Invariance formulas for stopping times of squared Bessel process

For any integer n = 1, 2, ⋯, limiting laws, as t→∞, for a Bessel process with dimension d (0 < d < 2) penalised by the nth-ranked length of its excursions up to t, or up to the last zero before t, or again up to the first zero after t, are shown to exist, and are characterized.Under these limiting laws Q (n), the canonical process admits a last zero g, and the sequence of the normalized ranked lengths of its excursions up to g, is described in terms of the Poisson-Dirichlet distribution studies, e.g., by Pitman-Yor [PY5]. As n → ∞, Q (n) is shown to converge to Q (∞) the distribution of the Bessel process penalised by an adequate function of its local time at 0, and the sequence of the normalized ranked lengths of the excursions up to g under Q (∞) is then precisely the Poisson-Dirichlet distribution PD1 − d/2 , 0 studied in Pitman-Yor [PY5].KeywordsRanked lengths of excursionsPoisson-Dirichlet distributionsBessel meandersPenalisations

We investigate the effects of noise reinforcement on a Bessel process of dimension d∈(0,2), and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.

A new global isomorphism theorem is obtained that expresses the local\ntimes of transient regular diffusions under $P^{x,y}$, in terms of related\nGaussian processes. This theorem immediately gives an explicit description\nof the local times of diffusions in terms of $0$th order squared Bessel\nprocesses similar to that of Eisenbaum and Ray's classical description\nin terms of certain randomized fourth order squared Bessel processes. The\nproofs given are very simple. They depend on a new version of Kac's lemma\nfor $h$-transformed Markov processes and employ little more than standard\nlinear algebra. The global isomorphism theorem leads to an elementary\nproof of the\nMarkov property of the local times of diffusions and to other recent\nresults about the local times of general strongly symmetric Markov\nprocesses. The new version of Kac's lemma gives simple, short proofs of\nDynkin's isomorphism theorem and an unconditioned isomorphism theorem due\nto Eisenbaum.

Lévy discovered that the fraction of time a standard one-dimensional Brownian motion B spends positive before time t has arcsine distribution, both for t a fixed time when Bt ≠ 0 almost surely, and for t an inverse local time, when Bt = 0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension d for 0 < d < 2.

We now take up the fine structure of the local time t(t) = t(t 0) and its inverse function t-1(t) for a persistent non-singular diffusion D* on an interval Q containing 0 as an inside point or as a left end point, with — u+(0) + m(0) (Gu)(0) = 0 in the second case. A number of the statements made below hold for transient diffusions also (see esp. 6.3, 6.5, 6.6); the necessary modifications of the proofs are left to the reader.

Polylinear Additive Functionals of Superprocesses

Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the process $s (X+Y)$ is a local martingale. We study the representation and the dynamics of $s(X+Y)$, projected on the filtration generated by $X$. This projection is a strict supermartingale if, and only if, $m<2$. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of $X$ at zero.

Squares of Bessel processes and Ray-Knight theorems for Brownian local times

We solve explicitly a two-dimensional singular control problem of finite fuel type for infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative and transient price impact. Liquidity is stochastic in that the volume effect process, which determines the inter-temporal resilience of the market in spirit of Predoiu, Shaikhet and Shreve (2011), is taken to be stochastic, being driven by own random noise. The optimal control is obtained as the local time of a diffusion process reflected at a non-constant free boundary. To solve the HJB variational inequality and prove optimality, we need a combination of probabilistic arguments and calculus of variations methods, involving Laplace transforms of inverse local times for diffusions reflected at elastic boundaries.

For a one dimensional diffusion process $\\mathbf{D}^{*}_{s,m}$ and the harmonic transformed process $\\mathbf{D}^{*}_{s_{h},m_{h}}$, the asymptotic behavior of the Lévy measure density corresponding to the inverse local time at the regular end point is investigated. The asymptotic behavior of $n^{*}$, the Lévy measure density corresponding to $\\mathbf{D}^{*}_{s,m}$, follows from asymptotic behavior of the speed measure $m$. However, that of $n^{h*}$, the Lévy measure density corresponding to $\\mathbf{D}^{*}_{s_{h},m_{h}}$, is given by a simple form, $n^{*}$ multiplied by an exponential decay function, for any harmonic function $h$ based on the original diffusion operator.