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Stavros Vakeroudis

Publications and source records attributed to Stavros Vakeroudis.

11 recordsLinked to original sources
arXiv2019

On Doney's striking factorization of the arc-sine law

R. Doney identifies a striking factorization of the arc-sine law in terms of the suprema of two independent stable processes of the same index by an elegant random walks approximation. In this paper, we provide an alternative proof and a generalization of this factorization based on the theory recently developed for the exponential functional of Lévy processes. As a by-product, we provide some interesting distributional properties for these variables and also some new examples of the factorization of the arc-sine law.

math.PR↗
arXiv2018

Windings of planar processes, Exponential Functionals and Asian options

Motivated by a common Mathematical Finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of an associated Brownian motion. We prove a conjecture by Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and we present a novel simple proof of De Blassie's result (1987-1988) about the asymptotic behaviour of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, for t large. Similar issues for the exponential functional of a Levy process are also discussed. We finally use the findings obtained by the windings approach in order to get results for quantities associated to the pricing of Asian options.

math.PR↗
arXiv2014

On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process

We deal with a complex-valued Ornstein-Uhlenbeck (OU) process with parameter $λ\in\mathbb{R}$starting from a point different from 0 and the way that it winds around the origin.The starting point of this paper is the skew product representation for an OU process which is associated to the skew product representationof its driving planar Brownian motion under a new deterministic time scale.We present the stochastic differential equations (SDEs)for the radial and for the winding process. Moreover, we obtain the large time (analogue of Spitzer's Theorem for Brownian motion in the complex plane) and the small time asymptotics for the winding and for the radialprocess, and we explore the exit time from a cone for a 2-dimensional OU process.Some Limit Theorems concerning the angle of the cone (when our process winds in a cone) and the parameter $λ$ are also presented.Furthermore, we discuss the decomposition of the winding process of a complex-valued OU process in "small" and "big" windings,where, for the "big" windings, we use some results already obtained by Bertoin and Werner in \cite{BeW94},and we show that only the "small" windings contribute in the large time limit.Finally, we study the windings of a complex-valued OU process driven by a Stable processand we obtain similar results for its (well-defined) winding and radial process.

math.PR↗
arXiv2012

A scaling proof for Walsh's Brownian motion extended arc-sine law

We present a new proof of the extended arc-sine law related to Walsh's Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams in the 1-dimensional Brownian case, which can be generalized to the multivariate case. A discussion concerning the time spent positive by a skew Bessel process is also presented.

math.PR↗
arXiv2012

Windings of planar stable processes

Using a generalization of the skew-product representation of planar Brownian motion and the analogue of Spitzer's celebrated asymptotic Theorem for stable processes due to Bertoin and Werner, for which we provide a new easy proof, we obtain some limit Theorems for the exit time from a cone of stable processes of index $α\in(0,2)$. We also study the case $t\rightarrow0$ and we prove some Laws of the Iterated Logarithm (LIL) for the (well-defined) winding process associated to our planar stable process.

math.PR↗
arXiv2012

Bougerol's identity in law and extensions

We present a list of equivalent expressions and extensions of Bougerol's celebrated identity in law, obtained by several authors. We recall well-known results and the latest progress of the research associated with this celebrated identity in many directions, we give some new results and possible extensions and we try to point out open questions.

math.PR↗
arXiv2011

On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity

Some identities in law in terms of planar complex valued Ornstein-Uhlenbeck processes $(Z_{t}=X_{t}+iY_{t},t\geq0)$ including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion:$(β_{t},t\geq0)$: for any fixed $u>0$: \sinh(β_{u}) \stackrel{(law)}{=} \hatβ_{(\int^{u}_{0}ds\exp(2β_{s}))}. These identities in law for 2-dimensional processes allow to study the distributions of hitting times $T^θ_{c}\equiv\inf\{t:θ_{t} =c \}, (c>0)$, $T^θ_{-d,c}\equiv\inf\{t:θ_{t}\notin(-d,c) \}, (c,d>0)$ and more specifically of $T^θ_{-c,c}\equiv\inf\{t:θ_{t}\notin(-c,c) \}, (c>0)$ of the continuous winding processes $θ_{t}=\mathrm{Im}(\int^{t}_{0}\frac{dZ_{s}}{Z_{s}}), t\geq0$ of complex Ornstein-Uhlenbeck processes.

math.PR↗
arXiv2011

The Mean First Rotation Time of a planar polymer

We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of n rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on the square root of n and, interestingly, depends weakly on the mean initial configuration. Our analytical results are confirmed by Brownian simulations.Our analytical results are confirmed by Brownian simulations.

math.PR↗