Given (Mt)t∈R+ and (M ∗ t )t∈R+ respectively a forward and a backward exponential martingale with jumps and a continuous part, we prove that E[φ(MtM∗ t )] is non-increasing in t when φ is a convex function, provided the local characteristics of the stochastic logarithms of (Mt)t∈R+ and of (M∗ t )t∈R+ satisfy some comparison inequalities. As an application, we deduce bounds on option prices in m...

We consider the exploration process associated to the continuous random tree (CRT) built using a Lévy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its in...

In this paper, we study the averaging principle for ψ-Capuo fractional stochastic delay differential equations (FSDDEs) with Poisson jumps. Based on calculus, Burkholder-Davis-Gundy’s inequality, Doob’s martingale and Ho¨lder prove that solution of averaged FSDDEs converges to standard in sense Lp. Our result extends some known results literature. Finally, an example simulation is performed sho...

the stochastic reaction diffusion systems may suffer sudden shocks‎, ‎in order to explain this phenomena‎, ‎we use markovian jumps to model stochastic reaction diffusion systems‎. ‎in this paper‎, ‎we are interested in almost sure exponential stability of stochastic reaction diffusion systems with markovian jumps‎. ‎under some reasonable conditions‎, ‎we show that the trivial solution of stocha...

The paper aims to establish a new sharp Burkholder-type maximal inequality in Lp for a class of stationary sequences that includes martingale sequences, mixingales and other dependent structures. The case when the variables are bounded is also addressed, leading to an exponential inequality for a maximum of partial sums. As an application we present an invariance principle for partial sums of c...

Using key tools such as Itô’s formula for general semimartingales, Kunita’s moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable. Keywords; stochastic differential equ...

For this paper, we consider the almost sure exponential stability of uncertain stochastic Hopfield neural networks based on subadditive measures. Firstly, deduce two corollaries, using Itô–Liu formula. Then, introduce concept for networks. Next, investigate networks, Lyapunov method, Liu inequality, lemma, and martingale inequality. In addition, prove sufficient conditions stability. Furthermor...

In this paper, we consider a class of stochastic pantograph differential equations with Lévy jumps (SPDEwLJs). By using the Burkholder-Davis-Gundy inequality and the Kunita’s inequality, we prove the existence and uniqueness of solutions to SPDEwLJs whose coefficients satisfying the Lipschitz conditions and the local Lipschitz conditions. Meantime, we establish the p-th exponential estimations ...

In this paper, we study the connections between three concepts — reverse Hölder inequality for matrix-valued martingales, well-posedness of linear BSDEs with unbounded coefficients, and quadratic BSDE systems. particular, show that a bmo coefficients is well-posed if only stochastic exponential related martingale satisfies inequality. Furthermore, give structural conditions under which these eq...

We derive results similar to Bo et al. (2010), but in the case of dynamics of the FX rate driven by a general Merton jump-diffusion process. The main results of our paper are as follows: 1) formulas for the Esscher transform parameters which ensure that the martingale condition for the discounted foreign exchange rate is a martingale for a general Merton jump-diffusion process are derived; usin...