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This thesis has been prepared in six chapters. In the first chapter, the necessary analytical preliminaries are revised. The second chapter is specified on the introducing of the fractional Brownian motion and the description of some of its properties. The subject of the third chapter is simulation. The practical utilization of stochastic models usually needs simulation; therefore the fractional Brownian motion and the processes derived from it are not exempted either. The fifth chapter is consisted of two major parts; the first part is the simulation of fractional Brownian motion, in which no new work has been done, and only one of the available methods has been explained. The second part... 

Large deviation principle (LDP) for stochastic differential equation is one of the interesting and modern topics in stochastic analysis. Principally, this theory gives the rate of convergence to the solution of the corresponding deterministic equation when the noise tends to zero. The study of LDP for SDE’s has been initiated by M. Freidlin and A. Wentzell and then has been considered by many other researchers. Freidlin andWentzell divided the interval [0, T] to small subintervals and considered the diffusion coefficient as a constant on any small subintervals. Then the problem is reduced to the additive noise case. But using the contraction principle, the study of LDP for an equation with... 

This thesis is devoted to study some asymptotic behaviors of Poisson Voronoi tessellation in the Euclidean space as dimension of the space tends to infinity. First we use Blaschke-Petkantschin formula to prove that the variance of volume of the typical cell tends to zero exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume converges in distribution to the constant . Next we consider the linear contact distribution function of the Poisson Voronoi tessellation and compute the limit when dimension goes to infinity. As a byproduct, the chord length distribution and the geometric covariogram of the typical cell are... 

Stochastic Partial Differential Equations have many applications in other area of science. In this thesis we investigate two pproaches in SPDE.The first approach is semigroup and the second is variational method.Our main purpose is stability of these equations  

We establish the existence and uniqueness of the mild solution for stochastic Volterra equation with a non-self-adjoint operator. The specific Volterra equation that we consider is a generalization of the fractional differential equation. To obtain the mild solution for the case of multiplicative problem, the resolvent property of the linear perturbation of a sectorial operator will be considered. Moreover, we establish the existence and uniqueness of the mild solution for semilinear stochastic Volterra equation involving a demicontinuous and semimonotone nonlinearity. The Volterra equation in this case, has a positive-type memory kernel. To obtain the mild solution of the multiplicative... 

Let (X; d) be a metric space and (X;) be a partially ordered Space. Let F, g be measurable mappings such that F has g-monotone property and satisfying in a contraction condition. Firstly, some extentions of Banach fixed point theorem was investigated in particular way that lead to random coupled and random fixed point for mentioned mappings. Then, linear stochastic evolution equation and semilinear dissipitive stochastic evolution equation driven by infinite dimentional fractional Brownian noise was evaluated. It has been shown these equations define random dynamical systems with exponentially attracting random fixed points that are stationary solution for them  

We study stochastically perturb the classical Lotka-Volterra model x ̇(t)=diag(x_1 (t),…,x_n (t))[b+Ax(t)] Into the stochastic differential equation dx(t)=diag(x_1 (t),…,x_n (t))[b+Ax(t)dt+σ(t)dw(t)]. The main aim is to study the asymptotic properties of the solution. We will show that if the noise is too large then the population may become extinct with probability one. We find out a sufficient condition for stochastic differential equation such that it has a unique global positive solution. Moreover, we will establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we discuss ultimate boundedness and extinction in population... 

Brownian motion played a central role throughout the twentieth century in probability theory. The same statement is even truer in finance, with the introduction in 1900 by the French mathematician Louis Bachelier of an arithmetic Brownian motion (or a version of it) to represent stock price dynamics. This process was pragmatically transformed by Samuelson in 1965 into a geometric Brownian motion ensuring the positivity of stock prices. More recently, the elegant martingale property under an equivalent probability measure derived from the no-arbitrage assumption combined with Monroe's theorem on the representation of semi martingales has led to write asset prices as time-changed Brownian... 

n this article, we find necessary conditions for the existence of Ito Integral in a Banach space with respect to compensated Poisson random measure (cPrm). Ito integrals with values on M-type 2 Banach spaces F of the above form exist for all measurable, adapted functions f square integrable w.r.t. β ⊗ d t (f ∈ M2T;_(E/F)),with β being Lévy measure associated with cPrm, and have strong second moments. We show that, for general separable Banach spaces F, an inequality of the type resulting for M-type 2 Banach spaces with constant depending on cPrm is necessary and sufficient for the existence of Ito integral having second moment finite for all f ∈ M2 T;_(E/F). It is shown that M2 T;_(E/F) is... 

