Introduzimos uma familia de trajetorias aleatorias coalescentes com certo tipo de comportamento radial a qual chamaremos de Teia Poissoniana radial discreta. Mostramos que o limite fraco na escala difusiva desta familia e uma familia de trajetorias aleatorias coalescentes que chamaremos de Teia Browniana radial. Por m, caraterizamos o objeto limite como um mapeamento continuo da Teia Browniana restrita num subconjunto de R2. / We introduce a family of coalescing random paths with certain kind of radial behavior. We call them the discrete radial Poisson Web. We show that under diusive scaling this family converges in distribution to a family of coalescing random paths which we call radial Brownian Web. Finally, we characterize the limiting object as a continuous mapping of the Brownian Web restricted to a subset of R2.

This thesis established the cubature method developed by Gyurkó & Lyons (2010) and Lyons & Victor (2004) for the Heath–Jarrow–Morton (HJM) model. The HJM model was first proposed by Heath, Jarrow, and Morton (1992) to model the evolution of interest rates through the dynamics of the forward rate curve. These dynamics are described by an infinite-dimensional stochastic equation with the whole forward rate curve as a state variable. To construct the cubature method, we first discretize the infinite dimensional HJM equation and thereafter apply stochastic Taylor expansion to obtain cubature formulae. We further used their results to construct cubature formulae to degree 3, 5, 7 and 9 in 1-dimensional space. We give, a considerable step by step calculation regarding construction of cubature formulae on Wiener space.

The maximum drawdown on a time interval [0, T] of a random process can be defined as the largest drop from a high water mark to a low water mark. In this project, expected maximum drawdowns are analyzed in two cases: maximum drawdowns under constant volatility and stochastic volatility. We consider maximum drawdowns of both generalized and geometric Brownian motions. Their paths are numerically simulated and their expected maximum drawdowns are computed using Monte Carlo approximation and plotted as a function of time. Only numerical representation is given for stochastic volatility since there are no analytical results for this case. In the constant volatility case, the asymptotic behavior is described by our simulations which are supported by theoretical findings. The asymptotic behavior can be logarithmic for positive mean return, square root for zero mean return, or linear for negative mean return. When the volatility is stochastic, we assume it is driven by a mean-reverting process, in which case we discovered that if one uses the effective volatility in the formulas obtained for the constant volatility case, the numerical results suggest that similar asymptotic behavior holds in the stochastic case.

This dissertation studies a collection of problems on trading assets and derivatives over finite and infinite horizons. In the first part, we analyze an optimal switching problem with transaction costs that involves an infinite sequence of trades. The investor's value functions and optimal timing strategies are derived when prices are driven by an exponential Ornstein-Uhlenbeck (XOU) or Cox-Ingersoll-Ross (CIR) process. We compare the findings to the results from the associated optimal double stopping problems and identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. Our results show that when prices are driven by a CIR process, optimal strategies for the switching problems are of the classic buy-low-sell-high type. On the other hand, under XOU price dynamics, the investor should refrain from entering the market if the current price is very close to zero. As a result, the continuation (waiting) region for entry is disconnected. In both models, we provide numerical examples to illustrate the dependence of timing strategies on model parameters. In the second part, we study the problem of trading futures with transaction costs when the underlying spot price is mean-reverting. Specifically, we model the spot dynamics by the OU, CIR or XOU model. The futures term structure is derived and its connection to futures price dynamics is examined. For each futures contract, we describe the evolution of the roll yield, and compute explicitly the expected roll yield. For the futures trading problem, we incorporate the investor's timing options to enter and exit the market, as well as a chooser option to long or short a futures upon entry. This leads us to formulate and solve the corresponding optimal double stopping problems to determine the optimal trading strategies. Numerical results are presented to illustrate the optimal entry and exit boundaries under different models. We find that the option to choose between a long or short position induces the investor to delay market entry, as compared to the case where the investor pre-commits to go either long or short. Finally, we analyze the optimal risk-averse timing to sell a risky asset. The investor's risk preference is described by the exponential, power or log utility. Two stochastic models are considered for the asset price -- the geometric Brownian motion (GBM) and XOU models to account for, respectively, the trending and mean-reverting price dynamics. In all cases, we derive the optimal thresholds and certainty equivalents to sell the asset, and compare them across models and utilities, with emphasis on their dependence on asset price, risk aversion, and quantity. We find that the timing option may render the investor's value function and certainty equivalent non-concave in price even though the utility function is concave in wealth. Numerical results are provided to illustrate the investor's optimal strategies and the premia associated with optimally timing to sell with different utilities under different price dynamics.

