Sharif Digital Repository

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  

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... 

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  

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  

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... 

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 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 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 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...