Non-independence of excursions of the Brownian sheet and of additive Brownian motion

A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.

     

The exact packing measure of Brownian double points

Let $D\subset {\mathbb{R}}^3$ be the set of double points of a three-dimensional Brownian motion. We show that, if ξ = ξ₃(2,2) is the intersection exponent of two packets of two independent Brownian motions, then almost surely, the ?-packing measure of D is zero if $$ \int_{0^+} r^{-1-\xi} \phi(r)^{\xi} \, dr < \infty,$$ and infinity otherwise. As an important step in the proof we show up-to-constants estimates for the tail at zero of Brownian intersection local times in dimensions two and three.     

Karhunen–Loève expansion for additive Brownian motions

For Gaussian random fields defined as additive processes based on standard Brownian motions and Brownian bridges, we find their Karhunen–Loève expansions and make connections with related mean centered processes in distribution. Moreover, Pythagorean type distribution identities are established for additive Brownian motions and Brownian bridges. As applications, the corresponding Laplace transform and small deviation estimates are given.     

The rate of convergence of Hurst index estimate for the stochastic differential equation

We consider a stochastic differential equation involving a pathwise integral with respect to fractional Brownian motion. The estimates for the Hurst parameter are constructed according to first- and second-order quadratic variations of observed values of the solution. The rate of convergence of these estimates to the true value of a parameter is established when the diameter of interval partition tends to zero.     

On singularity as a function of time of a conditional distribution of an exit time

We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability distribution that the sum of two one-dimensional Brownian motions will exit from the unit interval before time t, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.     

Ten penalisation results of Brownian motion involving its one-sided supremum until first and last passage times, VIII

We penalise Brownian motion by a function of its one-sided supremum considered up to the last zero before t, respectively first zero after t, of that Brownian motion. This study presents some analogy with penalisation by the longest length of Brownian excursions, up to time t.     

Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators.     

Attractiveness of Brownian queues in tandem

Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one-sided reflection.

     

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

We review the analytic transformations allowing to construct standard Brownian bridges from a Brownian motion. These are generalized and some of their properties are studied. The new family maps the space of continuous positive functions into a family of curves which is the topic of our study. We establish a simple and explicit formula relating the distributions of the first hitting times of each of these curves by a standard Brownian motion. To cite this article: L. Alili, P. Patie, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Oscillatory Fractional Brownian Motion

We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. A randomness property of the initial condition increases the long range dependence. We emphasize the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space.     

THE JOINT DISTRIBUTIONS OF SOME ACTUARIAL DIAGNOSTICS FOR THE JUMP-DIFFUSION RISK PROCESS

In this article, the joint distributions of several actuarial diagnostics which are important to insurers’ running for the jump-diffusion risk process are examined. They include the ruin time, the time of the surplus process leaving zero ultimately (simply, the ultimately leaving-time), the surplus immediately prior to ruin, the supreme profits before ruin, the supreme profits and deficit until it leaves zero ultimately and so on. The explicit expressions for their distributions are obtained mainly by the various properties of L′evy process, such as the homogeneous strong Markov property and the spatial homogeneity property etc, moveover, the many properties for Brownian motion.     

A note on convergence of option prices and their Greeks for Lévy models

We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.     

Passport options with stochastic volatility

A passport option is a call option on the profits of a trading account. In this article, the robustness of passport option pricing is investigated by incorporating stochastic volatility. The key feature of a passport option is the holders' optimal strategy. It is known that in the case of exponential Brownian motion the strategy is to be long if the trading account is below zero and short if the account is above zero. Here this result is extended to models with stochastic volatility where the volatility is defined via an autonomous SDE. It is shown that if the Brownian motions driving the underlying asset and the volatility are independent then the form of the optimal strategy remains unchanged. This means that the strategy is robust to misspecification of the underlying model. A second aim of this article is to investigate some of the biases which become apparent in a stochastic volatility regime. Using an analytic approximation, comparisons are obtained for passport option prices using the exponential Brownian motion model and some well-known stochastic volatility models. This is illustrated with numerical examples. One conclusion is that if volatility and price are uncorrelated, then prices are sometimes lower in a model with stochastic volatility than in a model with constant volatility.     

Bouncing Skew Brownian Motions

We consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. In Gloter and Martinez (Ann Probab 41(3A):1628–1655, 2013), the evolution of the distance between the two processes, in local timescale and up to their first hitting time, is shown to satisfy a stochastic differential equation with jumps driven by the excursion process of one of the two skew Brownian motions. In this article, we show that the distance between the two processes in local timescale may be viewed as the unique continuous Markovian self-similar extension of the process described in Gloter and Martinez (2013). This permits us to compute the law of the distance of the two skew Brownian motions at any time in the local timescale, when both original skew Brownian motions start from zero. As a consequence, we give an explicit formula for the entrance law of the associated excursion process and study the Markovian dependence on the skewness parameter. The results are related to an open question formulated initially by Burdzy and Chen (Ann Probab 29(4):1693–1715, 2001).     

Asymptotics for the Expected Lifetime of Brownian Motion on Thin Domains in ℝ n

We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in ? ⁿ , and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime. The approach is similar to that which we used previously to study the eigenvalues of the Dirichlet Laplacian and consists of scaling the domain in one direction and deriving the corresponding asymptotic expansions as the scaling parameter goes to zero. Apart from being dominated by the one-dimensional Brownian motion along the direction of the scaling, we also see that the symmetry of the perturbation plays a role in the expansion. As in the case of eigenvalues, these expansions may also be used to approximate the exit time for domains where the scaling parameter is not necessarily close to zero.     

Réflexion entre deux diffusions conjuguées

We observed, in a previous work, that Brownian motion reflected on an independent time-reversed Brownian motion is again Brownian motion. We present the generalisation of this result to pairs of conjugate diffusions (which are also dual, in the sense of Siegmund). To cite this article: F. Soucaliuc, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1119–1124.     

Distribution of functionals of Brownian motion stopped at a time that is inverse to local time

Integral functionals of Brownian motion and of Brownian local time, as well as the supremum of Brownian motion and the supremum of Brownian local time are considered. The obtained results allow the computation of the distributions of these functionals for a Brownian motion stopped at the moment when the local time attains first a given value at one of two levels. It has been established that for this stopping time the Brownian local time is a Markov process with respect to the space variable and the generating operator of the process has been found. Examples of the computation of the distributions of certain functionals are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 37–61, 1990.     

The Zero Set of Fractional Brownian Motion Is a Salem Set

The existence of sets supporting a Borel measure such that its Fourier transform tends to zero at infinity can be traced back to the problem of uniqueness of trigonometric series, studied extensively by Cantor. Given \(\alpha \in (0, 1)\), Beurling asked if there exists a subset of the real line of Hausdorff dimension \(\alpha \) supporting a Borel measure whose Fourier transform converges to zero at infinity with rate \(\alpha /2\). Salem answered the question in the affirmative and such sets are now called Salem sets or rounded sets. Kahane showed that images of compact sets by fractional Brownian motion are Salem sets and this was recently extended to Gaussian random fields with stationary increments and to multi-parameter Brownian sheets. He asked if the level sets of fractional Brownian motion are also Salem sets and the problem has remained open since. This paper answers Kahane’s question in the affirmative. The argument is based on the study of oscillatory integrals with non-smooth amplitudes and new properties of the generalised Euler spiral which have independent interest.     

Gene flow across geographical barriers — scaling limits of random walks with obstacles

We study a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. We obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at zero reaches an exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at zero reaches another exponential level, etc. These random walks are used in population genetics to trace the position of ancestors in the past near geographical barriers.