In this thesis, a novel framework for dynamic congestion control has been proposed. The study is about the congestion control in broadband communication networks. Congestion results when demand temporarily exceeds capacity and leads to severe degradation of Quality of Service (QoS) and possibly loss of traffic. Since traffic is stochastic in nature, high demand may arise anywhere in a network and possibly causing congestion. There are different ways to mitigate the effects of congestion, by rerouting, by aggregation to take advantage of statistical multiplexing, and by discarding too demanding traffic, which is known as admission control. This thesis will try to accommodate as much traffic as possible, and study the effect of routing and aggregation on a rather general mix of traffic types. Software Defined Networking (SDN) and Network Function Virtualization (NFV) are concepts that allow for dynamic configuration of network resources by decoupling control from payload data and allocation of network functions to the most suitable physical node. This allows implementation of a centralised control that takes the state of the entire network into account and configures nodes dynamically to avoid congestion. Assumes that node controls can be expressed in commands supported by OpenFlow v1.3. Due to state dependencies in space and time, the network dynamics are very complex, and resort to a simulation approach. The load in the network depends on many factors, such as traffic characteristics and the traffic matrix, topology and node capacities. To be able to study the impact of control functions, some parts of the environment is fixed, such as the topology and the node capacities, and statistically average the traffic distribution in the network by randomly generated traffic matrices. The traffic consists of approximately equal intensity of smooth, bursty and long memory traffic. By designing an algorithm that route traffic and configure queue resources so that delay is minimised, this thesis chooses the delay to be the optimisation parameter because it is additive and real-time applications are delay sensitive. The optimisation being studied both with respect to total end-to-end delay and maximum end-to-end delay. The delay is used as link weights and paths are determined by Dijkstra's algorithm. Furthermore, nodes are configured to serve the traffic optimally which in turn depends on the routing. The proposed algorithm is a fixed-point system of equations that iteratively evaluates routing - aggregation - delay until an equilibrium point is found. Three strategies are compared: static node configuration where each queue is allocated 1/3 of the node resources and no aggregation, aggregation of real-time (taken as smooth and bursty) traffic onto the same queue, and dynamic aggregation based on the entropy of the traffic streams and their aggregates. The results of the simulation study show good results, with gains of 10-40% in the QoS parameters. By simulation, the positive effects of the proposed routing and aggregation strategy and the usefulness of the algorithm. The proposed algorithm constitutes the central control logic, and the resulting control actions are realisable through the SDN/NFV architecture.

Este texto apresenta alguns dos elementos básicos envolvidos em um estudo introdutório das equações diferencias estocásticas. Tais equações modelam problemas a tempo contínuo em que as grandezas de interesse estão sujeitas a certos tipos de perturbações aleatórias. Em nosso estudo, a aleatoriedade nessas equações será representada por um termo que envolve o processo estocástico conhecido como Movimento Browniano. Para um tratamento matematicamente rigoroso dessas equações, faremos uso da Integral Estocástica de Itô. A construção dessa integral é um dos principais objetivos do texto. Depois de desenvolver os conceitos necessários, apresentaremos alguns exemplos e provaremos existência e unicidade de solução para equações diferenciais estocásticas satisfazendo certas hipóteses. / This text presents some of the basic elements involved in an introductory study of stochastic differential equations. Such equations describe certain kinds of random perturbations on continuous time models. In our study, the randomness in these equations will be represented by a term involving the stochastic process known as Brownian Motion. For a mathematically rigorous treatment of these equations, we use the Itô Stochastic Integral. The construction of this integral is one of the main goals of the text. After developing the necessary concepts, we present some examples and prove existence and uniqueness of solution of stochastic differential equations satisfying some hypothesis.

