This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on ...their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C∞-smooth density.
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Exit times of diffusion processes play an important role in the application of finance, insurance, biology, engineering, and so on. This article investigates the two-side exit time problems for the ...Vasicek model. By using Strong Markov property, the Laplace-Stieltjes transform (LST) of some exit times are obtained. And then the results are used to figure out the probability and mathematical expectation of the exit times. Finally, numerical examples are given to illustrate the applications of the LST of some exit times.
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The objective of this paper is to study the strong Markov property for the stochastic differential equations driven by G-Brownian motion (G-SDEs for short). We first extend the deterministic-time ...conditional G-expectation to optional times. The strong Markov property for G-SDEs is then obtained by Kolmogorov’s criterion for tightness. In particular, for any given optional time τ and G-Brownian motion B, the reflection principle for B holds and (Bτ+t−Bτ)t≥0 is still a G-Brownian motion.
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Motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics, we study existence, uniqueness, and the strong Markov property of weak ...solutions to a class of degenerate stochastic differential equations. The stochastic differential equations considered in our article admit solutions supported in the set 0,\infty )^n\times \mathbb{R}^m, and they are degenerate in the sense that the diffusion matrix is not strictly elliptic, as the smallest eigenvalue converges to zero at a rate proportional to the distance to the boundary of the domain, and the drift coefficients are allowed to have power-type singularities in a neighborhood of the boundary of the domain. Under suitable regularity assumptions on the coefficients, we establish existence of solutions that satisfy the strong Markov property, and uniqueness in law in the class of Markov processes.
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For a sequence in discrete time having stationary independent values (respectively, random walk) X, those random times R of X are characterized set-theoretically, for which the strict post-R sequence ...(respectively, the process of the increments of X after R) is independent of the history up to R. For a Lévy process X and a random time R of X, reasonably useful sufficient conditions and a partial necessary condition on R are given, for the process of the increments of X after R to be independent of the history up to R.
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The diffusion risk model is considered in the presence of liquid reserves, credit and debit interest. For arbitrary boundary and any interval, the Laplace-Stieltjes transform (LST) of some exit times ...of the risk process are derived. The results are then used to find the probability and mathematical expectation of the exit times. Finally, several numerical examples be discussed in order to illustrate the applications of the LST of some exit times.
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For a given quasi-regular positivity preserving coercive form, we construct a family of (σ-finite) distribution flows associated with the semigroup of the form. The canonical cadlag process equipped ...with the distribution flows behaves like a strong Markov process. Moreover, employing distribution flows we can construct optional measures and establish Revuz correspondence between additive functionals and smooth measures. The results obtained in this paper will enable us to perform a kind of stochastic analysis related to positivity preserving coercive forms.
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This paper studies a single-sever queue with disasters and repairs, in which after each service completion the server may take a vacation with probability
(0
1), or begin to serve the next customer, ...if any, with probability
(= 1
). The disaster only affects the system when the server is in operation, and once it occurs, all customers present are eliminated from the system. We obtain the stationary probability generating functions (PGFs) of the number of customers in the system by solving the balance equations of the system. Some performance measures such as the mean system length, the probability that the server is in different states, the rate at which disasters occur and the rate of initiations of busy period are determined. We also derive the sojourn time distribution and the mean sojourn time. In addition, some numerical examples are presented to show the effect of the parameters on the mean system length.
Noncolliding Brownian motion (Dyson’s Brownian motion model with parameter β=2) and noncolliding Bessel processes are determinantal processes; that is, their space–time correlation functions are ...represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes.
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