Simulation of First-Passage Times for Alternating Brownian Motions

The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.AMS 2000 Subject Classification: 60J65, 60G40, 93E30     

On Rescaled Poisson Processes and the Brownian Bridge

The process obtained by rescaling a homogeneous Poisson process by the maximum likelihood estimate of its intensity is shown to have surprisingly strong self-correcting behavior. Formulas for the conditional intensity and moments of the rescaled Poisson process are derived, and its behavior is demonstrated using simulations. Relationships to the Brownian bridge are explored, and implications for point process residual analysis are discussed.     

Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

We consider a model composed of a signal process X given by a classic stochastic differential equation and an observation process Y, which is supposed to be correlated to the signal process. We assume that process Y is observed from time 0 to s>0 at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process X crosses a fixed barrier after a given time t>0. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the involved quantities.     

A note about the Gaussian property of the Brownian bridge

The Gaussian property of the Brownian bridge is characterized as an application of Ramachandran's theorem in terms of the independence of the random variables that appear in the Karhunen-Loéve expansion of the process. A reference about the construction of the Brownian bridge by means of functional transformations is also included.     

次分数布朗运动局部时的研究

设X^H={X^H(t),t∈R+}是一个取值于R^d参数为H的次分数布朗运动.本文给出了X^H在单参数情况下局部时的Holder条件和尾概率估计.同时,还给出了X^H在多参数情况下局部时的存在性及L^2表示.     

标准布朗运动关于曲线边界通过概率

布朗运动是一种重要的随机过程,它的首出时的分布在很多方面有着重要的应用.该文讨论了布朗运动关于任意曲线边界的首出时的问题,求出了布朗运动停在双侧(单侧)曲线边界内的概率的分析表达式.     

标准布朗运动关于线性边界通过概率

该文讨论了布朗运动关于线性边界的首出时问题,求出了布朗运动停留在双侧(单侧)逐段线性边界内的概率的分析表达式.     

New Brownian bridge construction in quasi-Monte Carlo methods for computational finance

Quasi-Monte Carlo (QMC) methods have been playing an important role for high-dimensional problems in computational finance. Several techniques, such as the Brownian bridge (BB) and the principal component analysis, are often used in QMC as possible ways to improve the performance of QMC. This paper proposes a new BB construction, which enjoys some interesting properties that appear useful in QMC methods. The basic idea is to choose the new step of a Brownian path in a certain criterion such that it maximizes the variance explained by the new variable while holding all previously chosen steps fixed. It turns out that using this new construction, the first few variables are more “important” (in the sense of explained variance) than those in the ordinary BB construction, while the cost of the generation is still linear in dimension. We present empirical studies of the proposed algorithm for pricing high-dimensional Asian options and American options, and demonstrate the usefulness of the new BB.     

A Note on the Local Time of Fractional Brownian Motion

Asymptotic behavior of the local time at the origin of q-dimensional fractional Brownian motion is considered when the index approaches the critical value 1/q. It is proved that, under a suitable (temporally inhomogeneous) normalization, it converges in law to the inverse of an extremal process which appears in the extreme value theory.     

Optimal stopping of a Brownian bridge with an unknown pinning point

The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.     

The fractal dimensions of the level sets of the generalized iterated Brownian motion

Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.     

The Distribution of the Sojourn Time for the Brownian Excursion

The solutions of various problems in the theories of queuing processes, branching processes, random graphs and others require the determination of the distribution of the sojourn time (occupation time) for the Brownian excursion. However, no standard method is available to solve this problem. In this paper we approximate the Brownian excursion by a suitably chosen random walk process and determine the moments of the sojourn time explicitly. By using a limiting approach, we obtain the corresponding moments for the Brownian excursion. The moments uniquely determine the distribution, enabling us to derive an explicit formula.     

Distribution of maximum loss of fractional Brownian motion with drift

In this paper, we find bounds on the distribution of the maximum loss of fractional Brownian motion with $H\ge 1/2$ and derive estimates on its tail probability. Asymptotically, the tail of the distribution of maximum loss over $[0,t]$ behaves like the tail of the marginal distribution at time $t$.     

Asymptotics of the distribution of the integral of the positive part of the Brownian bridge for large arguments

The double Laplace transform of the distribution function of the integral of the positive part of the Brownian bridge was determined by M. Perman and J.A. Wellner, as well as the moments of this distribution. The purpose of the present paper is to determine the asymptotics of this distribution for large values of the argument, and the corresponding asymptotics of the moments.     

On the Wiener integral with respect to a sub-fractional Brownian motion on an interval

The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , k≠0, is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions.     

Singularity of generalized grey Brownian motions with different parameters

In this note we prove that the probability measures generated by two generalized grey Brownian motions with different parameters are singular with respect to each other. This result can be interpreted as an extension of the Feldman–Hájek dichotomy of Gaussian measures to a family of non-Gaussian measures.     

On the Asymptotic Behavior of the Storage Process Fed by a Markov Modulated Brownian Motion

Consider a storage model fed by a Markov modulated Brownian motion. We prove that the stationary distribution of the model exits and that the running maximum of the storage process over the interval [0, t] grows asymptotically like log t as t→∞.