We consider nonlinear parabolic stochastic equations of the form ∂ t u = Lu + λσ (u)ξ on the ball B(0, R), whereξ denotes some Gaussian noise and σ is Lipschitz continuous. Here L corresponds to a symmetric α-stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in
This paper considers the dynamic behaviours of a hybrid stochastic population model with impulsive perturbations. The existence of the global positive solution is studied in this paper. Moreover, under some conditions on the noises and impulsive perturbations, the properties of the persistence and extinction, stochastic permanence, global attractivity and stability in distribution are presented. Our results illustrate that impulsive perturbations play a crucial role in these properties. The bounded impulse term will not affect these properties, however, when the impulse term is unbounded, some of the properties, such as the persistence and extinction may be changed significantly. As a part of this paper, a couple of examples and numerical simulations are provided to illustrate our results.
We consider nonlinear parabolic stochastic equations of the form ∂tu = Lu + λσ(u) ξ on the ball B(0, R), where ξ denotes some Gaussian noise and σ is Lipschitz continuous. Here L corresponds to an α-stable process killed upon exiting B(0, R). We will consider two types of noise; spacetime white noise and spatially correlated noise. Under a linear growth condition on σ, we study growth properties of the second moment of the solutions. Our results are significant extensions of those in [8] and complement those of [11] and [10].
This article formulates and dissects a Black–Scholes model with regime switching that can be used to describe the performance of a complete market. An explicit integrand formula ϕ t , ω is obtained when the T -claim F ω is given for an attainable claim in this complete market. In addition, some perfect results are presented on how to hedge an attainable claim for this Black–Scholes model, and the price p of the European call and the self-financing portfolio θ t = θ 0 t , θ 1 t are given explicitly. Finally, some concluding remarks are provided to illustrate the theoretical results.
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