Stability analysis of a stochastic Gilpin–Ayala model driven by Lévy noise

A stochastic one-dimensional Gilpin–Ayala model driven by Lévy noise is presented in this paper. Firstly, we show that this model has a unique global positive solution under certain conditions. Then sufficient conditions for the almost sure exponential stability and moment exponential stability of the trivial solution are established. Results show that the jump noise can make the trivial solution stable under some conditions. Numerical example is introduced to illustrate the results.     

Reflected backward doubly stochastic differential equations driven by a Lévy process

We prove the existence and uniqueness of a solution for reflected backward doubly stochastic differential equations (RBDSDEs) driven by Teugels martingales associated with a Lévy process, in which the obstacle process is right continuous with left limits (càdlàg), via Snell envelope and the fixed point theorem.     

A note on the doubly reflected backward stochastic differential equations driven by a Lévy process

In this note, we study the doubly reflected backward stochastic differential equations driven by Teugels martingales associated with a Lévy process (DRBSDELs for short). In our framework, the reflecting barriers are allowed to have general jumps. Under the Mokobodski condition, by means of the Snell envelope theory as well as the fixed point theory, we show the existence and uniqueness of the solution of the DRBSDELs. Some known results are generalized.     

Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process

In this paper, a new concept of Poisson asymptotically almost automorphy for stochastic processes is introduced. And then, some fundamental properties including composition theorems for the space of such processes are proved. Subsequently, this concept is applied to investigate the existence and uniqueness of asymptotically almost automorphic solutions in distribution to some linear and semilinear stochastic differential equations driven by a Lévy process under some suitable conditions. Finally, an example is given to illustrate the main results.     

Invariant measure for the stochastic Cauchy problem driven by a cylindrical Lévy process

In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process and demonstrate their application in some examples.     

Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process

The Euler scheme is a well-known method of approximation of solutions of stochastic differential equations (SDEs). A lot of results are now available concerning the precision of this approximation in case of equations driven by a drift and a Brownian motion. More recently, people got interested in the approximation of solutions of SDEs driven by a general Lévy process. One of the problem when we use Lévy processes is that we cannot simulate them in general and so we cannot apply the Euler scheme. We propose here a new method of approximation based on the cutoff of the small jumps of the Lévy process involved. In order to find the speed of convergence of our approximation, we will use results about stability of the solutions of SDEs.     

How to test that a given process is an Ornstein–Uhlenbeck process

Statistical Inference for Stochastic Processes - We show asymptotic distributions of the residual process in Ornstein–Uhlenbeck model, when the model is true. These distributions are of...     

Orbits in a stochastic Goodwin–Lotka–Volterra model

This paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the Lotka–Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic highly non-linear model. We then study a stochastic version, with Brownian noise introduced via a heterogeneous productivity factor. Existence conditions for a solution to the system are provided. We prove that the system produces cycles around a unique equilibrium point in finite time for general volatility levels, using stochastic Lyapunov techniques for recurrent domains. Numerical insights are provided.     

Pseudo almost automorphic solution to stochastic differential equation driven by Lévy process

Almost automorphic is a particular case of the recurrent motion, which has been studied in differential equations for a long time. We introduce square-mean pseudo almost automorphic and some of its properties, and then study the pseudo almost automorphic solution in the distribution sense to stochastic differential equation driven by Lévy process.     

The threshold of a stochastic Susceptible–Infective epidemic model under regime switching

In this paper, we consider a stochastic Susceptible–Infective (SI) epidemic model under regime switching. Firstly, by constructing suitable Lyapunov functions, we establish sufficient criteria for the existence and uniqueness of an ergodic stationary distribution. Then we obtain the threshold which guarantees the extinction and the existence of the stationary distribution of the epidemic. Finally, some numerical simulations are introduced to illustrate our main results.     

The variational approach to Hamilton–Jacobi equations driven by a Gaussian noise

The global existence and uniqueness of viscosity solutions to the Cauchy problem for the Hamilton–Jacobi equations in R^N${\mathbb{R}}^N$ driven by additive and multiplicative Wiener processes are studied for convex Hamiltonians via variational techniques. The finite speed of propagation is also established in the multiplicative noise case for equations with Lipschitzian Hamiltonians.     

