Exponential Martingale Research Articles

Stability and rate of decay for solutions to stochastic differential equations with Markov switching

In this article, we present the almost sure asymptotic stability and a general rate of decay for solutions to stochastic differential equations (SDEs) with Markov switching. By establishing a suitable Lyapunov function and using an exponential Martingale inequality and the Borel-Cantelli theorem, we give sufficient conditions for the asymptotic stability. Also, we obtain sufficient conditions for the construction of two kinds of Lyapunov functions. Finally give two examples to illustrate the validity of our results. For more information see https://ejde.math.txstate.edu/Volumes/2024/01/abstr.html

Almost Sure Exponential Stability of Uncertain Stochastic Hopfield Neural Networks Based on Subadditive Measures

For this paper, we consider the almost sure exponential stability of uncertain stochastic Hopfield neural networks based on subadditive measures. Firstly, we deduce two corollaries, using the Itô–Liu formula. Then, we introduce the concept of almost sure exponential stability for uncertain stochastic Hopfield neural networks. Next, we investigate the almost sure exponential stability of uncertain stochastic Hopfield neural networks, using the Lyapunov method, Liu inequality, the Liu lemma, and exponential martingale inequality. In addition, we prove two sufficient conditions for almost sure exponential stability. Furthermore, we consider stabilization with linear uncertain stochastic perturbation and present some exceptional examples. Finally, our paper provides our conclusion.

Moderate deviations of generalized N-urn Ehrenfest models

This paper is a further investigation of the generalized N-urn Ehrenfest model introduced in Xue [Hydrodynamics of the generalized N-urn Ehrenfest model, Potential Anal. 2022. Online. Available at https://rdcu.be/cFzrj]. A moderate deviations principle from the hydrodynamic limit of the model is derived. The proof of this main result follows an exponential martingale approach introduced in C. Kipnis et al. [Hydrodynamics and large deviation for simple exclusion processes, Commun. Pure Appl. Math. 42 (1989), pp. 115–137], where a replacement lemma plays the key role. To prove the replacement lemma, the large deviations principle of the model given in Xue [Hydrodynamics of the generalized N-urn Ehrenfest model, Potential Anal. 2022. Online. Available at https://rdcu.be/cFzrj] is utilized. As applications of our main result, we further give moderate deviations of hitting times of our process and moderate deviations of our process at given moments.

Stochastic stabilization of hybrid neural networks by periodically intermittent control based on discrete-time state observations

This paper is concerned with stabilization of hybrid neural networks by intermittent control based on continuous or discrete-time state observations. By means of exponential martingale inequality and the ergodic property of the Markov chain, we establish a sufficient stability criterion on hybrid neural networks by intermittent control based on continuous-time state observations. Meantime, by M-matrix theory and comparison method, we show that hybrid neural networks can be stabilized by intermittent control based on discrete-time state observations. Finally, two examples are presented to illustrate our theory.

PARTIAL PRACTICAL STABILITY AND ASYMPTOTIC STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY LÉVY NOISE WITH A GENERAL DECAY RATE

In this paper, we mainly study the almost sure partial practical stability of stochastic differential equations driven by Lévy noise with a general decay rate. By establishing a suitable Lyapunov function and using Exponential Martingale inequality and Borel-Cantelli theorem, giving some sufficient conditions that can guarantee the almost sure partial practical stability of equations. At the same time, we also study general conditions that guarantee the almost sure asymptotic stability of the equation. Finally, we also give two examples to illustrate our theoretical results.

A study on analytical solutions for stochastic differential equations via martingale processes

In this paper, we propose some analytical solutions of stochastic differential equations related to Martingale processes. In the first resolution, the answers of some stochastic differential equations are connected to other stochastic equations just with diffusion part (or drift free). The second suitable method is to convert stochastic differential equations into ordinary ones that it is tried to omit diffusion part of stochastic equation by applying Martingale processes. Finally, solution focuses on change of variable method that can be utilized about stochastic differential equations which are as function of Martingale processes like Wiener process, exponential Martingale process and differentiable processes.

Forests, cumulants, martingales

This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability space. We establish a “broken exponential martingale” expansion that generalizes and unifies the exponentiation result of Alòs, Gatheral, and Radoičić (SSRN’17; Quant. Finance 20 (2020) 13–27) and the cumulant recursion formula of Lacoin, Rhodes, and Vargas (arXiv; (2019)). Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula, as required by many of our examples, including iterated stochastic integrals, Lévy area, Bessel processes, KPZ with smooth noise, Wiener–Itô chaos, and “rough” stochastic (forward) variance models.

