Existence of optimal controls for SPDE with locally monotone coefficients

The aim of this paper is to investigate the existence of optimal controls for systems described by stochastic partial differential equations (SPDEs) with locally monotone coefficients controlled by external forces which are feedback controls. To attain our objective we adapt the argument of Lisei (2002) where the existence of optimal controls to the stochastic Navier–Stokes equation was studied. The results obtained in the present paper may be applied to demonstrate the existence of optimal controls to various types of controlled SPDEs such as: a stochastic nonlocal equation and stochastic semilinear equations which are locally monotone equations; we also apply the result to a monotone equation such as the stochastic reaction–diffusion equation and to a stochastic linear equation.     

Stochastic elastic–plastic finite elements

A computational framework has been developed for simulations of the behavior of solids and structures made of stochastic elastic–plastic materials. Uncertain elastic–plastic material properties are modeled as random fields, which appear as the coefficient term in the governing partial differential equation of mechanics. A spectral stochastic elastic–plastic finite element method with Fokker–Planck–Kolmogorov equation based probabilistic constitutive integrator is proposed for solution of this non-linear (elastic–plastic) partial differential equation with stochastic coefficient. To this end, the random field material properties are discretized, in both spatial and stochastic dimension, into finite numbers of independent basic random variables, using Karhunen–Loève expansion. Those random variables are then propagated through the elastic–plastic constitutive rate equation using Fokker–Planck–Kolmogorov equation approach, to obtain the evolutionary material properties, as the material plastifies. The unknown displacement (solution) random field is then assembled - using polynomial chaos - as a function of known basic random variables and unknown deterministic coefficients, which are obtained by minimizing the error of finite representation, by Galerkin technique.     

Stochastic differential games and inverse optimal control and stopper policies

In this paper, we consider a two-player stochastic differential game problem over an infinite time horizon where the players invoke controller and stopper strategies on a nonlinear stochastic differential game problem driven by Brownian motion. The optimal strategies for the two players are given explicitly by exploiting connections between stochastic Lyapunov stability theory and stochastic Hamilton–Jacobi–Isaacs theory. In particular, we show that asymptotic stability in probability of the differential game problem is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Isaacs equation, and hence, guaranteeing both stochastic stability and optimality of the closed-loop control and stopper policies. In addition, we develop optimal feedback controller and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the stochastic differential game problem. These results are then used to provide extensions of the linear feedback controller and stopper policies obtained in the literature to nonlinear feedback controllers and stoppers that minimise and maximise general polynomial and multilinear performance criteria.     

Nonlinear–nonquadratic optimal and inverse optimal control for stochastic dynamical systems

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for nonlinear stochastic dynamical systems. Specifically, we provide a simplified and tutorial framework for stochastic optimal control and focus on connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory. In particular, we show that asymptotic stability in probability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady‐state form of the stochastic Hamilton–Jacobi–Bellman equation and, hence, guaranteeing both stochastic stability and optimality. In addition, we develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the stochastic stabilization problem. These results are then used to provide extensions of the nonlinear feedback controllers obtained in the literature that minimize general polynomial and multilinear performance criteria. Copyright © 2017 John Wiley & Sons, Ltd.     

Dempster–Shafer structure based fuzzy logic system for stochastic modeling

The Dempster–Shafer (D–S) theory of evidence is introduced to improve fuzzy inference under the complex stochastic environment. The Dempster–Shafer based fuzzy set (DFS) is first proposed, together with its union and intersection operations, to capture the principal stochastic uncertainties. Then, the fuzzy inference will be modified based on the extensional Dempster rule of combination. This new approach is able to capture the stochastic disturbance acting on fuzzy membership function, and provide a more effective inference under strong stochastic uncertainty. Finally, the numerical simulation and the experimental prediction of the wind speed are conducted to show the potential of the proposed method in stochastic modeling.     

Numerical solution of continuous-time mean–variance portfolio selection with nonlinear constraints

An investment problem is considered with dynamic mean–variance (M–V) portfolio criterion under discontinuous prices described by jump-diffusion processes. Some investment strategies are restricted in the study. This M–V portfolio with restrictions can lead to a stochastic optimal control model. The corresponding stochastic Hamilton–Jacobi–Bellman equation of the problem with linear and nonlinear constraints is derived. Numerical algorithms are presented for finding the optimal solution in this article. Finally, a computational experiment is to illustrate the proposed methods by comparing with M–V portfolio problem which does not have any constraints.     

