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#### Fractional Brownian Motion and Application in Mathematical Finance

, M.Sc. Thesis Sharif University of Technology ; Zohuri Zangeneh, Bijan (Supervisor) ; Farhadi, Hamid Reza (Co-Advisor)
Abstract

Farctional Brownian motion (fBm) is a Gaussian Stochastic process B={B_t ∶t ≥0} With zero mean and Covariance function given by RH (t,s)=1/2 (t^2H+ S^2H-├|t-├ s┤|┤ 〖^2H〗) Where 0

#### Reaction-diffusion equations with polynomial drifts driven by fractional brownian motions

, Article Stochastic Analysis and Applications ; Volume 28, Issue 6 , Oct , 2010 , Pages 1020-1039 ; 07362994 (ISSN) ; Sharif University of Technology
2010

Abstract

A reaction-diffusion equation on [0, 1]d with the heat conductivity k > 0, a polynomial drift term and an additive noise, fractional in time with H > 1/2, and colored in space, is considered. We have shown the existence, uniqueness and uniform boundedness of solution with respect to k Also we show that if k tends to infinity, then the corresponding solutions of the equation converge to a process satisfying a stochastic ordinary differential equation

#### Applications and outlook

, Article Understanding Complex Systems ; 2019 , Pages 243-260 ; 18600832 (ISSN) ; Sharif University of Technology
Springer Verlag
2019

Abstract

The method outlined in the Chaps. 15 – 21 has been used for revealing nonlinear deterministic and stochastic behaviors in a variety of problems, ranging from physics, to neuroscience, biology and medicine. In most cases, alternative procedures with strong emphasis on deterministic features have been only partly successful, due to their inappropriate treatment of the dynamical fluctuations [1]. In this chapter, we provide a list of the investigated phenomena using the introduced reconstruction method. In the “outlook” possible research directions for future are discussed. © 2019, Springer Nature Switzerland AG

#### Applications and Outlook

, Article Understanding Complex Systems ; 2019 , Pages 243-260 ; 18600832 (ISSN) ; Sharif University of Technology
Springer Verlag
2019

Abstract

The method outlined in the Chaps. 15 – 21 has been used for revealing nonlinear deterministic and stochastic behaviors in a variety of problems, ranging from physics, to neuroscience, biology and medicine. In most cases, alternative procedures with strong emphasis on deterministic features have been only partly successful, due to their inappropriate treatment of the dynamical fluctuations [1]. In this chapter, we provide a list of the investigated phenomena using the introduced reconstruction method. In the “outlook” possible research directions for future are discussed. © 2019, Springer Nature Switzerland AG

#### Stochastic Calculus with Respect to Fractional Brownian Motion

, M.Sc. Thesis Sharif University of Technology ; Zohuri Zangeneh, Bijan (Supervisor)
Abstract

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...

#### Hyperbolic Branching Brownian Motion

, M.Sc. Thesis Sharif University of Technology ; Esfahani Zade, Mostafa (Supervisor)
Abstract

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...

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...

#### Non-Parametric Analysis of Positional data of a Micro-Nano Sphere Trapped by Optical Tweezers

, M.Sc. Thesis Sharif University of Technology ; Reihani, Nader (Supervisor) ; Rahimi Tabar, Mohammad Reza (Supervisor)
Abstract

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...

#### , M.Sc. Thesis Sharif University of Technology ; Zohuri Zangeneh, Bijan (Supervisor)

Abstract

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....

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....

#### Existence and measureability of the solution of the stochastic differential equations driven by fractional brownian motion

, Article Bulletin of the Iranian Mathematical Society ; Volume 35, Issue 2 , 2009 , Pages 47-68 ; 10186301 (ISSN) ; Zohori Zangeneh, B ; Sharif University of Technology
2009

Abstract

Here, the existence and measurability of solutions for stochastic differential equations driven by fractional Brownian noise with Hurst parameter greater than 1 2 is proved. Our method is based on approximating the main equation by delayed equations as in Peano's method in ODEs. This method makes the proofs easier and needs weaker assumptions for the existence part, compared with the previous works as in [25]. In addition the constructive nature of the proofs helps to develop some numerical methods for solving such SDEs. © 2009 Iranian Mathematical Society

#### Experimental and Theoretical Investigation on Thermal Conductivity of Combined Nanofluids

, M.Sc. Thesis Sharif University of Technology ; Shafii, Mohammad Behshad (Supervisor)
Abstract

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...

