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Exponential mixtures of exponentials are Pareto

Did you know that if [latex]X\sim\mathcal{E}(\lambda)[/latex] and [latex]\mathcal{L}(Y\,\vert\,X=x)=\mathcal{E}(x)[/latex] for all [latex]x\geq0[/latex] then [latex]Y[/latex] follows a Pareto distribution with  probability density function [latex]x\mapsto 1/(\lambda+x)^2[/latex]? Funny!

Consider now the kinetic diffusion process [latex](X_t,Y_t)_{t\geq0}[/latex] on [latex]\mathbb{R}^2[/latex] where

[latex]\displaystyle\begin{cases}dX_t&=dB_t-s(X_t)\lambda dt\\dY_t&=dW_t-s(Y_t)|X_t|dt\end{cases}[/latex]

where [latex](B_t)_{t\geq0}[/latex] and [latex](W_t)_{t\geq0}[/latex] are independent standard Brownian motions and [latex]s[/latex] is the sign function… Can you guess the invariant measure and control the speed of convergence?

2 Comments

  1. Djalil 2010-05-08

    Hello Arthur,

    merci pour ton message et ton lien. Pour Captcha, il y a une petite icône sur le côté de l’image qui permet de générer un code. Il faut appuyer dessus jusqu’à l’obtention de quelque chose de lisible.

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