Optimal control problems for stochastic delay evolution equations in Banach spaces

We consider the optimal control for a Banach space valued stochastic delay evolution equation. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of backward stochastic differential equations. An application to optimal control of stochastic delay partial differential equations is also given.     

LSM Algorithm for Pricing American Option Under Heston–Hull–White’s Stochastic Volatility Model

In this paper, we present American option pricing under Heston–Hull–White’s stochastic volatility and stochastic interest rate model. To do this, we first discretize the stochastic processes with Euler discretization scheme. Then, we price American option by using least-squares Monte Carlo algorithm. We also compare the numerical results of our model with the Heston-CIR model. Finally, numerical results show the efficiency of the proposed algorithm for pricing American option under the Heston–Hull–White model.     

Choice of θ and its effects on stability in the stochastic θ-method of stochastic delay differential equations

The second author's work [F. Wu, X. Mao, and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math. 115 (2010), pp. 681–697] and Mao's papers [D.J. Higham, X. Mao, and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal. 45 (2007), pp. 592–607; X. Mao, Y. Shen, and G. Alison, Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations, J. Comput. Appl. Math. 235 (2011), pp. 1213–1226] showed that the backward Euler–Maruyama (BEM) method may reproduce the almost sure stability of stochastic differential equations (SDEs) without the linear growth condition of the drift coefficient and the counterexample shows that the Euler–Maruyama (EM) method cannot. Since the stochastic θ-method is more general than the BEM and EM methods, it is very interesting to examine the interval in which the stochastic θ-method can capture the stability of exact solutions of SDEs. Without the linear growth condition of the drift term, this paper concludes that the stochastic θ-method can capture the stability for θ∈(1/2, 1]. For θ∈[0, 1/2), a counterexample shows that the stochastic θ-method cannot reproduce the stability of the exact solution. Finally, two examples are given to illustrate our conclusions.     

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.     

A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients

In this note, we study the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations under some Carathéodory-type conditions on the coefficients by means of the successive approximation. In particular, we generalize and improve the results that appeared in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139–154] and Bao and Hou [J. Bao, Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl. 59 (2010) 207–214].     

Treatment of uncertain material interfaces in compressible flows

To treat uncertain interface position is an important issue for complex applications. In this paper, we address the characterization of randomly perturbed interfaces between fluids thanks to stochastic modeling and uncertainty quantification through the 2D Euler system. The perturbed interface is modeled as a random field and represented by a Karhunen–Loève expansion. The stochastic 2D Euler system is solved applying Polynomial Chaos theory through the Intrusive Polynomial Moment Method (IPMM). This stochastic resolution method is fully explained and studied (theoretically and numerically). Stochastic Richtmyer–Meshkov unstable flows are solved and presented for several configurations of the uncertain interface (different rugosities) between the fluids. The probability density functions of the mass density of the fluid in the vicinity of the interface are computed built and compared for the different simulations: the system exhibits strong sensitivity with respect to the stochastic initially leading modes.     

Nonlinear stochastic passivity,feedback equivalence and global stabilization

This paper addresses several important issues including stochastic passivity, feedback equivalence and global stabilization for a class of nonlinear stochastic systems. Based on a nonlinear stochastic Kalman–Yacubovitch–Popov (KYP) lemma, we investigate the relationship between a stochastic passive system and the corresponding zero‐output system. Different from the deterministic case, it is shown for the first time that feedback equivalence to a stochastic passive system requires a strong minimum‐phase condition, not the minimum‐phase one. Following the stochastic passivity theory, global stabilization results are established for a class of nonlinear stochastic systems with relative degree 1≤r<n. An example is presented to illustrate the effectiveness of our results. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal control of a stochastic delay heat equation with boundary-noise and boundary-control

In this paper, a controlled stochastic delay heat equation with Neumann boundary-noise and boundary-control is considered. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of the backward stochastic differential equations, which is applied to the optimal control problem.     

Stability of stochastic dynamic random-structure systems with aftereffect and Markov switchings

Using the second Lyapunov–Krasovskii method, sufficient conditions are obtained for the asymptotic stochastic global stability, for the global stability, and mean-square stability of trivial solutions to systems of diffusion stochastic functional–differential equations with Markov switchings, and the theory is illustrated by two model problems.     