#### A review of thermal conductivity of various nanofluids

, Article Journal of Molecular Liquids ; Volume 265 , 2018 , Pages 181-188 ; 01677322 (ISSN) ; Mirlohi, A ; Alhuyi Nazari, M ; Ghasempour, R ; Sharif University of Technology
Abstract

In the present paper, several experimental and theoretical studies conducted on the thermal conductivity of nanofluids are represented and investigated. Based on the reviewed studies, various factors affect thermal conductivity of nanofluids such as temperature, the shape of nanoparticles, concentration and etc. Results indicated the increase in temperature and concentration of nanoparticles usually leads to the higher thermal conductivity of nanofluids. In addition, it is concluded that there are some novel approaches in order to obtain nanofluids with more appropriate thermal properties including using binary fluids as the base fluid or utilizing hybrid nanofluids. © 2018 Elsevier B.V

#### Stochastic Maximum Principle for Fractional Brownian Motion

, M.Sc. Thesis Sharif University of Technology ; Zohoori Zangeneh, Bijan (Supervisor) ; Tahmasebi, Mahdieh ($item.subfieldsMap.e)
Abstract

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...

#### Optimal Interaction between Shareholders and Employees on Issuing Employee Stock Options within a Stackelberg Game Framework

, M.Sc. Thesis Sharif University of Technology ; Modarres Yazdi, Mohammad (Supervisor)
Abstract

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...

#### Fractional Brownian Motion and Stochastic Differential Equations Driven by Fractional Noise

, Ph.D. Dissertation Sharif University of Technology ; Zohori Zangeneh, Bijan (Supervisor)
Abstract

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...

#### First-passage-time processes and subordinated Schramm-Loewner evolution

, Article Physical Review E - Statistical, Nonlinear, and Soft Matter Physics ; Volume 84, Issue 1 , July , 2011 ; 15393755 (ISSN) ; Rajabpour, M. A ; Rouhani, S ; Sharif University of Technology
2011

Abstract

We study the first-passage-time processes of the anomalous diffusion on the self-similar curves in two dimensions. The scaling properties of the mean-square displacement and mean first passage time of the fractional Brownian motion and subordinated walk on the different fractal curves (loop-erased random walk, harmonic explorer, and percolation front) are derived. We also define natural parametrized subordinated Schramm-Loewner evolution (NS-SLE) as a mathematical tool that can model diffusion on fractal curves. The scaling properties of the mean-square displacement and mean first passage time for NS-SLE are obtained by numerical means

#### Stationary Solutions of Semilinear Differential Equations Driven by Fractional Brownian Motions

, M.Sc. Thesis Sharif University of Technology ; Zohori Zangene, Bijan (Supervisor)
Abstract

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

#### Stochastic Clock and Financial Mathematics

, M.Sc. Thesis Sharif University of Technology ; Zohuri Zangeneh, Bijan (Supervisor)
Abstract

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...

#### A Framework to Forecast Highway Construction Materials Prices

, M.Sc. Thesis Sharif University of Technology ; Kashani, Hamed (Supervisor)
Abstract

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...

#### Modified Buongiorno's model for fully developed mixed convection flow of nanofluids in a vertical annular pipe

, Article Computers and Fluids ; Vol. 89 , 2014 , pp. 124-132 ; ISSN: 00457930 ; Moshizi, S. A ; Soltani, E. G ; Ganji, D. D ; Sharif University of Technology
Abstract

This paper deals with the mixed convective heat transfer of nanofluids through a concentric vertical annulus. Because of the non-adherence of the fluid-solid interface in the presence of nanoparticle migrations, known as slip condition, the Navier's slip boundary condition was considered at the pipe walls. The employed model for nanofluid includes the modified two-component four-equation non-homogeneous equilibrium model that fully accounts for the effects of nanoparticles volume fraction distribution. Assuming the fully developed flow and heat transfer, the basic partial differential equations including continuity, momentum, and energy equations have been reduced to two-point ordinary...

#### Analysis of nanoparticles migration on natural convective heat transfer of nanofluids

, Article International Journal of Thermal Sciences ; Volume 68 , June , 2013 , Pages 79-93 ; 12900729 (ISSN) ; Yaghoubi, M ; Sharif University of Technology
2013

Abstract

Both experimental and numerical studies are unanimous for enhancing Nusselt number for forced convection of nanofluids with slight difference, but there is inconsistency for natural convection heat transfer of nanofluids. In this paper attempt is made to study the effects of nanoparticles migration on the natural convection behavior of nanofluids. For analysis, a mixture model is used by including important phenomena such as Brownian motion and thermophoresis effects. These two mechanisms are taken into account to compute the slip velocities between the base fluid and nanoparticles. The governing equations are solved numerically and good agreements are observed in comparison with...