Sharif Digital Repository

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... 

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  

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 purpose of this thesis is to introduce a very important and recent branch of probability, ”SLE”, its relation to some discrete models in statistical physics, motivation, geometrical intuition and some of its important properties. In chapter 1, we introduce three discrete models: Self Avoiding Walk (SAW), Loop Erased Random Walk (LERW), and Percolation that have an important role in the thesis. In chapter 2, we set up the concept of SLE on two of these models. Required preliminaries from complex analysis and Loewner equations will be studied in chapter 3, and then, in chapter 4 we will define Chordal and Radial SLE. Studying some important properties of SLE is the subject of chapter 5. In... 

In this thesis we study sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater than one-half. In addition, the mean-square numerical approximations of such problems are investigated, performed by the spectral Galerkin method in space and the linear implicit Euler method in time. We see that by using the obtained sharp regularity properties of the problems one can identify optimal mean-square convergence rates of the full discrete scheme. At the end, these theoretical findings are accompanied by several numerical... 

In this thesis, we study first passage times of a double exponential jump diffusion process to boundaries. This process consists of a continuous part which includes brownian motion and a jump part with jump sizes which have a double exponential distribution. We study explicit solutions obtained for the laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima. Additionally, several interesting probabilistic results are provided. Its results have finance applications, including pricing barrier and lookback options  

In this thesis we study Galerkin methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. The strong error of convergence for spatially semidiscrete approximations as well as a spatio-temporal discretization which is based on a linear implicit Euler–Maruyama method, are also investigated. We see that the obtained error estimates in both cases as well as the regularity results for the mild solution of the SPDE are optimal. The results hold for different Galerkin methods such as the standard finite element method or spectral Galerkin. At the end, these theoretical findings are accompanied by several numerical... 

In this thesis we illustrate the Cox,Ingersoll and Ross (CIR) model for the term structure of the interest rate as a Stochastic Differential Equation (SDE) and provide a numerical solution of that.
In this way, we first study the numerical Euler method for SDEs with Lipschitz coefficients. Then we introduce the CIR model and will see that the main body of this model is a SDE whose diffusion coefficient is not Lipschitz but only (1=2 + α)-Hölder continuous for some α>0. With this motivation, we go to study the existence and the uniqueness of the solution for SDEs with Hölder continuous diffusion coefficient,then characterize some Euler approximations for this kind of SDEs and then... 

In the past decades, an extensive mathematical theory has been developed for the problems of derivative pricing. One of the salient features of this theory is its assumption of a common information flow on which the portfolio decisions of all economic agents are based. In this thesis, we attempt to widen the scope of the pricing financial derivatives by studying two important classes of contingent claims in a financial market with two types of investors on different information levels. Following the well-known link between optimal stopping problems and reflected backward stochastic differential equations (RBSDE) in El Karoui et al. [31], we also investigate the problem of information... 

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... 

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... 

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  

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.... 

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 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... 

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... 

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