Large deviations of mean-field interacting particle systems in a fast varying environment

This paper studies large deviations of a “fully coupled” finite state mean-field interacting particle system in a fast varying environment. The empirical measure of the particles evolves in the slow time scale and the random environment evolves in the fast time scale. Our main result is the path-space large deviation principle for the joint law of the empirical measure process of the particles and the occupation measure process of the fast environment. This extends previous results known for two time scale diffusions to two time scale mean-field models with jumps. Our proof is based on the method of stochastic exponentials. We characterise the rate function by studying a certain variational problem associated with an exponential martingale.

Dynamics and intermittent stochastic stabilization of a rumor spreading model with guidance mechanism in heterogeneous network

We propose a novel rumor propagation model with guidance mechanism in heterogeneous complex networks. Firstly, the sharp threshold of rumor propagation, global stability of the information-equilibrium and information-prevailing-equilibrium under R 0 < 1 and R 0 > 1 is carried out by Lyapunov method and LaSalle’s invariant principle. Next, we design an aperiodically intermittent stochastic stabilization method to suppress the rumor propagation. By using the Itô formula and exponential martingale inequality, the expression of the minimum control intensity is calculated. This method can effectively stabilize the rumor propagation by choosing a suitable perturb intensity and a perturb time ratio, while minimizing the control cost. Finally, numerical examples are given to illustrate the analysis and method of the paper.

STOCHASTICALLY PERMANENT ANALYSIS OF A NON-AUTONOMOUS HOLLING Ⅱ PREDATOR-PREY MODEL WITH A COMPLEX TYPE OF NOISES

This paper is considered a non-autonomous stochastic Holling II predator-prey model with a complex type of noises. By constructing a Lyapunov function and applying the dominated convergence theorem, stochastically permanent is proved. More importantly, two values λ₁, λ₂ are expressed by using the density function of the Falk Planck equation and some parameters in the system. Among them, λ₁ > 0 is proved to be the sufficient condition for the persistence in mean. Then, applying the strong law of large number and exponential martingale inequality, two necessary lemmas are introduced. Furthermore, utilizing the lemmas and λ₂ < 0, the sufficient conditions for extinction of the system are obtained. Actually, the two sufficiency conditions obtained are approached to the necessary conditions. Finally, some numerical simulations are carried out to verify the influence of the complex type of noises on the system.

Optimal time-consistent investment-reinsurance strategy for state-dependent risk aversion with delay and common shocks

This article investigates the optimal investment-reinsurance problem for state-dependent risk aversion with delay and common shock dependence. We suppose that the insurer manages insurance risk which is characterized by a two-dimensional dependent claim process by purchasing proportional reinsurance, and the insurer can invest his wealth in a financial market that consists of a risk-free asset and a risky asset. On the basis of the consideration of the performance-related capital inflow/outflow, the wealth process of the insurer is modeled by a stochastic delay differential equation. Then we formulate the optimal investment and reinsurance mean-variance problem within a game-theoretic framework and then applying the stochastic control theory with delay, we derive the corresponding extended HJB equations with delay. Next by solving the extended HJB equations and constructing the exponential martingale process, the closed-form solution to the optimal investment-reinsurance strategy and the corresponding equilibrium value function are derived. Besides, we use a numerical example to analyze the influence of the model parameters on the optimal time-consistent strategy and give an economic explanation.

Asymptotic behavior of the weak approximation to a class of Gaussian processes

AbstractWe study, under mild conditions, the weak approximation constructed from a standard Poisson process for a class of Gaussian processes, and establish its sample path moderate deviations. The techniques consist of a good asymptotic exponential approximation in moderate deviations, the Besov–Lèvy modulus embedding, and an exponential martingale technique. Moreover, our results are applied to the weak approximations associated with the moving average of Brownian motion, fractional Brownian motion, and an Ornstein–Uhlenbeck process.

Moderate deviations of density-dependent Markov chains

A density dependent Markov chain (DDMC) introduced in Kurtz (1978) is a special continuous time Markov process. Examples are considered in fields like epidemics and processes which describe chemical reactions. Moreover the Yule process is a further example. In this paper we prove a moderate deviation principle for the paths of a certain class of DDMC. The proofs of the bounds utilize an exponential martingale as well as a generalized version of Girsanov’s theorem. The exponential martingale is defined according to the generator of the DDMC.

