A Class of Jump-Diffusion Stochastic Differential System Under Markovian Switching and Analytical Properties of Solutions

2020 ◽  
Vol 8 (1) ◽  
pp. 17-32
Author(s):  
Xiangdong Liu ◽  
Zeyu Mi ◽  
Huida Chen
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
Stochastic Differential Equation ◽  
Differential System ◽  
Numerical Solutions ◽  
Jump Diffusion ◽  
Markovian Switching ◽  
Analytical Properties ◽  
Global Nature ◽  
First Moment

AbstractOur article discusses a class of Jump-diffusion stochastic differential system under Markovian switching (JD-SDS-MS). This model is generated by introducing Poisson process and Markovian switching based on a normal stochastic differential equation. Our work dedicates to analytical properties of solutions to this model. First, we give some properties of the solution, including existence, uniqueness, non-negative and global nature. Next, boundedness of first moment of the solution to this model is considered. Third, properties about coefficients of JD-SDS-MS is proved by using a right continuous markov chain. Last, we study the convergence of Euler-Maruyama numerical solutions and apply it to pricing bonds.

scholarly journals Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Dan Li ◽  
Jing’an Cui ◽  
Guohua Song
Keyword(s):  
Differential Equation ◽  
Initial Data ◽  
Stochastic Differential Equation ◽  
Asymptotic Behaviour ◽  
Average Rate ◽  
Sufficient Conditions ◽  
Jump Diffusion ◽  
Volterra Model ◽  
Environmental Perturbations ◽  
Stochastically Ultimate Boundedness

This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associated with our model has a unique global positive solution and give sufficient conditions that ensure stochastically ultimate boundedness, moment average boundedness in time, and asymptotic polynomial growth of our model; (c) the sufficient conditions for the extinction of the system are obtained, which generalized the former results and showed that the sufficiently large random jump magnitudes and intensity (average rate of jump events arrival) may lead to extinction of the population.

scholarly journals Portfolio problems based on jump-diffusion models

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 573-583 ◽  
Author(s):  
Tiantian Liu ◽  
Jun Zhao ◽  
Peibiao Zhao
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Asset Price ◽  
Efficient Frontier ◽  
Jump Diffusion ◽  
Natural Generalization ◽  
Optimal Portfolio ◽  
Linear Quadratic ◽  
Price Process ◽  
Portfolio Problem

Zhou and Li [49] by virtue of stochastic linear-quadratic control theory studied the optimal portfolio problems with the asset price process satisfying a diffusion stochastic differential equation, and proposed the celebrated LQ framework and the efficient frontier for the given portfolio problem. In this paper, we consider the optimal portfolio problems based on the asset price process satisfying a jump-diffusion stochastic differential equation. Similarly, we also arrive at the efficient frontier of the optimal portfolio selection problem. The conclusions obtained here can be regarded as a natural generalization of the work by Zhou and Li [49].

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

2015 ◽  
Vol 5 (2) ◽  
pp. 192-208 ◽  
Author(s):  
Ning Li ◽  
Bo Meng ◽  
Xinlong Feng ◽  
Dongwei Gui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Finite Difference ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Polynomial Chaos ◽  
Numerical Solutions ◽  
Numerical Comparison ◽  
Fe Method

AbstractA numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

scholarly journals A Stochastic Model with Jumps for Smoking

2019 ◽  
pp. 1-16
Author(s):  
Mohamed Coulibaly ◽  
Modeste N'Zi
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Diffusion Coefficient ◽  
Stochastic Differential Equation ◽  
Equilibrium State ◽  
Deterministic Model ◽  
Jump Diffusion ◽  
Epidemiological Models ◽  
Sirs Model ◽  
Theoretical Results

Some stochastic epidemiological models are less significant. They do not take into account some sudden events that could disrupt the behavior of the studied phenomenon. In this work, we introduce a white noise and jumps in a deterministic SIRS model for smoking to take into account of the effects of randomly fluctuation and such sudden factors respectively. First of all we prove that the solution of the stochastic differential equation with jumps of the new modelis positive. Then we study the asymptotic behavior around the smoking-free equilibrium state and the smoking-present equilibrium state of the original deterministic model. Under certain conditions, we show that the solution oscillate respectively around these equilibrium states. We prove that the intensity of these oscillations depends on the magnitude of noise and the jump diffusion coefficient of our stochastic differential equation with jumps. To support our theoretical results, we realise numerical simulations. The observations confirm our conclusions.

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