Gaussian moving averages,semimartingales and option pricing

We provide a characterization of the Gaussian processes with stationary increments that can be represented as a moving average with respect to a two-sided Brownian motion. For such a process we give a necessary and sufficient condition to be a semimartingale with respect to the filtration generated by the two-sided Brownian motion. Furthermore, we show that this condition implies that the process is either of finite variation or a multiple of a Brownian motion with respect to an equivalent probability measure. As an application we discuss the problem of option pricing in financial models driven by Gaussian moving averages with stationary increments. In particular, we derive option prices in a regularized fractional version of the Black–Scholes model.     

Large Deviations from the Almost Everywhere Central Limit Theorem

We prove large deviation principles for the almost everywhere central limit theorem, assuming that the i.i.d. summands have finite moments of all orders. The level 3 rate function is a specific entropy relative to Wiener measure and the level 2 rate the Donsker-Varadhan entropy of the Ornstein-Uhlenbeck process. In particular, the rate functions are independent of the particular distribution of the i.i.d. process under study. We deduce these results from a large deviation theory for Brownian motion via Skorokhod's representation of random walk as Brownian motion evaluated at random times. The results for Brownian motion come from the well-known large deviation theory of the Ornstein-Uhlenbeck process, by a mapping between the two processes.     

Performance sampling and multiple choices: Brow ni an motion models

This paper extends the Brownian motion model for a decision making process that was developed in (Romanow, 1984). Several of the assumptions of the basic model are relaxed here. It is no longer assumed that performance is always observed, but rather that observation is intermittent. Models are developed for several different data collection schemes. The original model allowed two choice alternatives. Here, a third choice is included by considering a model in which a Brownian motion may cross either of two mutually exclusive boundaries.     

Path Space Large Deviations of a Large Buffer with Gaussian Input Traffic

We consider a queue fed by Gaussian traffic and give conditions on the input process under which the path space large deviations of the queue are governed by the rate function of the fractional Brownian motion. As an example we consider input traffic that is composed of of independent streams, each of which is a fractional Brownian motion, having different Hurst indices.     

Small and Large Scale Behavior of the Poissonized Telecom Process

The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well.     

Karhunen-Loeve expansions for the m-th order detrended Brownian motion

The m-th order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion onto the subspace spanned by polynomials of degree up to m. We obtain the Karhunen-Loeve expansion for the process and establish a connection with the generalized (m-th order) Brownian bridge developed by MacNeill (1978) in the study of distributions of polynomial regression. The resulting distribution identity is also verified by a stochastic Fubini approach. As applications, large and small deviation asymptotic behaviors for the L ₂ norm are given.     

Professor V. Lakshmikantham,Editor Stochastic Analysis and Applications (1983–2012)

We use the stochastic calculus of variations for the fractional Brownian motion to derive formulas for the replicating portfolios for a class of contingent claims in a Bachelier and a Black–Scholes markets modulated by fractional Brownian motion. An example of such a model is the Black–Scholes process whose volatility solves a stochastic differential equation driven by a fractional Brownian motion that may depend on the underlying Brownian motion.     

A Set-indexed Fractional Brownian Motion

We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.      

Laplace approximation of transition densities posed as Brownian expectations

We construct the Laplace approximation of the Lebesgue density for a discrete partial observation of a multi-dimensional stochastic differential equation. This approximation may be computed integrating systems of ordinary differential equations. The construction of the Laplace approximation begins with the definition of the point of minimum energy. We show how such a point can be defined in the Cameron–Martin space as a maximum a posteriori estimate of the underlying Brownian motion given the observation of a finite-dimensional functional. The definition of the MAP estimator is possible via a renormalization of the densities of piecewise linear approximations of the Brownian motion. Using the renormalized Brownian density the Laplace approximation of the integral over all Brownian paths can be defined. The developed theory provides a method for performing approximate maximum likelihood estimation.     

A version of Pitman's 2M — X theorem for geometric Brownian motions

We show that geometric Brownian motion with parameter μ, i.e., the exponential of linear Brownian motion with drift μ, divided by its quadratic variation process is a diffusion process. Taking logarithms and an appropriate scaling limit, we recover the Rogers-Pitman extension to Brownian motion with drift of Pitman's representation theorem for the three-dimensional Bessel process. Time inversion and generalized inverse Gaussian distributions play crucial roles in our proofs.     