Introduzimos uma familia de trajetorias aleatorias coalescentes com certo tipo de comportamento radial a qual chamaremos de Teia Poissoniana radial discreta. Mostramos que o limite fraco na escala difusiva desta familia e uma familia de trajetorias aleatorias coalescentes que chamaremos de Teia Browniana radial. Por m, caraterizamos o objeto limite como um mapeamento continuo da Teia Browniana restrita num subconjunto de R2. / We introduce a family of coalescing random paths with certain kind of radial behavior. We call them the discrete radial Poisson Web. We show that under diusive scaling this family converges in distribution to a family of coalescing random paths which we call radial Brownian Web. Finally, we characterize the limiting object as a continuous mapping of the Brownian Web restricted to a subset of R2.

We present a new pathwise approximation method for stochastic differential equations driven by Brownian motion which does not require simulation of the stochastic integrals. The method is developed to give Wasserstein bounds O(h3/2) and O(h2) which are better than the Euler and Milstein strong error rates O(√h) and O(h) respectively, where h is the step-size. It assumes nondegeneracy of the diffusion matrix. We have used the Taylor expansion but generate an approximation to the expansion as a whole rather than generating individual terms. We replace the iterated stochastic integrals in the method by random variables with the same moments conditional on the linear term. We use a version of perturbation method and a technique from optimal transport theory to find a coupling which gives a good approximation in Lp sense. This new method is a Runge-Kutta method or so-called derivative-free method. We have implemented this new method in MATLAB. The performance of the method has been studied for degenerate matrices. We have given the details of proof for order h3/2 and the outline of the proof for order h2.

Particle-particle interactions are of paramount importance in every multi-body system as they determine the collective behaviour and coupling strength. Many well-known interactions like electro-static, van der Waals or screened Coulomb, decay exponentially or with negative powers of the particle spacing r. Similarly, hydrodynamic interactions between particles undergoing Brownian motion decay as 1/r in bulk, and are assumed to decay in small channels. Such interactions are ubiquitous in biological and technological systems. Here I confine multiple particles undergoing Brownian motion in narrow, microfluidic channels and study their coupling through hydrodynamic interactions. Our experiments show that the hydrodynamic particle-particle interactions are distance-independent in these channels. We also show that these interactions affect actively propelled particles via electrophoresis or gravity, resulting in non-linear transport phenomena. These findings are of fundamental importance for understanding transport of dense mixtures of particles or molecules through finite length, water-filled channels or pore networks.

Barrier options have become increasingly popular financial instruments due to the lower costs and the ability to more closely match speculating or hedging needs. In addition, barrier options play a significant role in modeling and managing risks in insurance and finance as well as in refining insurance products such as variable annuities and equity-indexed annuities. Motivated by these immediate applications arising from actuarial and financial contexts, the thesis studies the pricing of barrier options and some exotic variations, assuming that the underlying asset price follows the Black-Scholes model or jump-diffusion processes. Barrier options have already been well treated in the classical Black-Scholes framework. The first part of the thesis aims to develop a new valuation approach based on the technique of exponential stopping and/or path counting of Brownian motions. We allow the option's boundaries to vary exponentially in time with different rates, and manage to express our pricing formulas properly as combinations of the prices of certain binary options. These expressions are shown to be extremely convenient in further pricing some exotic variations including sequential barrier options, immediate rebate options, multi-asset barrier options and window barrier options. Many known results will be reproduced and new explicit formulas will also be derived, from which we can better understand the impact on option values of various sophisticated barrier structures. We also consider jump-diffusion models, where it becomes difficult, if not impossible, to obtain the barrier option value in analytical form for exponentially curved boundaries. Our model assumes that the logarithm of the underlying asset price is a Brownian motion plus an independent compound Poisson process. It is quite common to assign a particular distribution (such as normal or double exponential distribution) for the jump size if one wants to pursue closed-form solutions, whereas our method permits any distributions for the jump size as long as they belong to the exponential family. The formulas derived in the thesis are explicit in the sense that they can be efficiently implemented through Monte Carlo simulations, from which we achieve a good balance between solution tractability and model complexity.