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In this paper, we establish an existence and uniqueness theorem for solutions to backward stochastic differential equations driven by time-changed Lévy noises, in which the generator is monotonic and general growth with respect to y. Our conclusion extends the corresponding result in Di Nunno and Sjursen (Stoch. Process. Appl. 124:1679-1709, 2014). [ABSTRACT FROM AUTHOR]

This paper deals with a predator-prey model of Beddington-DeAngelis type functional response with Lévy jumps. The proposed mathematical model consists of a system of two stochastic differential equations to stimulate the interactions between predator population and prey population. The dynamics of the system is discussed mainly from the point of view of persistence and extinction. To begin with, the global positivity, stochastically boundedness and other asymptotic properties have been derived. In addition, sufficient conditions for extinction, nonpersistence in the mean and weak persistence are obtained. It is proved that the variation of Lévy jumps can affect the asymptotic property of the system. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of neutral stochastic differential equations (NSDEs) with the cylindrical Brownian motion and Lévy noises in an infinite-dimensional Hilbert space. The existence and uniqueness of the mild solutions to these stochastic differential equations are discussed under assumptions of linear growth on the coefficients. The results of Taniguchi (J. Math. Anal. Appl. 360:245-253, 2009) are generalized and improved as a special case of our theory. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two approximations in law of the complex fractional Brownian motion by processes constructed from a Poisson process and a Lévy process, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic Lotka-Volterra model with additive jump noises. We show some desired properties of the solution such as existence and uniqueness of positive strong solution, unique stationary distribution, and exponential ergodicity. After that, we investigate the maximum likelihood estimation for the drift coefficients based on continuous time observations. The likelihood function and explicit estimator are derived by using semimartingale theory. In addition, consistency and asymptotic normality of the estimator are proved. Finally, computer simulations are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract In the present paper, we employ a wavelets optimization method is employed for the elucidations of fractional partial differential equations of pricing European option accompanied by a Lévy model. We apply the Legendre wavelets optimization method (LWOM) to optimize the governing problem. The novelty of the proposed method is the inclusion of differential evolution algorithm (DE) in the Legendre wavelets method for the optimized approximations of the unknown terms of the Legendre wavelets. Sequentially, the functions and components of the pricing models are discretized by utilizing the operational matrix of fractional integration of Legendre wavelets. Illustratively, the implementation of the LWOM is exemplified on a pricing European option Lévy model and successfully depicted the stock paths. Moreover, comparison analysis of the Black-Scholes model with a class of Lévy model and LWOM with q-homotopy analysis transform method (q-HATM) is also deliberated out. Accordingly, the technique is found to be appropriate for financial models that can be expressed as partial differential equations of integer and fractional orders, subjected to initial or boundary conditions.

The objective of this work is to show a new kind of mean-field anticipated backward stochastic differential equation (in short MF-ABSDE) driven by time-changed Lévy noises. We give two methods to prove the existence and uniqueness of the solution of those equations by the fixed point theorem and the Picard iterative sequence. Finally, we obtain a comparison theorem for the solutions. [ABSTRACT FROM AUTHOR]