Showing papers in "Stochastic Analysis and Applications in 2011"

Vector Random Fields with Second-Order Moments or Second-Order Increments

Abstract: This article is concerned with vector (multivariate, or multidimensional) random fields with second-order moments or second-order increments. Two crucial questions for such a random field are what kind of the square matrix function can be employed as its covariance matrix or variogram matrix, and, in particular, what type of the functions can be employed as its cross covariances or cross variograms. We attempt to explore the relationships between the direct covariance and the cross covariance in a covariance matrix and the relationships between the direct variogram and the cross variogram in a variogram matrix. Necessary and sufficient conditions are obtained for a given square matrix function to be the covariance matrix or variogram matrix of a vector Gaussian or elliptically contoured random field, and some parametric or nonparametric examples are given for stationary and nonstationary cases in a temporal, spatial, or spatio-temporal domain.

A Girsanov Type Theorem Under G-Framework

Abstract: This article establishes a Girsanov type theorem under the G-Framework of Peng [15]. Our result generalizes the classical Girsanov theorem for Brownian motion [10]. As an application, we price the European call option when the underlying asset's price follows the Geometric G-Brownian motion.

Fractional Lévy Processes as a Result of Compact Interval Integral Transformation

Abstract: Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional Levy processes are defined by integrating the infinite interval kernel w.r.t. a general Levy process. In this article we define fractional Levy processes using the com pact interval representation. We prove that the fractional Levy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions. Also, we present relations between different fractional Levy processes and analyze the properties of such processes. A financial example is introduced as well.

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

Abstract: We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.

On the Strong Rates of Convergence for Arrays of Rowwise Negatively Dependent Random Variables

Abstract: The strong convergence rate and complete convergence results for arrays of rowwise negatively dependent random variables are established. The results presented generalize the results of Chen et al. [1] and Sung et al. [2]. As applications, some well-known results on independent random variables can be easily extended to the case of negatively dependent random variables.

Stochastic Two-Dimensional Hydrodynamical Systems: Wong-Zakai Approximation and Support Theorem

Abstract: We deal with a class of abstract nonlinear stochastic models with multiplicative noise, which covers many 2D hydrodynamical models including the 2D Navier–Stokes equations, 2D MHD models and 2D magnetic Benard problems as well as some shell models of turbulence. Our main result describes the support of the distribution of solutions. Both inclusions are proved by means of a general Wong–Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process.

Uniform Convergence of Wavelet Expansions of Gaussian Random Processes

Abstract: New results on uniform convergence in probability for the most general classes of wavelet expansions of stationary Gaussian random processes are given.

Brownian Motion Indexed by a Time Scale

Abstract: In this article, we generalize Wiener's existence result for one-dimensional Brownian motion by constructing a suitable continuous stochastic process where the index set is a time scale. We construct a countable dense subset of a time scale and use it to prove a generalized version of the Kolmogorov–Centsov theorem. As a corollary, we obtain a local Holder-continuity result for the sample paths of generalized Brownian motion indexed by a time scale.

A Generalization of Weak Law of Large Numbers

Abstract: Kolmogorov's weak law of large numbers for i.i.d. random variables is generalized to a larger class distributions and to a wide class of normalizing sequences. The result is extended to maximal sums of negatively associated identically distributed random variables.

L p -Solutions for Doubly Reflected Backward Stochastic Differential Equations

Abstract: This article deals with the problem of existence and uniqueness of a solution for a backward stochastic differential equation with two strictly separated continuous reflecting barriers in the case when the terminal value, the generator and the obstacle process are L p -integrable with p ∈ (1, 2). The main idea is to use the concept of local solution to construct the global one. As applications, we obtain new results in the domains of zero-sum Dynkin games and in double obstacle variational inequalities.

