Some Properties of the Wishart Processes and a Matrix Extension of the Hartman-Watson Laws †

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Subfields: Finance

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Multidimensional Yamada-Watanabe theorem and its applications to particle systems (2011, methodology): “In the Wishart case we contribute in this way to realization of a programme started by Donati-Martin, Doumerc, Matsumoto and Yor [9], claiming that Wishart processes have similar properties as classical 1-dimensional squared Bessel processes.”
Affine Processes on Positive Semidefinite Matrices (2009, unknown): “By a well-known result due to Paul Lévy, these Wishart distributions are not infinitely divisible if d ≥ 2 (see [15], Section 2.”
On strong solutions for positive definite jump diffusions (2009, unknown): “…Brownian motion B, d × d matrices Q and β, a symmetric constant drift b, and a positive linear drift Γ : S+d → S+d , weak global solutions exist to the stochastic differential equation (SDE) dXt = √ XtdBtQ+Q ⊤dB⊤t √ Xt + (Xtβ + β ⊤Xt + Γ(Xt) + b)dt,(1.1) X0 = x ∈ S+d , whenever b− (d− 1)Q⊤Q ∈ S+d .”
Some new examples of Markov processes which enjoy the time-inversion property (2005, result): “Bru [2], and further results have been obtained in [3], [4], [ 7 ]; in particular, the transition probability density qδ (t, x, dy) with respect to the Lebesgue measure dy = � i≤j (dyij ) is given by:”