A stable Cox-Ingersoll-Ross model with restart (Q739517)

A stable Cox-Ingersoll-Ross (CIR) model \(\{X_t: t\geq0\}\) defined by \(dX_t = (a-bX_t)dt +\sigma\root\alpha\of{X_{t-}}dZ_t\), where \(\{Z_t: t\geq0\}\) is a spectrally positive stable (\(\mathcal{F}_t\))-Lévy process with index \(1<\alpha\leq2\) is considered. Let the operator \(L\) be defined by \(f\in C_0^2(R+)\) \[ Lf(x)=(a-bx)f'(x) + \frac{\sigma^\alpha x}{\alpha\Gamma( -\alpha)} \int_0^\infty[f(x+y)-f(x)-yf'(x)]\frac {dy} {y^{\alpha+1}}. \] Consider a Markov process \(X(t)\) in \(D = [0,\infty)\) which performs a stable CIR process in \(D\) generated by the operator \(L\) and stopped at 0 and \(\infty\), and which while running, jumps instantaneously at an exponentially distributed random time with intensity \(\lambda>0\) to a new point, according to a distribution \(\nu\). Proposition 2.1 states: Let \(p(t, x, y)\) be the transition density of the stable CIR process \(X(t)\), then the Markov process \(\{\hat X(t): t\in[0,\infty)\}\) that restart from a distribution \(\nu\) after independent exponentially distributed random times with mean \(1/\lambda\) has the following transition density \[ \hat p_\nu(t,x,y) = e^{-\lambda t}p(t,x,y) + \int_0^t\lambda e^{-\lambda s}p(s,\nu,y)\,ds. \] In Section 2, the infinitesimal generator of the process \(\{\hat X(t): t\in[0,\infty)\}\) (Theorem 2.2) and the uniqueness of the martingale problem (Proposition 2.4) are given. In Section 3, the exponential decay of a stable CIR model with restart as \(t\to\infty\) is considered. The compactness of the corresponding semigroup is proved in Proposition 3.1 and thus the spectrum of the generator consists of exclusively of eigenvalues. Further, in Theorem 3.2 it is shown that the principal eigenvalue gives the exponential rate of decay of not exiting the domain by time \(t\). Finally, in Proposition 3.3 some properties of the principal eigenvalue are studied.