Some time change representations of stable integrals, via predictable transformations of local martingales (Q1190166)

It is well-known that a continuous local martingale \(M\) becomes a Brownian motion \(B\) when viewed on the time scale given by its quadratic variation \([M]\) and that, likewise, a simple, quasi-left-continuous point process \(N\) is a Poisson process \(\pi\) when viewed on the time scale determined by its compensator \(\hat N\): with appropriate constructions \(M=B\circ[M]\) a.s. and \(N=\pi\circ\hat N\) a.s.. Another fact in the same spirit is that if a predictable process \(V\) is locally integrable with respect to a strictly \(p\)-stable Lévy process \(X\) and if either \(V\) is non-negative or \(X\) is symmetric, then there exists a process \(X'\) with the same distribution as \(X\), such that \(\int^ t_ 0 V dX=X'\circ\int^ t_ 0| V|^ p\) a.s., \(t\geq 0\). The present paper establishes various extensions, the flavour of which is given by the representation \(\int^ t_ 0 V dX=X'\circ\int^ t_ 0 (V^ +)^ p-X''\circ\int^ t_ 0(V^ -)^ p\), valid without the stated restriction on \(V\) or \(X\); here \(X'\) and \(X''\) are independent processes having the same distribution as \(X\). The independence of \(X'\) and \(X''\) is a manifestation of the general principle (``Knight phenomenon'') that in a multivariate context orthogonal processes become independent after the time change. The author discusses a variety of representations, his point of departure being a predictable reduction of marked point processes to Poisson processes and a similar reduction of purely discontinuous martingales to processes with independent increments.