The Maximum of Randomly Weighted Sums with Long Tails in Insurance and Finance

Abstract: In risk theory we often encounter stochastic models containing randomly weighted sums. In these sums, each primary real-valued random variable, interpreted as the net loss during a reference period, is associated with a nonnegative random weight, interpreted as the corresponding stochastic discount factor to the origin. Therefore, a weighted sum of m terms, denoted as S (w) m , represents the stochastic present value of aggregate net losses during the rst m periods. Suppose that the primary random variables are independent of each other with long-tailed distributions and are independent of the random weights. We show conditions on the random weights under which the tail probability of max1 m n S (w)

On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering

Abstract: We analyze the exponential stability properties of a class of measure-valued equations arising in nonlinear multi-target filtering problems. We also prove the uniform convergence properties w.r.t. the time parameter of a rather general class of stochastic filtering algorithms, including sequential Monte Carlo type models and mean field particle interpretation models. We illustrate these results in the context of the Bernoulli and the Probability Hypothesis Density filter, yielding what seems to be the first results of this kind in this subject.

Sequential Monte Carlo Methods for Option Pricing

Abstract: In the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.

Stochastic Nonlinear Equations of Schrödinger Type

Abstract: In this article, the solution for a stochastic nonlinear equation of Schrodinger type, which is perturbed by an infinite dimensional Wiener process, is investigated. The existence of the solution is proved by using the Galerkin method. Moment estimates for the solution are also derived. Examples from physics are given in the final part of the article.

A Criterion for Stationary Solutions of Retarded Linear Equations with Additive Noise

Abstract: In this work, we shall consider stationary (mild) solutions for a class of retarded functional linear differential equations with additive noise in Hilbert spaces. We first introduce a family of Green operators for the stochastic systems and establish stability results which will play an important role in the investigation of stationary solutions. A criterion imposed on the Green operators is presented to identify a unique stationary solution for the systems considered. Under strong quasi-Feller property, it is shown that this criterion is a sufficient and necessary condition to guarantee a unique stationary solution, based on a method having its origins in optimal control theory.

The Euler Scheme for Feller Processes

Abstract: We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented. In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Levy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Levy processes.

The Geometric Markov Renewal Processes with Application to Finance

Abstract: We introduce the geometric Markov renewal processes as a model for a security market and study this processes in a series scheme. We consider its approximations in the form of averaged, merged and double averaged geometric Markov renewal processes. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes are presented. Martingale properties, infinitesimal operators of geometric Markov renewal processes are presented and a Markov renewal equation for expectation is derived. As an application, we consider the case of two ergodic classes. Moreover, we consider a generalized binomial model for a security market induced by a position dependent random map as a special case of a geometric Markov renewal process.

A Note on the Rate of Complete Convergence for Weighted Sums of Arrays of Banach Space Valued Random Elements

Abstract: A rate of complete convergence for weighted sums of arrays of rowwise independent Banach space valued random elements was obtained by Ahmed et al. [1]. Recently, Sung and Volodin [2], Chen et al. [3], and Kim and Ko [4] solved an open question posed by Ahmed et al. In this article, we improve and complement the result of Ahmed et al. The method used in this article is simpler than those in Ahmed et al., Sung and Volodin, Chen et al., and Kim and Ko.

Stochastic Processes with Age-Dependent Transition Rates

Abstract: We study stochastic processes with age-dependent transition rates. A typical example of such a process is a semi-Markov process which is completely determined by the holding time distributions in each state and the transition probabilities of the embedded Markov chain. The process we construct generalizes semi-Markov processes. One important feature of this process is that unlike semi-Markov processes the transition probabilities of this process are age-dependent. Under certain condition we establish the Feller property of the process. Finally, we compute the limiting distribution of the process.

The First Attempt on the Stochastic Calculus on Time Scale

Abstract: The aim of this article is to study the Doob–Meyer decomposition theorem, ∇-stochastic integration and Ito's formula for stochastic processes defined on time scale. The obtained results can be considered as a first attempt on the stochastic calculus on time scale.

A Wiener Chaos Approach to Hyperbolic SPDEs

Abstract: We construct generalized weighted Wiener chaos solutions for hyperbolic linear SPDEs driven by a cylindrical Brownian motion. Explicit conditions for the existence, uniqueness, and regularity of generalized (Wiener Chaos) solutions are established in Sobolev spaces. An equivalence relation between the Wiener Chaos solution and the traditional one is established. The Heath–Jarrow–Morton (HJM) forward rate model is used as an example to illustrate the general construction.

Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

Abstract: We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ℝ d , like, for example, (B 1(|C(t)|),…, B d (|C(t)|)) and (C 1(|C(t)|),…, C d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B 1(|B 2(…|B n+1(t)|…)|)|) are governed by fractional diffusion equations.

