Computational approaches to estimation in the principal component analysis of a stochastic process

After performing a review of the classical procedures for estimation in the principal component analysis (PCA) of a second order stochastic process, two alternative procedures have been developed to approach such estimates. The first is based on the orthogonal projection method and uses cubic interpolating splines when the data are discrete. The second is based on the trapezoidal method. The accuracy of both procedures is tested by simulating approximated sample-functions of the Brownian motion and the Brownian bridge. The real principal factors of these stochastic processes, which can be evaluated directly, are compared with those estimated by means of the two mentioned algorithms. An application for estimation in the PCA of tourism evolution in Spain from real data is also included.     

On an Identity in Law for the Variance of the Brownian Bridge

An explanation of a duplication identity in law involving thevariances of the Brownian bridge and Brownian motion is given,with the help of an elementary transformation in time and spaceof a two-dimensional Brownian motion. 1991 Mathematics SubjectClassification 60J65.     

Brownian bridge in quantum probability

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.     

On the maximum of the generalized Brownian bridge

We present some extensions of the distributions of the maximum of the Brownian bridge in [0,t] when the conditioning event is placed at a future timeu>t or at an intermediate timeu<t. The standard distributions of Brownian motion and Brownian bridge are obtained as limiting cases. These results permit us to derive also the distribution of the first-passage time of the Brownian bridge. Similar generalizations are carried out for the Brownian bridge with drift μ; in this case, it is shown that the maximal distribution is independent of μ (whenu≥t). Finally, the case of the two-sided maximal distribution of Brownian motion in [0,t], conditioned onB(u)=η (for bothu>t andu<t), is considered. Dip. di Statistica, Probabilità e Stat. Applicate, Università di Roma “La Sapienza,” Piazzale Aldo Moros, 00185 Roma, Italy. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 200–213, April–June, 1999.     

On Striking Identities About the Exponential Functionals of the Brownian Bridge and Brownian Motion

The negative moment of order 1, resp. of order 1/2, for the integral on (0, 1) of the exponential of α times the Brownian bridge, resp. the Brownian motion, does not depend on α. We give a simple explanation and a reinforcement of this property in the case of the Brownian bridge. We then discuss how different the case of the Brownian motion is.     

An excursion approach to maxima of the Brownian bridge

Distributions of functionals of Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the local time has been used widely in the literature. In this paper it is used to give a unified derivation of a number of known distributions, and a few new ones. Particular cases of calculations include the distribution of the Kolmogorov–Smirnov statistic and the Kuiper statistic.     

Test to distinguish a Brownian motion from a Brownian bridge using Polya tree process

The problem of distinguishing a Brownian bridge from a Brownian motion, both with possible drift, on the closed unit interval, is investigated via a pair of hypothesis tests. The first, tests for observations obtained at n discrete time points to be arising from a Brownian bridge with drift by embedding the Brownian bridge into a mixture of Polya trees which represents the non-parametric alternative. The second test, tests in an identical manner, for the observations to be coming from a Brownian motion with drift. The Bayes factors for the two tests are derived and then combined to obtain the Bayes factor for the test to distinguish between the two Gaussian processes. The Tierney-Kadane approximation of the Bayes factor is derived with an error approximation of order O(n⁻⁴).     

An Extension of Vervaat's Transformation and Its Consequences

Vervaat⁽¹⁸⁾ proved that by exchanging the pre-minimum and post-minimum parts of a Brownian bridge one obtains a normalized Brownian excursion. Let s (0, 1), then we extend this result by determining a random time m _s such that when we exchange the pre-m _s-part and the post-m _s-part of a Brownian bridge, one gets a Brownian bridge conditioned to spend a time equal to s under 0. This transformation leads to some independence relations between some functionals of the Brownian bridge and the time it spends under 0. By splitting the Brownian motion at time m _s in another manner, we get a new path transformation which explains an identity in law on quantiles due to Port. It also yields a pathwise construction of a Brownian bridge conditioned to spend a time equal to s under 0.     

Local spectral gaps on loop spaces

We prove Poincaré inequalities w.r.t. the distributions of Brownian bridges on sets of loops with jumps of limited size over compact Riemannian manifolds. Moreover, we study the asymptotic behavior of the second Dirichlet eigenvalues as the time parameter T of the underlying Brownian bridge tends to 0. This behavior depends crucially on the geodesics contained in the set of loops considered. In particular, for different choices of a Riemannian metric on the base manifold, qualitatively different asymptotic behaviors can occur. The proof of the basic Poincaré inequality is based on the construction of the Brownian bridge by consecutive bisection of the parametrization interval.     

