Fractional stochastic integration and Black–Scholes equation for fractional Brownian model with stochastic volatility

We modify the Hu-Øksendal and Elliot-van der Hoek approach to arbitrage-free financial markets driven by a fractional Brownian motion that is defined on a white noise space. We deduce and solve a Black–Scholes fractional equation for constant volatility and outline the corresponding equation with stochastic volatility. As an auxiliary result, we produce some simple conditions implying the existence of the Wick integral w.r.t. fractional noise.     

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.     

Algebraic solution of the Stein–Stein model for stochastic volatility

We provide an algebraic approach to the solution of the Stein–Stein model for stochastic volatility which arises in the determination of the Radon–Nikodym density of the minimal entropy of the martingale measure. We extend our investigation to the case in which the parameters of the model are time-dependent. Our algorithmic approach obviates the need for Ansätze for the structure of the solution.     

Positivity-preserving symplectic methods for the stochastic Lotka–Volterra predator-prey model

BIT Numerical Mathematics - In this paper, positivity-preserving symplectic numerical approximations are investigated for the 2d-dimensional stochastic Lotka–Volterra predator-prey model...     

A transient stochastic simulation–optimization model for operational fuel planning in-theater

Army fuel planners are responsible for developing daily loading plans that specify which tankers to load, with what fuel, and where to send the loaded tankers. The tools used to accomplish this task are custom built spreadsheets which require large amounts of time and effort to use, update, and keep free of errors. This research presents a transient stochastic simulation–optimization model of the in-theater bulk fuel supply chain, where the simulation model is used to simulate the performance of the fuel supply chain under a particular fuel distribution policy and the optimization portion is used to update the policy so that it results in the performance desired by the Army fuel planner. The fuel distribution policy can then be used to derive the daily loading plan. Due to the multi-objective nature of the problem, the set of policies that form the efficient frontier are all candidate policies for the Army fuel planner to select from. Results of experimentation with a wide variety of supply chain scenarios indicate that, for a given supply chain scenario, the optimization portion of the model identifies a set of fuel distribution policies that address the objectives of the Army fuel planner. In addition, the simulation–optimization model comfortably solves the largest supply chain scenarios the Army fuel planner would reasonably be expected to encounter.     

Asymptotic behavior of non-autonomous stochastic Gilpin–Ayala competition model with jumps

In this paper, a non-autonomous stochastic Gilpin–Ayala competition model with jumps is studied. We show that this model has a unique global positive solution under certain conditions, and establish sufficient conditions for stochastic ultimate boundedness. Asymptotic behavior of stochastic Gilpin–Ayala competition model with jumps is also discussed. Numerical examples are introduced to illustrate the results.     

Analysis of a delayed stochastic predator–prey model in a polluted environment

In this paper, we investigate two-species Lotka–Volterra delayed stochastic predator–prey systems, with and without pollution, denoted by (M)$(M)$ and (_M0)$(M_0)$, respectively. We show that there exists a unique non-negative solution in each system that is permanent in time average under certain conditions. Moreover, the non-permanence of model (M)$(M)$ is studied. Finally, computer simulations are carried out to verify our results.     

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.     

Unsteady flows of a viscoelastic fluid with the fractional Burgers’ model

The aim of this work is to discuss some unidirectional flows of a viscoelastic fluid between two parallel plates with fractional Burgers’ fluid model. The exact analytical solutions for Plane Poiseuille and Plane Couette flows are obtained by using the finite Fourier sine transform and the Laplace transform. Moreover, the graphs are plotted to show the effects of different parameters on the velocity field.     

Uniform stabilization to equilibrium of a nonlinear fluid–structure interaction model

We consider uniform stability to a nontrivial equilibrium of a nonlinear fluid–structure interaction (FSI) defined on a two or three dimensional bounded domain. Stabilization is achieved via boundary and/or interior feedback controls implemented on both the fluid and the structure. The interior damping on the fluid combining with the viscosity effect stabilizes the dynamics of fluid. However, this dissipation propagated from the fluid alone is not sufficient to drive uniformly to equilibrium the entire coupled system. Therefore, additional interior damping on the wave component or boundary porous like damping on the interface is considered. A geometric condition on the interface is needed if only boundary damping on the wave is active. The main technical difficulty is the mismatch of regularity of hyperbolic and parabolic component of the coupled system. This is overcome by considering special multipliers constructed from Stokes solvers. The uniform stabilization result obtained in this article is global for the fully coupled FSI model.     

Stationary distribution and extinction of a stochastic predator–prey model with distributed delay

In this paper, we focus on a stochastic predator–prey model with distributed delay. We first obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Then we establish sufficient conditions for extinction of the predator population, that is, the prey population is survival and the predator population is extinct.     

Generalised soft binomial American real option pricing model (fuzzy–stochastic approach)

The stochastic discrete binomial models and continuous models are usually applied in option valuation. Valuation of the real American options is solved usually by the numerical procedures. Therefore, binomial model is suitable approach for appraising the options of American type. However, there is not in several situations especially in real option methodology application at to disposal input data of required quality. Two aspects of input data uncertainty should be distinguished; risk (stochastic) and vagueness (fuzzy). Traditionally, input data are in a form of real (crisp) numbers or crisp-stochastic distribution function. Therefore, hybrid models, combination of risk and vagueness could be useful approach in option valuation. Generalised hybrid fuzzy–stochastic binomial American real option model under fuzzy numbers (T-numbers) and Decomposition principle is proposed and described. Input data (up index, down index, growth rate, initial underlying asset price, exercise price and risk-free rate) are in a form of fuzzy numbers and result, possibility-expected option value is also determined vaguely as a fuzzy set. Illustrative example of equity valuation as an American real call option is presented.     

Asymptotic behavior of a Lotka–Volterra food chain stochastic model in the Chemostat

This article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results.     

A steady Navier–Stokes model for compressible fluid with partially strong solutions

In this Note, we prove the existence of a partially strong solution to the steady Navier–Stokes equations for viscous barotropic compressible fluids, in a bounded simply connected domain of ${\mathbb{R}}^3$ with the prescribed generalized impermeability conditions ${\mathbf{curl}}^k\mathbf{u}?\mathbf{n}=0$, $k=0,1,2$ on the boundary. We call the solution “partially strong” because only the divergence-free part of the velocity field and the associated effective pressure have regularity typical for strong solution, while the density and the gradient part of the velocity have regularity typical for weak solution.     

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.     

A multiscale Darcy–Brinkman model for fluid flow in fractured porous media

The aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures. The Darcy’s law is assumed in the porous domain while the Stokes–Brinkman equations are considered in the fractures. We address the case of fractures whose thickness is very small compared to the characteristic diameter of the computational domain, and describe the fracture as if it were an interface between porous regions. We derive the corresponding interface model governing the fluid flow in the fracture and in the porous media, and establish the well-posedness of the coupled problem. Further, we introduce a finite element scheme for the approximation of the coupled problem, and discuss solution strategies. We conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non-reduced one.     

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.