scholarly journals Solutions to the jump-diffusion linear stochastic differential equations

2019 ◽  
Vol 3 (2) ◽  
pp. 115-119
Author(s):  
Dang Kien Cuong ◽  
Duong Ton Dam ◽  
Duong Ton Thai Duong ◽  
Du Thuan Ngo
Keyword(s):  
Differential Equations ◽  
Diffusion Process ◽  
Stochastic Differential Equations ◽  
Wiener Process ◽  
Lévy Process ◽  
Random Measure ◽  
Jump Diffusion ◽  
Continuous Functions ◽  
Levy Process ◽  
Jump Diffusion Process

The jump-diffusion stochastic process is one of the most common forms in reality (such as wave propagation, noise propagation, turbulent flow, etc.), and researchers often refer to them in models of random processes such as Wiener process, Levy process, Ito-Hermite process, in research of G. D. Nunno, B. Oksendal, F. B. Hanson, etc. In our research, we have reviewed and solved three problems: (1) Jump-diffusion process (also known as the Ito-Levy process); (2) Solve the differential equation jump-diffusion random linear, in the case of one-dimensional; (3) Calculate the Wiener-Ito integral to the random Ito-Hermite process. The main method for dealing with the problems in our presentation is the Ito random-integrable mathematical operations for the continuous random process associated with the arbitrary differential jump by the Poisson random measure. This study aims to analyse the basic properties of jump-diffusion process that are solutions to the jump-diffusion linear stochastic differential equations: dX(t) = [a (t)X (t􀀀)+A(t)]dt + [b (t)X (t􀀀 ∫ )+B(t)]dW (t) + R0 [g (t; z)X (t􀀀)+G(t; z)] ¯N (dt;dz) with a set of stochastic continuous functions fa;b ;g ;A;B;Gg and assuming that the compensated Poisson process ¯N (t; z) is independent of the Wiener process W(t). Derived from the Ito-Hermite formulas for the Ito-Hermite process and for the Ito-Levy process class we presented the results for the differential and multiple stochastic integration for the Ito- Hermite process. We also provided a separation method to solve jump-diffusion linear differential equations.  

scholarly journals REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY A LÉVY PROCESS

2009 ◽  
Vol 50 (4) ◽  
pp. 486-500 ◽  
Author(s):  
YONG REN ◽  
XILIANG FAN
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Process ◽  
Backward Stochastic Differential Equations ◽  
Levy Process ◽  
Penalization Method ◽  
Lower Semi ◽  
Uniqueness Of The Solution ◽  
Teugels Martingales ◽  
Continuous Convex Function

AbstractIn this paper, we deal with a class of reflected backward stochastic differential equations (RBSDEs) corresponding to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness of the solution for RBSDEs by means of the penalization method. As an application, we give a probabilistic interpretation for the solutions of a class of partial differential-integral inclusions.

scholarly journals Monte Carlo Methods and New Jump Diffusion Processes and Their Application in Gold Price

2017 ◽  
Author(s):  
Kutluk Kağan Sümer
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Standard Deviation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Mean Reversion ◽  
Jump Diffusion ◽  
Gold Market ◽  
Jump Diffusion Process ◽  
Standard Deviations

This study aimed to execute Monte Carlo simulation method with Wiener Process, Generalized Wiener Process, Mean Reversion Process and Mean Reversion Jump Diffusion Process and to compare them and then expended with the idea of how to include negative and positive news shocks in the gold market to the Monte Carlo simulation. By enhancing the determination of the 3 standard deviation shocks within the process of Classic Mean Jump Diffusion Process, an enchanted model for the 1,96 and 3 standard deviation shocks were being used and additionally positive and negative shocks were added to the system in a different way. This new Mean Reversion Jump Diffusion Process that have been developed by Sümer, executes Monte Carlo simulation regarding the gold market return with five random variables that are chosen from Poisson distribution and one random variable chosen from the normal distribution. Additionally, by accepting volatilities as outlies over the 1,96 and 3 standard deviations with the effect of the new and good news and the standard deviations on the traditional approximate return and the standard deviations (volatility) and the obtained new approximate return and the new standard deviation (volatility) and compares them with the Monte Carlo simulations.

Reflected generalized BSDEs with discontinuous barriers driven by a Lévy process

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed El Otmani
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Process ◽  
Backward Stochastic Differential Equations ◽  
Levy Process ◽  
Uniqueness Of The Solution ◽  
Teugels Martingales

Abstract This article deals with the reflected and doubly reflected generalized backward stochastic differential equations when the noise is given by Brownian motion and Teugels martingales associated with an independent pure jump Lévy process. We prove the existence and the uniqueness of the solution for these equations with monotone generators and right continuous left limited obstacles.

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