The Brownian Bridge Does Not Offer a Consistent Advantage in Quasi-Monte Carlo Integration

The Brownian bridge has been suggested as an effective method for reducing the quasi-Monte Carlo error for problems in finance. We give an example of a digital option where the Brownian bridge performs worse than the standard discretization. Hence, the Brownian bridge does not offer a consistent advantage in quasi-Monte Carlo integration. We consider integrals of functions of d variables with Gaussian weights such as the ones encountered in the valuation of financial derivatives and in risk management. Under weak assumptions on the class of functions, we study quasi-Monte Carlo methods that are based on different covariance matrix decompositions. We show that different covariance matrix decompositions lead to the same worst case quasi-Monte Carlo error and are, therefore, equivalent.     

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.     

On a One-Parameter Generalization of the Brownian Bridge and Associated Quadratic Functionals

A one-parameter generalization of the Brownian bridge is studied. These processes are then used to compute the laws of some quadratic functionals of Brownian motion, and to obtain identities in law involving local time of modified Bessel processes up to their first hitting time.     

Efficient Monte Carlo simulation for integral functionals of Brownian motion

In the paper, we develop a variance reduction technique for Monte Carlo simulations of integral functionals of a Brownian motion. The procedure is based on a new method of sampling the process, which combines the Brownian bridge construction with conditioning on integrals along paths of the process. The key element in our method is the identification of a low-dimensional vector of variables that reduces the dimension of the integration problem more effectively than the Brownian bridge. We illustrate the method by applying it in conjunction with low-discrepancy sequences to the problem of pricing Asian options.     

Asymptotics of an Efficient Monte Carlo Estimation for the Transition Density of Diffusion Processes

Discretized simulation is widely used to approximate the transition density of discretely observed diffusions. A recently proposed importance sampler, namely modified Brownian bridge, has gained much attention for its high efficiency relative to other samplers. It is unclear for this sampler, however, how to balance the trade-off between the number of imputed values and the number of Monte Carlo simulations under a given computing resource. This paper provides an asymptotically efficient allocation of computing resource to the importance sampling approach with a modified Brownian bridge as importance sampler. The optimal trade-off is established by investigating two types of errors: Euler discretization error and Monte Carlo error. The main results are illustrated with two simulated examples.      

Probability density function of the local score position

We calculate the probability density function of the local score position on complete excursions of a reflected Brownian motion. We use the trajectorial decomposition of the standard Brownian bridge to derive two different expressions of the density: the first one is based on a series and an integral while the second one is free off the series.     

The empirical distribution function and partial sum process of residuals from a stationary arch with drift process

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 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.     

Estimates of derivatives of the heat kernel on a compact Riemannian manifold

We give global estimates on the covariant derivatives of the heat kernel on a compact Riemannian manifold on a fixed finite time interval. The proof is based on analyzing the behavior of the heat kernel along Riemannian Brownian bridge.

     

Une extension des théorèmes de Ray et Knight sur les temps locaux Browniens

Summary We introduce two classes of random variablesV such that the Brownian local time process at timeV is distributed as a 0 or 2 dimensional Bessel bridge. Moreover we obtain new decompositions of the Brownian path on the interval [0,V], which generalize Williams' results.     

Bridge representation and modal-path approximation

The article shows a bridge representation for the joint density of a system of stochastic processes consisting of a Brownian motion with drift coupled with a correlated fractional Brownian motion with drift. As a result, a small time approximation of the joint density is readily obtained by substituting the conditional expectation under the bridge measure by a single path: the modal-path from the initial point to the terminal point.     

A Local Refinement Strategy for Constructive Quantization of Scalar SDEs

We present a fully constructive method for quantization of the solution X of a scalar SDE in the path space L _p [0,1] or C[0,1]. The construction relies on a refinement strategy which takes into account the local regularity of X and uses Brownian motion (bridge) quantization as a building block. Our algorithm is easy to implement, its computational cost is close to the size of the quantization, and it achieves strong asymptotic optimality provided this property holds for the Brownian motion (bridge) quantization.