Strong Solutions to a Beta-Wishart Particle System

Abstract

The purpose of this paper is to study the existence and uniqueness of solutions to a stochastic differential equation (SDE) coming from the eigenvalues of Wishart processes. The coordinates are non-negative, evolve as Cox–Ingersoll–Ross (CIR) processes and repulse each other according to a Coulombian like interaction force. We show the existence of strong and pathwise unique solutions to the system until the first multiple collision and give a necessary and sufficient condition on the parameters of the SDEs for this multiple collision not to occur in finite time.

Appendix

The next lemma deals with the existence and uniqueness to the CIR SDE and with the probability for the solution to hit zero. It is proved, for instance, in [21, Theorem 6.2.2 and Proposition 6.2.3]. The point 4. comes directly from [5].

Lemma 6.1

Let $a\ge 0,b,\sigma \in {\mathbb {R}}$. Suppose that W is a standard Brownian motion defined on ${\mathbb {R}}_+$. For any real number $x\ge 0$, there is a unique continuous, adapted process X, taking values in ${\mathbb {R}}_+$, satisfying $X_0=x$ and

$$\begin{aligned} dX_t=(a-bX_t)dt+\sigma \sqrt{X_t}dW_t \text { on } [0,\infty ). \end{aligned}$$

Moreover, if we denote by $X^x$ the solution to this SDE starting at x and by $\tau _0^x=\inf \{t\ge 0 : X_t^x=0\}$,

• 1. If $a\ge \sigma ^2/2$, we have ${\mathbb {P}}(\tau ^x_0=\infty )=1$, for all $x>0$.

• 2. If $0\le a <\sigma ^2/2$ and $b\ge 0$, we have ${\mathbb {P}}(\tau ^x_0<\infty )=1$, for all $x>0$.

• 3. If $0\le a <\sigma ^2/2$ and $b<0$, we have $0<{\mathbb {P}}(\tau ^x_0<\infty )<1$, for all $x>0$.

• 4. For all $s>t$,

  $$\begin{aligned}{\mathbb {E}}[r_s|r_t] = r_te^{-b(s-t)}+\frac{a}{b}(1-e^{-b(s-t)}). \end{aligned}$$

The following result is the Ikeda–Watanabe Theorem, which allows to compare two Itô processes if their starting points and their drift coefficients are comparable, and if their diffusion coefficients are regular enough. It is proved, for instance, in [25, Theorem V.43.1 p.269].

Theorem 6.2

(Ikeda–Watanabe) Suppose that, for $i=1,2$,

$$\begin{aligned} X^i_t = X^i_0+\int _0^t\sigma (X^i_s)dB_s+\int _0^t\beta _s^ids, \end{aligned}$$

(39)

and that there exist $b:{\mathbb {R}}\mapsto {\mathbb {R}}$, such that

$$\begin{aligned} \beta ^1_s\ge b(X^1_s)\text {, }b(X^2_s)\ge \beta ^2_s. \end{aligned}$$

Suppose also that

• 1. $\sigma$ is measurable and there exists an increasing function $\rho :{\mathbb {R}}_+\mapsto {\mathbb {R}}_+$ such that

  $$\begin{aligned} \int _{0^+}\rho (u)^{-1}du=\infty , \end{aligned}$$

  and for all $x,y\in {\mathbb {R}},$

  $$\begin{aligned} (\sigma (x)-\sigma (y))^2\le \rho (|x-y|); \end{aligned}$$

• 2. $X^1_0\ge X^2_0$ a.s.;

• 3. b is Lipschitz.

Then, $X^1_t\ge X^2_t$ for all t a.s.