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Register a new userIntegrated Wishart bridge processes and generalised Hartman-Watson law
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Jason Leung
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This article is concerned with the joint law of an integrated Wishart bridge process and the trace of an integrated inverse Wishart bridge process over the interval $ left[0,tright] $. Its Laplace transform is obtained by studying the Wishart bridge processes and the absolute continuity property of Wishart laws.
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The Hartman-Watson distribution with density $f_r(t)$ is a probability distribution defined on $t geq 0$ which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral $theta(r,t)$ which is difficult to evaluate numerically for small $tto 0$. Using saddle point methods, we obtain the first two terms of the $tto 0$ expansion of $theta(rho/t,t)$ at fixed $rho >0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $rho$. As an application we obtain the leading asymptotics of the density of the time average of the geometric Brownian motion as $tto 0$. This has the form $mathbb{P}(frac{1}{t} int_0^t e^{2(B_s+mu s)} ds in da) = (2pi t)^{-1/2} g(a,mu) e^{-frac{1}{t} J(a)} (1 + O(t))$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory.
Let $X^{(delta)}$ be a Wishart process of dimension $delta$, with values in the set of positive matrices of size $m$. We are interested in the large deviations for a family of matrix-valued processes ${delta^{-1} X_t^{(delta)}, t leq 1 }$ as $delta$ tends to infinity. The process $X^{(delta)}$ is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.
We study survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an a.s. constant. We also shed some light on the way the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parametrized by the retention probability $p$. We provide growth rates, uniformly in $p$, of the percolation clusters, and also show uniform convergence of the survival probability from the $n$-th level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalisations of results by Lyons (1992).
This paper deals with branching processes in varying environment, namely, whose offspring distributions depend on the generations. We provide sufficient conditions for survival or extinction which rely only on the first and second moments of the offspring distributions. These results are then applied to branching processes in varying environment with selection where every particle has a real-valued label and labels can only increase along genealogical lineages; we obtain analogous conditions for survival or extinction. These last results can be interpreted in terms of accessibility percolation on Galton-Watson trees, which represents a relevant tool for modeling the evolution of biological populations.
We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholms determinant and resolvent. Furthermore , we link the characteristic exponents to a system of non-standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear-quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.
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