Backward Stochastic Differential Equations for a Single Jump Process

Abstract: We consider backward stochastic differential equations (BSDEs) related to a finite continuous time single jump process. We prove the existence and uniqueness of solutions when the coefficients satisfy Lipschitz continuity conditions. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are then investigated.

From Brownian-Time Brownian Sheet to a Fourth Order and a Kuramoto–Sivashinsky-Variant Interacting PDEs Systems

Abstract: We introduce n-parameter ℝ d -valued Brownian-time Brownian sheet (BTBS): a Brownian sheet where each “time” parameter is replaced with the modulus of an independent Brownian motion. We then connect BTBS to a new system of n linear, fourth order, and interacting PDEs and to a corresponding fourth order interacting nonlinear PDE. The coupling phenomenon is a result of the interaction between the Brownian sheet, through its variance, and the Brownian motions in the BTBS; and it leads to an intricate, intriguing, and random field generalization of our earlier Brownian-time-processes (BTPs) connection to fourth order linear PDEs. Our BTBS does not belong to the classical theory of random fields; and to prove our new PDEs connections, we generalize our BTP approach in [4, 5] and we mix it with the Brownian sheet connection to a linear PDE system, which we also give along with its corresponding nonlinear second order PDE and a 2nth order linear PDE that we also connect to Brownian sheet. In addition, we introdu...

The Stochastic Dirichlet Problem Driven by the Ornstein–Uhlenbeck Operator: Approach by the Fredholm Alternative for Chaos Expansions

Abstract: In this article, we prove the Fredholm alternative theorem for mappings defined on spaces of generalized stochastic processes given by their Wiener–It chaos expansion form. We apply the result to solve the stochastic Dirichlet problem with a perturbation term driven by the Ornstein–Uhlenbeck operator.

Municipal Water Demand Forecasting: Tools for Intervention Time Series

Abstract: This article introduces some approaches to common issues arising in real cases of water demand prediction. Occurrences of negative data gathered by the network metering system and demand changes due to closure of valves or changes in consumer behavior are considered. Artificial neural networks (ANNs) have a principal role modeling both circumstances. First, we propose the use of ANNs as a tool to reconstruct any anomalous time series information. Next, we use what we call interrupted neural networks (I-NN) as an alternative to more classical intervention ARIMA models. Besides, the use of hybrid models that combine not only the modeling ability of ARIMA to cope with the time series linear part, but also to explain nonlinearities found in their residuals, is proposed. These models have shown promising results when tested on a real database and represent a boost to the use and the applicability of ANNs.

Convergence of Numerical Approximation for Jump Models Involving Delay and Mean-Reverting Square Root Process

Abstract: The mean-reverting square root process with jump has been widely used as a model on the financial market. Since the diffusion coefficient in the model does not satisfy the linear growth condition and local Lipschitz condition, we can not examine its properties by traditional techniques. To overcome the difficulties, we develop several new techniques to examine the numerical method of jump models involving delay and mean-reverting square root. We show that the numerical approximate solutions converge to the true solutions. Finally, we apply the convergence to examine a path-dependent option price and a bond in the financial pricing.

Almost Sure Local Limit Theorems with Rate

Abstract: We establish almost sure versions, with rate, of the local limit theorem for lattice distributed random variables. We also prove a new delicate correlation inequality for sums of i.i.d. lattice distributed random variables.

Nonparametric Estimation of Linear Multiplier for Fractional Diffusion Processes

Abstract: We study the problem of nonparametric estimation of linear multiplier function θ(t) for processes satisfying stochastic differential equations of the type where is a standard fractional Brownian motion with known Hurst index H ∈ (1/2, 3/4) and study the asymptotic behaviour of the estimator as e → 0.

On a Class of Nonlocal Stochastic Functional Differential Equations with Infinite Delay

Abstract: This article considers a class of nonlocal stochastic functional differential equations with infinite delay whose coefficients are dependent the pth moment and establishes the existence-and-uniqueness theorem under the conditions that are similar to the classical linear growth condition and the Lipschitz condition. Compared with the existing results, the conditions of this article are easier to test.