Sharif Digital Repository

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.... 

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  

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... 

The application of Stochastic Differential Equations in branches like nonlinear control,robatic systems, financial mathematics and etc. has grown impressively nowadays. In this thesis, we are going to introduce this equations and study their stability. First, we will present methods to check the stability of nonlinear systems dx(t) = f(x(t); t)dt using Lyapunov’s theorem.Then we will study stability of autonomous systems using it’s results. In the case of instability, we will study the possibility of stabilizing the stochastic equations dx(t) = Ax(t)dt + Σn i=1 Bix(t)dWi(t) by using Brownian motions and these methods. In the end, we will study the stabilizablity conditions of instable... 

We discuss, in the setting of UMD Banach spaces E, existence, uniqueness and the continuous dependence on the data A, F, G and of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form dX(t) = [AX(t) + F(t;X(t))]dt + G(t;X(t))dWH(t); t 2 [0; T]; X(0) = ; Where A generates an analytic C0-semigroup on a UMD Banach space E and WH is a cylindrical Brownian motion with values in a Hilbert space H. We can see that if the mappings F : [0; T] E ! E and G : [0; T] E ! L(H;E) satisfy suitable Lipschitz conditions and is F0-measurable and bounded, then this problem has a unique mild solution.And We arw going to review continuous dependence of the solutions... 

In the past decades, option pricing has become one of the major areas in modern financial theory and practice. The Black-Scholes-Merton method is a type of option pricing, which is an appropriate and very important model in financial markets due to the pricing process under the assumption of no arbitrage and the recognition of the appropriate discount rate.Inspite of its advantages, this model is not appropriate for pricing the options which need to be investigated before the maturity.To overcome this limitation, some discrete extension of Black Scholes model were introduced such as binomial and trinomial trees.In all of these models during the contract period, volatility is considered... 

In this article we investigate the strong convergence of the Euler-Maruyama method and stochastic theta method for stochastic differential delay equation with jump .Under a global lipschitts condition we not only prove the strong convergence but also obtain the rate of convergence.We show stronge convergence under a local lipschits condition and linear growth condition.Moreover it is the first time we obtain the rate of strong convergence under a local lipschits condition and a linear growth condition.i.e if the local lipschitsz constants for balls of radius R are supposed to grow not faster than logR  

We study linear and semilinear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. The equation is discretized in space by a standard piecewise linear finite element method. We show how to obtain error estimates when the truncated expansion is used in the equation. We show that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing theasymptotic order of the method, provided that the kernel of the covariance operator of the Wiener process is... 

In this monograph, we investigate the long-time behavior of stochastic delay equations. Our approach is random dynamical systems, and we solve our equation in the rough path point of view. Namely, we deal with the singular case, i.e., when the delay terms also are appearing in the diffusion part. Although we can solve the equation using the classical tools of stochastic analysis, the main obstacle is the lack of flow property. More precisely, the solution does not depend continuously on the initial value. To solve this problem, we define this property differently. We will show how we can generate a flow property on fields of Banach spaces using rough path theory. As a consequence, we prove... 

e solution for fully non-linear equations do not exists in general and for this reason the notion of viscosity solutions has been defined in this context. In this dissertation, a Monte carlo method is introduced for parabolic fully non-linear together with its asymptotics. In the proof of convergence, the method of [6] is used. en the rate of convergence from both sides is introduced: Finally, the error due to estimation of conditional expectation is derived for the estimation whose error is known with respect to sample size  

By means of classical Itô calculus, we decompose option prices as the sum of the classical Black–Scholes formula, with volatility parameter equal to the root-meansquare future average volatility, plus a term due to correlation and a term due to the volatility of the volatility. This decomposition allows us to develop first- and second-order Approximation formulas for option prices and implied volatilities in the Heston volatility framework, as well as to study their accuracy for short maturities.Numerical examples are given  

Option Pricing is one of the most challenging topics in the world of Finance. There are a lot of option pricing models such as Black-Scholes model, Binomial Trees model, Monte Carlo method and Stochastic Volatility model. The last one is the most famous among all of them. The profound financial crisis generated by the collapse of Lehman Brothers and the European sovereign debt crisis in 2011 have caused negative values of government bond yields both in the USA and in the EURO area. Therefore, Option Pricing Models should consider this fact. The stochastic volatility model of Oosterlee et al, notice a dynamic for interest rate and heeds interest rate as a stochastic factor. However, it does...