Stabilization of Stochastic Hybrid Linear Systems with Noise

Abstract

This article deals with the class of uncertain stochastic hybrid linear systems with noise. The uncertainties we are considering are of norm bounded type. The stochastic stabilization and robust stabilization problems are treated. Linear matrix inequality (LMI)-based sufficient conditions are developed to design the state feedback controller with constant gain that stochastically (robust stochastically) stabilizes the studied class of systems. Our results are mode independent and require only the complete access to the state vector. Numerical examples are given to show the effectiveness of the proposed results.     

Stochastic Correlation and Volatility Mean-reversion – Empirical Motivation and Derivatives Pricing via Perturbation Theory

Abstract

The dependence structure is crucial when modelling several assets simultaneously. We show for a real-data example that the correlation structure between assets is not constant over time but rather changes stochastically, and we propose a multidimensional asset model which fits the patterns found in the empirical data. The model is applied to price multi-asset derivatives by means of perturbation theory. It turns out that the leading term of the approximation corresponds to the Black–Scholes derivative price with correction terms adjusting for stochastic volatility and stochastic correlation effects. The practicability of the presented method is illustrated by some numerical implementations. Furthermore, we propose a calibration methodology for the considered model.     

Stochastic delay Lotka-Volterra model

We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka-Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model into the Itô form , and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A.     

Perpetual Options on Multiple Underlyings

Abstract

We study three classes of perpetual option with multiple uncertainties and American-style exercise boundaries, using a partial differential equation-based approach. A combination of accurate numerical techniques and asymptotic analyses is implemented, with each approach informing and confirming the other. The first two examples we study are a put basket option and a call basket option, both involving two stochastic underlying assets, whilst the third is a (novel) class of real option linked to stochastic demand and costs (the details of the modelling for this are described in the paper). The Appendix addresses the issue of pricing American-style perpetual options involving (just) one stochastic underlying, but in which the volatility is also modelled stochastically, using the Heston (1993) framework.     

Robust stochastic stability and H∞ performance for a class of uncertain impulsive stochastic systems

In this paper, we discuss the problem of robust stochastic stability and H_∞ performance for a class of uncertain impulsive stochastic systems under sampled measurements. The parameter uncertainties are assumed to be time-varying and value-bounded. We give a sufficient condition in terms of certain linear matrix inequalities (LMIs) to guarantee the uncertain impulsive stochastic system to be robustly stochastically stable. Furthermore, we discuss a stochastically stable filter, using the locally sampled measurements, which ensures both the stochastic stability and a prescribed level of H_∞ performance for the filtering error system for all admissible uncertainties. We give a sufficient condition for the existence of such a filter and an explicit expression of a desired filter if relevant conditions are satisfied.     

Stochastic Asymptotic Stability of Nowak-May Model with Variable Diffusion Rates

We consider a stochastically perturbed Nowak-May model of virus dynamics within a host. We prove the global existence of unique strong solution. Using the Lyapunov method, we found sufficient conditions for the stochastic asymptotic stability of equilibrium solutions of this model.     

Power processes and their application to reliability

We introduce the stochastic process we call a power process and show that under certain conditions it is stochastically monotone. We use it to model a deteriorative system where operating times and repair times are general discrete random variables, and illustrate our results with numerical examples.     

Resilient and robust finite-time H∞ control for uncertain discrete-time jump nonlinear systems

This paper studies the robust and resilient finite-time H_∞ control problem for uncertain discrete-time nonlinear systems with Markovian jump parameters. With the help of linear matrix inequalities and stochastic analysis techniques, the criteria concerning stochastic finite-time boundedness and stochastic H_∞ finite-time boundedness are initially established for the nonlinear stochastic model. We then turn to stochastic finite-time controller analysis and design to guarantee that the stochastic model is stochastically H_∞ finite-time bounded by employing matrix decomposition method. Applying resilient control schemes, the resilient and robust finite-time controllers are further designed to ensure stochastic H_∞ finite-time boundedness of the derived stochastic nonlinear systems. Moreover, the results concerning stochastic finite-time stability and stochastic finite-time boundedness are addressed. All derived criteria are expressed in terms of linear matrix inequalities, which can be solved by utilizing the available convex optimal method. Finally, the validity of obtained methods is illustrated by numerical examples.     

A stochastic model for interactions of hot gases with cloud droplets and raindrops

In this paper we analyze a stochastic model for interactions of hot gases with cloud droplets and raindrops. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic hot gases modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any hot gas dynamics model. We also carry out analysis on the stochastically ultimate boundedness, extinction and stability of the hot gases model.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

On Weak Solutions to Stochastic Differential Inclusions Driven by Semimartingales

Abstract

In this paper we consider weak solutions to stochastic inclusions driven by a general semimartingale. We prove the existence of weak solutions and equivalence with the existence of solutions to the martingale problem formulated to such inclusion. Using this we then analyze compactness property of solutions set. Presenting results extend some of those being known for stochastic differential inclusions of Itô's type.     

Second Order Stochastic Inclusion

Abstract

The purpose of the paper is to consider some stochastic control problems as a particular case of a more general theory, the stochastic inclusions theory. We discuss the existence of weak solutions to a stochastic inclusion of second order, driven by two general semimartingales. Finally we present some examples.     

The Upper and Lower Solutions Method for Stochastic Inclusions with Discontinuous Multivalued Mappings

Abstract

In this article, we consider a stochastic integral inclusion driven by semimartingale with discontinuous multivalued right hand side. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered sets. The presented studies extend some recent results both for deterministic differential inclusions and stochastic differential equations for increasing operators.     

随机时滞 Lotka-Volterra 模型

该文揭示了关于生物动态过程中的一类重要的模型, 随机时滞Lotka-Volterra模型的渐近行为, 这种随机过程的解具有很好的逼近性质:如, 解的轨道估计, 渐近性质, 而且它的解还具有随机有界性.     

Nonlinear Stochastic Difference Equations Driven by Martingales

Abstract

We provide in this paper a systematic development of nonlinear stochastic difference equations driven by martingales (that depend on a spatial parameter); three such equations are considered. We begin with the existence and uniqueness of solutions and continue with the study of stochastic properties, such as the martingale and Markov properties, along with ? irreducibility and recurrence. We discuss in the final section the discrete-time flow and asymptotic flow properties of the solution process.     

The obstacle problem for quasilinear stochastic integral-partial differential equations

ABSTRACT

We prove the existence and uniqueness of solutions to a kind of quasilinear stochastic integral-partial differential equations with obstacles. Our method is based on the probabilistic interpretation of the solutions so that penalization method can be applied to a sequence of backward doubly stochastic differential equations with jumps. Relations between regular potentials and regular measures play an important role.     

Dynamical behavior of a stochastic predator-prey model with stage structure for prey

Abstract

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings.     

Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

Abstract

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field.     

Stability in Terms of Two Measures for Stochastic Differential Equations with Markovian Switching

Abstract

In this paper, we investigate the stability in terms of two measures for stochastic differential equations with Markovian switching by using the method of Lyapunov functions. Our new theory can not only be used to show a given system to be stochastically stable in the classical sense, but can also be used to deal with some situations where the classical stability theory is not applicable.