data from Duolingo, a popular language-learning online platform, and show that learners who follow a reviewing schedule determined by our algorithm memorize more effectively than learners who follow alternative schedules determined by several heuristics.

Limit theorems and the support of SDES with oblique reflections on nonsmooth domains

In this paper we consider the Wong–Zakai type approximation and the support problem of stochastic differential equations with oblique reflections in nonsmooth domains satisfying conditions introduced by Dupuis and Ishii [4]. We first establish the limit theorem for reflected diffusions by proving that the sequence of solutions of approximating equations converge uniformly in Lp to the unique solution of the reflected stochastic differential equation. We also prove the approximate continuity and thus characterize the support of diffusions described by reflected SDEs.

Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations

In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well.

Successive approximation of neutral functional stochastic differential equations with variable delays

By using successive approximation, we prove the existence and uniqueness of initial value problem for stochastic differential equations driven by both the cylindrical Brownian motion and by the variable delays in a Hilbert space with non-Lipschitzian coefficients. Moreover, the numerical solutions are shown to converge uniformly to the analytical solutions of the stochastic differential equation.

The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

The optimal control related to Riemannian manifolds and the viscosity solutions to Hamilton–Jacobi–Bellman equations

In this paper we study the optimal stochastic control problem for stochastic differential equations on Riemannian manifolds. The cost functional is specified by controlled backward stochastic differential equations in Euclidean space. Under some suitable assumptions, we conclude that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation which is a fully nonlinear parabolic partial differential equation on Riemannian manifolds.

Successive approximation to solutions of stochastic differential equations with jumps in local non-Lipschitz conditions

In this paper, by using successive approximation, the existence and uniqueness of initial value problem for stochastic differential equations driven by both Wiener process and Poisson process are studied under a local non-Lipschitz conditions. Moreover, the numerical solutions are shown to converge uniformly to the analytical solutions of the stochastic differential equation with jumps.

Multi-dimensional BSDE with oblique reflection and optimal switching

In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimates. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.

Optimal Stopping Problem for Stochastic Differential Equations with Random Coefficients

An optimal stopping problem for stochastic differential equations with random coefficients is considered. The dynamic programming principle leads to a Hamiltion-Jacobi-Bellman equation, which, for the current case, is a backward stochastic partial differential variational inequality (BSPDVI, for short) for the value function. Well-posedness of such a BSPDVI is established, and a verification theorem is proved.

An inverse problem for stochastic differential equations

We discuss the problem of reconstructing the drift coefficient of a diffusion from the knowledge of the transition probabilities outside a given bounded region in ℝ d ,d>1. We also give an interpretation of the solution of this inverse problem in the framework of stochastic mechanics.

Optimal Stopping of a Markov Process

Optimal Stopping of a Markov Process