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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... 

Optical tweezers consist of a tightly focused laser beam. A particle with refractive index greater than that of the surrounding medium can be trapped at the focus of the laser. The trapped object experiences a three-dimensional Hookean restoring force towards the focus. Nano (micron)-sized spheres can produce forces in the range of few pico-newton to few nano-newton. This range covers a large number of the forces, which contribute in biological processes; therefore, optical tweezers are very often used for micromanipulation of biological tissues. In a typical micromanipulation experiment it is crucial to perform proper positional calibration prior to use. There are several calibrate methods... 

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.... 

Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian paths in the hyperbolic plane H2, and undergo binary fission(s) at rate λ > 0. It is shown that there is a phase transition in λ : For λ ≤ 1/8 the number
of particles in any compact region of H2 is eventually 0, w.p.1, but for λ > 1/8 the number of particles in any open set grows to ∞ w.p.1. In the subcritical case (λ ≤ 1/8) the set Λ of all limit points in ∂H2 (the boundary circle at ∞) of particle trails is a Cantor set, while in the supercritical case (λ > 1/8) the set Λ has full Lebesgue measure. For λ ≤ 1/8 it is shown that w.p.1 the Hausdorff dimension of Λ... 

Application of nanotechnology in the field of heat transfer has increased recently. The need to increase heat transfer rate yet decrease the size of cooling equipment, brought about lots of attention to thermal properties of Nanofluids. Nanofluid is the suspension of nanometer-sized solid particles in base liquid. Research on convective heat transfer of nanofluids which is only two decades old, shows great potential in increasing heat transfer rate. Although there is a remarkable research on thermal conductivity of Nanofluids, negligible research was conducted on combined Nanofluids . Developed Theory for thermal conductivity of combined nanofluid can be used to modeling the thermal... 

Portfolio optimization is one of the most important issues in capital market and Mathematical Finance. Also in simiulations of financial instruments, in many cases the fluctuations are not independed so we can’t use standard Brownian motion for portfolio optimization and simiulations. In these cases, we should use another kind of Brownian motion which is called fractional Brownian motion. After introducing fractional Brownian motion in chapter 1, we will present its properties in chapter 2 , then at chapter 3 we’ll study stochastic calculus in fractional case and finally in chapter 4 after presenting Stochastic maximum Principle and applying it on a portfolio optimization problem, we will... 

This paper investigates the interaction between the beneficiaries of an employee stock option plan within a Stackelberg game framework. The beneficiaries are shareholders and employees. In the proposed model, shareholders, as the leaders of the Stackelberg game, determine the optimal features of employee stock option grants. In response, employees, the followers of the proposed Stackelberg game, maximize their own profits by determining their own effort level by considering that every effort level of employees has an associated cost and expected income for employees. It is assumed that the stock price follow Geometric Brownian Motion process with a known drift rate and volatility. Also, it... 

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... 

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  

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... 

Due¬ to the volatile nature of material prices, projecting highway construction costs are proven to be very difficult, leading to many challenges in cost estimation process. Bid preparations as well as project planning and control processes are also negatively affected by the concerns about the inaccuracy of cost projections. There is a growing body of evidence that suggests the use of inaccurate cost estimate can result in bid loss or profit loss for contractors and hidden price contingencies, delayed or cancelled projects, inconsistency in budgets and unsteady flow of projects for owner organizations. Analysis of historical data indicates that a relationship between construction materials...