The exit distribution for iterated Brownian motion in cones

We study the distribution of the exit place of iterated Brownian motion in a cone, obtaining information about the chance of the exit place having large magnitude. Along the way, we determine the joint distribution of the exit time and exit place of Brownian motion in a cone. This yields information on large values of the exit place (harmonic measure) for Brownian motion. The harmonic measure for cones has been studied by many authors for many years. Our results are sharper than any previously obtained.     

n重分数布朗运动的Strassen泛函重对数律

本文给出了由两个不同的分数布朗运动组成的重分数布朗运动的Strassen型泛函重对数律和局部Strassen型泛函重对数律.我们的结果也适用于由两个布朗运动组成的重布朗运动及由一个分数布朗运动和一个布朗运动组成的重过程.最后将上述结果推广到n重分数布朗运动中.推广了已有文献的相应结果.     

An identity of hitting times and its application to the valuation of guaranteed minimum withdrawal benefit

In this paper we explore an identity in distribution of hitting times of a finite variation process (integrated geometric Brownian motion) and a diffusion process (geometric Brownian motion with affine drift), both of which arise from various applications in financial mathematics. We develop semi-analytical solutions to fair charges of variable annuity guaranteed minimum withdrawal benefit from both a policyholder’s perspective and an insurer’s perspective. The pricing framework from the policyholder’s perspective was known previously in the literature only by numerical methods, whereas the insurer’s pricing method was used in the industry but only with Monte Carlo simulations. While comparing their similarities and differences, we prove under the assumption of no friction cost the two pricing approaches are equivalent. In the presence of friction cost, the semi-analytic solutions in this paper lead to a fast and accurate algorithm for determining rider charges and other management fees.     

Harmonic functions on Walsh’s Brownian motion

We examine a variation of two-dimensional Brownian motion introduced by Walsh that can be described as Brownian motion on the spokes of a (rimless) bicycle wheel. We construct the process by randomly assigning angles to excursions of reflecting Brownian motion. Hence, Walsh’s Brownian motion behaves like one-dimensional Brownian motion away from the origin, but differently at the origin as it is immediately sent off in random directions. Given the similarity, we characterize harmonic functions as linear functions on the rays satisfying a slope-averaging property. We also classify superharmonic functions as concave functions on the rays satisfying extra conditions.     

How Big Are the Increments of a Two-Parameter Gaussian Process?

Some limit theorems on the increments of a two-parameter Gaussian process are obtained via estimating large deviation probability inequalities on the suprema of the Gaussian process which is a generalization of a two-parameter Lévy Brownian motion.     

Ergodicity and First Passage Probability of Regime-Switching Geometric Brownian Motions

A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov chain. In this work, the author gives a complete characterization of the recurrent property of this process. The long time behavior of this process such as its p-th moment is also studied. Moreover, the quantitative properties of the regime-switching geometric Brownian motion with two-state switching are investigated to show the difference between geometric Brownian motion with switching and without switching. At last, some estimates of its first passage probability are established.     

General approach to filtering with fractional brownian noises — application to linear systems

At first a general approach is proposed to filtering in systems where the observation noise is a fractional Brownian motion. It is shown that the problem can be handled in terms of some appropriate semimartingale and analogs of the classical innovation process and fundamental filtering theorem are obtained. Then the problem of optimal filtering is completely solved for Gaussian linear systems with fractional Brownian noises. Closed form simple equations are derived both for the mean of the optimal filter and the variance of the filtering error. Finally the results are explicited in various specific cases     

Finite dimensional characteristic functions of Brownian rough path

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2: Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in ?^d and obtain an explicit formula for the case when d = 2     

The rate of quasi sure convergence in the functional limit theorem for increments of a Brownian motion

We investigate the quasi sure convergence of the functional limit for increments of a Brownian motion. The rate of quasi sure convergence in the functional limit for increments of a d-dimensional Brownian motion is derived. The main tool in the proof is large deviation and small deviation for Brownian motion in terms of (r,p)-capacity.