Absolute continuity of joint laws of multiple stable stochastic integrals (Q1776116)

Let \(0<\alpha<2\), \(d\in {\mathbb N}\), \[ L^\alpha(\log_+)^{d-1}([0,1]^d):=\biggl\{f: [0,1]^d\to {\mathbb R}: \int_{[0,1]^d}| f|^\alpha(1+\log_+| f|)^{d-1}\,d\lambda^d \biggr\}, \] where \(\log_+ x:=\log(x\lor 1)\), and \(M\) be \(\alpha\)-stable random measure on \(([0,1],{\mathcal B}(0,1),\lambda)\) with control measure the Lebesgue measure \(\lambda\) and skewness intensity \(\beta\colon\;[0,1]\to [-1,1]\). The \(d\)-multiple \(\alpha\)-stable integral \(\int_{[0,1]^d}f\,dM^d\) is denoted by \(I_d(f)\). The author obtains conditions for the joint law of \((I_{d_1}(f_1),\ldots,I_{d_p}(f_p))\) with \(f_1\in L^\alpha(\log_+)^{d_1-1}([0,1]^{d_1}), \ldots, f_p\in L^\alpha(\log_+)^{d_p-1}([0,1]^{d_p}) \) to be absolutely continuous with respect to measure \(\lambda^p\). This result is a generalization of the author's previous result [J. Theor. Probab. 15, No. 4, 877--901 (2002; Zbl 1011.60031)].