Persistence and extinction of a stochastic non-autonomous Gilpin–Ayala system driven by Lévy noise

In this paper, we consider the persistence and extinction of a stochastic non-autonomous Gilpin–Ayala system driven by Lévy noise. Sufficient criteria for extinction, non-persistence in the mean and weak persistence of the system are established. The threshold between weak persistence and extinction is obtained. From the results we can see that both persistence and extinction have close relationships with Lévy noise. Some simulation figures are introduced to demonstrate the analytical findings.     

A new approach to the stabilization of a fluid–structure interaction model

This article is a continuation of our work on a linear fluid–structure interaction model [Grobbelaar-Van Dalsen, On a fluid–structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech. 10(3) (2008), pp. 388–401; Grobbelaar-Van Dalsen, Strong stability for a fluid––structure model, Math. Methods Appl. Sci., 32(2009) pp. 1452–1466]. The model describes the interaction between a 3-D incompressible fluid and a 2-D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with the inclusion of the shear stress that the fluid exerts on the plate. A dissipative damping mechanism of Kelvin–Voigt type is applied to the interior of the plate. While our earlier work shows that weak solutions in a space of finite energy are strongly asymptotically stable under no-slip transmission conditions at the interface with uniform exponential stability only attainable under an additional domination condition, the present research is directed at achieving uniform exponential stability of weak solutions without imposing the domination condition. Using energy methods we establish uniform exponential decay under a modified transmission condition at the interface. This condition entails that the fluid velocity at the interface is coupled to a linear combination of the plate velocity and displacement.     

The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid

The velocity field corresponding to the Rayleigh–Stokes problem for an edge, in an incompressible generalized Oldroyd-B fluid has been established by means of the double Fourier sine and Laplace transforms. The fractional calculus approach is used in the constitutive relationship of the fluid model. The obtained solution, written in terms of the generalized G-functions, is presented as a sum of the Newtonian solution and the corresponding non-Newtonian contribution. The solution for generalized Maxwell fluids, as well as those for ordinary Maxwell and Oldroyd-B fluids, performing the same motion, is obtained as a limiting case of the present solution. This solution can be also specialized to give the similar solution for generalized second grade fluids. However, for simplicity, a new and simpler exact solution is established for these fluids. For β → 1, this last solution reduces to a previous solution obtained by a different technique.      

Optimal harvesting policy for a stochastic predator–prey model

Optimal harvesting of a stochastic predator–prey model is considered in this paper. Sufficient and necessary criteria for the existence of optimal harvesting strategy are obtained. At the same time, the optimal harvest effort and the maximum of sustainable yield are given.     

The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid

The velocity field corresponding to the Rayleigh–Stokes problem for an edge, in an incompressible generalized Oldroyd-B fluid has been established by means of the double Fourier sine and Laplace transforms. The fractional calculus approach is used in the constitutive relationship of the fluid model. The obtained solution, written in terms of the generalized G-functions, is presented as a sum of the Newtonian solution and the corresponding non-Newtonian contribution. The solution for generalized Maxwell fluids, as well as those for ordinary Maxwell and Oldroyd-B fluids, performing the same motion, is obtained as a limiting case of the present solution. This solution can be also specialized to give the similar solution for generalized second grade fluids. However, for simplicity, a new and simpler exact solution is established for these fluids. For β → 1, this last solution reduces to a previous solution obtained by a different technique.     

Parameter estimation of Ornstein–Uhlenbeck process generating a stochastic graph

Given Y a graph process defined by an incomplete information observation of a multivariate Ornstein–Uhlenbeck process X, we investigate whether we can estimate the parameters of X. We define two statistics of Y. We prove convergence properties and show how these can be used for parameter inference. Finally, numerical tests illustrate our results and indicate possible extensions and applications.     

An optimal investment strategy for a stream of liabilities generated by a step process in a financial market driven by a Lévy process

In this paper we investigate an asset–liability management problem for a stream of liabilities written on liquid traded assets and non-traded sources of risk. We assume that the financial market consists of a risk-free asset and a risky asset which follows a geometric Lévy process. The non-tradeable factor (insurance risk or default risk) is driven by a step process with a stochastic intensity. Our framework allows us to consider financial risk, systematic and unsystematic insurance loss risk (including longevity risk), together with possible dependencies between them. An optimal investment strategy is derived by solving a quadratic optimization problem with a terminal objective and a running cost penalizing deviations of the insurer’s wealth from a specified profit-solvency target. Techniques of backward stochastic differential equations and the weak property of predictable representation are applied to obtain the optimal asset allocation.