The aim of this thesis is to examine different perspectives on stochastic integrals of fractional Brownian motion. We examine two main perspectives. In the first perspective, we present Mallivan idea in general and in the second idea Riemannian calculus perspective in briefly.In first, we explain basic idea in Mallivan calculus for example Hida spaces, operator δ and we try as ordinary Brownian motion, in this work follow the same trend. The next step, as conventional stochastic integrals Martingle Dob inequality, we introduce torques to find an upper bound for this integral.In Mallivan perspective, we are looking for a formula to maintain Ito formula in a certain space.In the following... 

Theoretical investigation of stochastic delay differential equation driven by fractional Brownian motion is important issue because of its application in the modeling. In this thesis, after defining of the stochastic integral with respect to fractional Brownian motion and describing the delay differential equation, we prove existence and
uniqueness of solution of stochastic delay differential equation driven by fractional Brownian motion with Hurst parameter H>1/2 and we show that the solution has finite moments from each order. Moreover we show when the delay goes to zero, thesolutions to these equations converge, almost surely and in Lp, to the solution for the equation without delay.... 

We obtain a large deviation principle describing the small time asymptotics of the solution of a stochastic evolution equation with multiplicative noise. Our assumptions are a condition on the linear drift operator that is satisfied by generators of analytic semigroups and Lipschitz continuity of the nonlinear coefficient functions. Methods originally used by Peszat.
For the small noise asymptotics problem are adapted to solve the small time asymptotics problem. The results obtained in this way improve on some results of Zhang  

The six-vertex model is one of the lattice models of two dimensional statistical physics. In this model, like other models in statistical physics, the probability of occurrence of any configuration is proportional to the product of some local weights.The partition function of the model is the sum of products of local weights over all of the allowable configurations. The partition function has important physical interpretations and computing it is regarded as the first step toward the understanding of the model. In this thesis, we give a survey on different methods of calculating the partition functions. The important point is the generality of these methods such as employing Yang-Baxter... 

n this thesis we have investigated stochastic evolution equations by variational method. For these equations, explicit and implicit numerical schemes are presented. We have performed these numerical schemes for stochastic heat equation. We have investigated 2-D Navier-Stokes equation too  

In this thesis, we consider an implicit approximation scheme for the stochastic heat equation with additive and multiplicative space-time white noise. we use the spectral Galerkin method in space combined with the linear implicit Euler method in time to simulate weak approximation error  

In some pricing methods like European Option Price there are some deterministic exercise times but in American Option Price it is Stochastic Process. So, it would be very difficult to calculate the exact formula for it. So, we can use some approximation for this goal. The main purpose of the thesis is to consider of weak convergence for a special approximation  

Semilinear stochastic evolution equations with multiplicative Lévy noise and monotone nonlinear drift are considered. A novel method of proof for establishing existence and uniqueness of the mild solution is proposed. We also prove the continuous dependence of the mild solution with respect to initial conditions and also on coefficients. As corollaries of the continuity result, we derive sufficient conditions for asymptotic stability of the solutions, we show that Yosida approximations converge to the solution and we prove that solutions have Markov property. Examples on stochastic partial differential equations and stochastic delay differential equations are provided to demonstrate the... 

A sequence of Markov chains is said to exhibit (total variation) cutoff if the convergence to stationarity in total variation distance is abrupt. We consider reversible lazy chains. We prove a necessary and sufficient condition for the occurrence of the cutoff phenomena in terms of concentration of hitting time of “worst” (in some sense) sets of stationary measure at least , for some ... (0; 1). As an application of these methods, we prove that a sequence of lazy chains on finite trees exhibits cutoff if and only if the product of the mixing time and the spectral gap tends to  

It is well-known that the components of solution to a system of N interacting stochastic differential equations with an averaged sum of interaction terms and with independent identically distributed (chaotic) initial values , as $N$ tends to infinity , converge to the solutions of Vlasov-McKean equations , in which the averaged sum is replaced by the expectation . Since the solutions to the corresponding Vlasov-McKean equations are independent , this phenomenon is called propagation of chaos . This thesis is about well-posedness of path-dependent stochastic differential equations , propagation of chaos for spatially structured neural network with delay and existence and uniqueness of...