Solution to Stein’s Equation for Self-Decomposable Laws

Abstract

Having found in Chap. 3 that the operator $\mathcal{A}_{{\text {gen}}}$ given for all $f\in BLip(\mathbb R)$, by

$$\begin{aligned}\mathcal{A}_{{\text {gen}}} f(x)=xf(x)-bf(x)-\int ^{+\infty }_{-\infty } (f(x+u)-f(x) {1\!\!1}_{|u|\le 1})u\nu (du), \end{aligned}$$

characterizes $X\sim ID(b, 0,\nu )$, the usual next step in Stein’s method is now to show that for any $h\in \mathcal{H}$ (a class of nice functions), the equation

$$\begin{aligned} \mathcal{A}_{{\text {gen}}} f(x)=h(x)-\mathbb Eh(X) \end{aligned}$$

has a solution $f_h$ which also belongs to a class of nice functions. Of course for $X\sim ID(b,\sigma ^2,\nu )$, the integral operator $\mathcal{A}_{gen}$ becomes an integro-differential operator given by

$$\begin{aligned}\mathcal{A}_{{\text {gen}}} f(x)=xf(x)-\sigma ^2f'(x)-bf(x)- \int ^{+\infty }_{-\infty }(f(x+u)-f(x) {1\!\!1}_{|u|\le 1})u\nu (du)). \end{aligned}$$