Convex Hull of Brownian Motion and Brownian Bridge

Karhunen–Loève expansion for additive Brownian motions

The weak convergence of the empirical process and partial sum process of the residuals from a stationary ARCH-M model is studied. It is obtained for and\(\sqrt n \) consistent estimate of the ARCH-M parameters. We find that the limiting Gaussian processes are no longer distribution free and hence residuals cannot be treated as i.i.d. In fact the limiting Gaussian process for the empirical process is a standard Brownian bridge plus an additional term, while the one for partial sum process is a standard Brownian motion plus an additional term. In the special case of a standard ARCH process, that is an ARCH process with no drift, the additional term disappears. We also study a sub-sampling technique which yields the limiting Gaussian processes for the empirical process and partial sum process as a standard Brownian bridge and a standard Brownian motion respectively.

The goal of the present paper is to show that the intersection exponents for planar Brownian motions are analytic. Let k~>l be a positive integer and let X 1 , . . . , X k be independent Brownian motions in the complex plane C started from 0. Let Y, y1, y2, ... denote other independent planar Brownian motions started from 1, and let St be the random variable (measurable with respect to X 1, ..., X k)

1 - Wiener and some Related Gaussian Processes

We consider a system of noncolliding Brownian motions introduced in our previous paper, in which the noncolliding condition is imposed in a finite time interval $(0,T]$. This is a temporally inhomogeneous diffusion process whose transition probability density depends on a value of $T$, and in the limit $T \to \infty$ it converges to a temporally homogeneous diffusion process called Dyson's model of Brownian motions. It is known that the distribution of particle positions in Dyson's model coincides with that of eigenvalues of a Hermitian matrix-valued process, whose entries are independent Brownian motions. In the present paper we construct such a Hermitian matrix-valued process, whose entries are sums of Brownian motions and Brownian bridges given independently of each other, that its eigenvalues are identically distributed with the particle positions of our temporally inhomogeneous system of noncolliding Brownian motions. As a corollary of this identification we derive the Harish-Chandra formula for an integral over the unitary group.

How do path generation methods affect the accuracy of quasi-Monte Carlo methods for problems in finance?

Let {η(t), 0 ≤ t} ≤ 1} be a standard Brownian bridge. We have for 0 < t < 1, where $$ \Phi (x) = \frac{1}{{\sqrt {{2\pi }} }}\int_{{ - \infty }}^{x} {{{e}^{{ - u \frac{2}{2} }}}} du $$ (4.1) is the normal distribution function. We define $$ \tau (\alpha ) = \mathop{{\lim }}\limits_{{\varepsilon \to 0}} \frac{1}{\varepsilon }{\text{measure\{ }}t:\alpha \leqslant \eta (t) < \alpha + \varepsilon ,0 \leqslant t \leqslant 1{\text{\} }} $$ (4.2) for any real α. The limit (4.2) exists with probability one, and τ(α) is a nonnegative random variable that is called the local time at level α. We have $$ P\{ \tau (\alpha ) \leqslant x\} = 1 - {{e}^{{ - (2|\alpha | + x) \frac{2}{2} }}} $$ (4.3) for x ≥ 0. The notion of local time was introduced by P. Levy [9,10] in 1939 (see also [15] and [6]).

We provide a representation of the maximal difference between a standard Brownian bridge and its concave majorant on the unit interval, from which we deduce expressions for the distribution and density functions and moments of this difference. This maximal difference has an application in nonparametric statistics, where it arises in testing monotonicity of a density or regression curve.

We introduce a new class of stochastic processes called fractional Wiener–Weierstraß bridges. They arise by applying the convolution from the construction of the classical, fractal Weierstraß functions to an underlying fractional Brownian bridge. By analyzing the p p -th variation of the fractional Wiener–Weierstraß bridge along the sequence of b b -adic partitions, we identify two regimes in which the processes exhibit distinct sample path properties. We also analyze the critical case between those two regimes for Wiener–Weierstraß bridges that are based on a standard Brownian bridge. We furthermore prove that fractional Wiener–Weierstraß bridges are never semimartingales, and we show that their covariance functions are typically fractal functions. Some of our results are extended to Weierstraß bridges based on bridges derived from a general continuous Gaussian martingale.

On the first crossing times of a Brownian motion and a family of continuous curves

This article on Brownian motion starts with a short historical survey on its discovery and importance and continues with its most relevant properties. Mathematically detailed formulas are given and connections between the properties are indicated, however no derivations or proofs are provided. The properties covered start with the mathematical modeling of Brownian motion via the Wiener process; equivalent definitions and constructions such as the Lévy characterization, the midpoint displacement, and the rescaled random walks are discussed. Basic properties given include uniqueness, symmetry, restarted Brownian motion, and time inversion. More advanced topics treated cover stopping times, the strong Markov property, the reflection principle, extreme values, escape times from a strip, and last visits. Path oscillations, stationarity, Gaussian white noise, and Karhunen‐Löwe expansions are also discussed. Brownian motion with drift, the Brownian bridge, as well as intersections, recurrence, and transience of Brownian motion in higher dimensions, are mentioned briefly.

This article on Brownian motion starts with a short historical survey on its discovery and importance and continues with its most relevant properties. Mathematically detailed formulas are given and connections between the properties are indicated, however no derivations or proofs are provided. The properties covered start with the mathematical modeling of Brownian motion via the Wiener process; equivalent definitions and constructions such as the Lévy characterization, the midpoint displacement, and the rescaled random walks are discussed. Basic properties given include uniqueness, symmetry, restarted Brownian motion, and time inversion. More advanced topics treated cover stopping times, the strong Markov property, the reflection principle, extreme values, escape times from a strip, and last visits. Path oscillations, stationarity, Gaussian white noise, and Karhunen‐Löwe expansions are also discussed. Brownian motion with drift, the Brownian bridge, as well as intersections, recurrence, and transience of Brownian motion in higher dimensions, are mentioned briefly.

For a random process $X$ consider the random vector defined by the values of $X$ at times $0 \lt U_{n,1} \lt ... \lt U_{n,n} \lt 1$ and the minimal values of $X$ on each of the intervals between consecutive pairs of these times, where the $U_{n,i}$ are the order statistics of $n$ independent uniform $(0,1)$ variables, independent of $X$. The joint law of this random vector is explicitly described when $X$ is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at $n$ independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae are Brownian representions of various special functions, including Bessel polynomials, some hypergeometric polynomials, and the Hermite function. Various combinatorial identities involving random partitions and generalized Stirling numbers are also obtained.

The Classical Brownian Bridge is constructed in Symmetric Fock space over an appropriate base Hilbert space. While the representation of the classical Ito-Wiener integral with respect to the increments of the Brownian bridge implements the unitary isomorphism between the Fock space and the (classical) L2 space of the Brownian bridge (as is the case with the standard Brownian motion (SBM)), the quantum Ito-integrals with respect to the associated creation and annihilation bridge processes give different left-and right-integrals. This essentially displays the feature that the Brownian Bridge is not a process of independent increments.

We consider all two times iterated Ito integrals obtained by pairing m independent standard Brownian motions. First we calculate the conditional joint characteristic function of these integrals, given the Brownian increments over the integration interval, and show that it has a form entirely similar to what is obtained in the univariate case. Then we propose an algorithm for the simultaneous simulation of the m^2 integrals conditioned on the Brownian increments that achieves a mean square error of order 1/n^2, where n is the number of terms in a truncated sum. The algorithm is based on approximation of the tail-sum distribution, which is a multivariate normal variance mixture, by a multivariate normal distribution.