Asymptotic moment estimation for stochastic Lotka–Volterra model driven by G-Brownian motion

A stochastic Lotka–Volterra model disturbed by G-Brownian motion (G-LVM for short) in the framework of non-linear expectation is proposed in this paper. This model takes into account the uncertainty of variance of the noise. We prove the G-LVM exists a unique solution and the solution does not tend to infinity when the time is finite under some constraints, and obtain many asymptotic moment estimations which depend on the variance of G-Brownian motion by capacity theory, exponential martingale inequality and analytical skills.

Forests, Cumulants, Martingales

This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability spaces. We establish a “broken exponential martingale” expansion that generalizes and unifies the exponentiation result of Alos, Gatheral, and Radoicic (SSRN’17; [AGR20]) and the cumulant recursion formula of Lacoin, Rhodes, and Vargas (arXiv; [LRV19]). Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula, as required by many of our examples, including iterated stochastic integrals, Levy area, Bessel processes, KPZ with smooth noise, Wiener-Ito chaos and “rough” stochastic (forward) variance models.

Almost sure stability of hybrid stochastic systems under asynchronous Markovian switching

This paper presents two novel Lyapunov methodologies for almost sure stability of hybrid stochastic systems under asynchronous Markovian switching. One is established by a concave composite Lyapunov function with exponential martingale inequality. The other is derived by the strong law of large numbers, which can explore the coupling between the drift part and diffusion part of the systems, thus fully capturing the stabilizing effect of the stochastic noise. Both of these stability conditions give a quantitative relationship between the size of the detected delay of switching signal, the stationary distribution and the generator of Markov chain. As applications, easy-to-check stability and stabilization criteria are further provided for one-sided growth nonlinear systems and linear systems. Numerical examples illustrate the proposed theoretical results.

Dynamics analysis of a stochastic non-autonomous one-predator\u2013two-prey system with Beddington\u2013DeAngelis functional response and impulsive perturbations

In this paper, we explore a stochastic non-autonomous one-predator–two-prey system with Beddington–DeAngelis functional response and impulsive perturbations. First, by using Itô’s formula, exponential martingale inequality, Chebyshev’s inequality and other mathematical skills, we establish some sufficient conditions for extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the solution of the stochastic system. Then the limit of the average in time of the sample path of the solution is estimated by two constants. Afterwards, the lower-growth rate and the upper-growth rate of the positive solution are estimated. In addition, sufficient conditions for global attractivity of the system are established. Finally, we carry out some simulations to verify our main results and explain the biological implications: the large stochastic interference is disadvantageous for the persistence of the population and the strong impulsive harvesting can lead to extinct of the population.

Quantum Martingale Theory and Entropy Production.

We employ martingale theory to describe fluctuations of entropy production for open quantum systems in nonequilbrium steady states. Using the formalism of quantum jump trajectories, we identify a decomposition of entropy production into an exponential martingale and a purely quantum term, both obeying integral fluctuation theorems. An important consequence of this approach is the derivation of a set of genuine universal results for stopping-time and infimum statistics of stochastic entropy production. Finally, we complement the general formalism with numerical simulations of a qubit system.

Analysis of a stochastic predator\u2013prey population model with Allee effect and jumps

This paper is concerned with a stochastic predator–prey model with Allee effect and Lévy noise. First, by the comparison theorem of stochastic differential equations, we prove that the model has a unique global positive solution starting from the positive initial value. Then we investigate the asymptotic pathwise behavior of the model by the generalized exponential martingale inequality and the Borel–Cantelli lemma. Next, we establish the conditions under which predator and prey populations are extinct. Furthermore, we show that the global positive solution is stochastically ultimate bounded under some conditions by using the Bernoulli equation and Chebyshev’s inequality. At last, we introduce some numerical simulations to support the main results obtained. The results in this paper generalize and improve the previous related results.

Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

We introduce a family of real random variables $(\beta,\theta)$ arising from the supersymmetric nonlinear sigma model and containing the family $\beta$ introduced by Sabot, Tarr\`es, and Zeng [STZ17] in the context of the vertex-reinforced jump process. Using this family we construct an exponential martingale generalizing the one considered in [DMR17]. Moreover, using the full supersymmetric nonlinear sigma model we also construct a generalization of the exponential martingale involving Grassmann variables.