Stochastic Optimal Control of Finite Ensembles of Nanomagnets

We control ferromagnetic N-spin dynamics in the presence of thermal fluctuations by minimizing a quadratic functional subject to the stochastic Landau–Lifshitz–Gilbert equation. Existence of a weak solution of the stochastic optimal control problem is shown. The related first order optimality conditions consist of a coupled forward–backward SDE system, which is numerically solved by a structure-inheriting discretization, the least squares Monte-Carlo method to approximate related conditional expectations, and the new stochastic gradient method. Computational experiments are reported which motivate optimal controls in the case of interacting anisotropy, stray field, exchange energies, and acting noise.     

Continuous-time mean–variance portfolio selection with value-at-risk and no-shorting constraints

An investment problem is considered with dynamic mean–variance(M-V) portfolio criterion under discontinuous prices which follow jump–diffusion processes according to the actual prices of stocks and the normality and stability of the financial market. The short-selling of stocks is prohibited in this mathematical model. Then, the corresponding stochastic Hamilton–Jacobi–Bellman(HJB) equation of the problem is presented and the solution of the stochastic HJB equation based on the theory of stochastic LQ control and viscosity solution is obtained. The efficient frontier and optimal strategies of the original dynamic M-V portfolio selection problem are also provided. And then, the effects on efficient frontier under the value-at-risk constraint are illustrated. Finally, an example illustrating the discontinuous prices based on M-V portfolio selection is presented.     

Exponentially incremental dissipativity for nonlinear stochastic switched systems

ABSTRACT

In this paper, we investigate the exponentially incremental dissipativity for nonlinear stochastic switched systems by using the designed state-dependent switching law and multiple Lyapunov functions approach. Specifically, using incremental supply rate as well as a state dissipation inequality in expectation, a stochastic version of exponentially incremental dissipativity is presented. The sufficient conditions for nonlinear stochastic switched systems to be exponentially incrementally dissipative are given by the designed state-dependent switching law. Furthermore, the extended Kalman–Yakubovich–Popov conditions are derived by using two times continuously differentiable storage functions. Moreover, the incremental stability conditions in probability for nonlinear stochastic switched systems are derived based on exponentially incremental dissipativity. The exponentially incremental dissipativity is preserved for the feedback-interconnected nonlinear stochastic switched systems with the composite state-dependent switching law; meanwhile, the incremental stability in probability is preserved under some certain conditions. A numerical example is given to illustrate the validity of our results.     

Robust exponential stability of uncertain stochastic systems with probabilistic time‐varying delays

This paper is concerned with the problem of delay‐distribution–dependent robust exponential stability for uncertain stochastic systems with probabilistic time‐varying delays. Firstly, inspired by a class of networked systems with quantization and packet losses, we study the stabilization problem for a class of network‐based uncertain stochastic systems with probabilistic time‐varying delays. Secondly, an equivalent model of the resulting closed‐loop network‐based uncertain stochastic system is constructed. Different from the previous works, the proposed equivalent system model enables the controller design of the network‐based uncertain stochastic systems to enjoy the advantage of probability distribution characteristic of packet losses. Thirdly, by applying the Lyapunov‐Krasovskii functional approach and the stochastic stability theory, delay‐distribution–dependent robust exponential mean‐square stability criteria are derived, and the sufficient conditions for the design of the delay‐distribution–dependent controller are then proposed to guarantee the stability of the resulting system. Finally, a case study is given to show the effectiveness of the results derived. Moreover, the allowable upper bound of consecutive packet losses will be larger in the case that the probability distribution characteristic of packet losses is taken into consideration.     

Stochastic Evolutionary Game Dynamics and Their Selection Mechanisms

An underlying assumption of deterministic evolutionary game dynamics is that all individuals interact with each other in infinite populations, which seems unrealistic since in reality populations are always finite in size and even disturbed by stochastic effects and random drift. Developed in the context of finite populations and described by finite state Markov processes, stochastic evolutionary game dynamics have received much attention recently. However, the relationship between two types of evolutionary dynamics so far has failed to be thoroughly understood. In this paper, we establish several classes of selection mechanisms in large populations under which corresponding stochastic evolutionary dynamics approach the imitative dynamic (including the replicator dynamic), the impartial pairwise comparison dynamic (including the Smith dynamic) and the separable excess payoff dynamic (including the Brown–von Neumann–Nash dynamic) respectively in adjusted forms. In other words, we present intuitive interpretations from a statistical perspective for these deterministic dynamics by constructing their microscopic foundations in settings with finite but large populations.