Continuity of the occupation density for anticipating stochastic integral processes (Q1261225)

Let \(X=\{X_ t\), \(0\leq t\leq 1\}\) be a stochastic process defined over the standard Wiener space and of the form \[ X_ t=\int^ t_ 0 u_ s dW_ s+\int^ t_ 0v_ s ds, \tag{1} \] where \(U=\{u_ s\), \(0\leq s\leq 1\}\) and \(V=\{V_ s\), \(0\leq s\leq 1\}\) are processes anticipating the driving standard Wiener process \(W=\{W_ s\), \(0\leq s\leq 1\}\), and the stochastic integral is understood in Skorokhod's sense. Defining the occupation measure \(\mu\) of \(X\) by \(\mu(B)=\int^ 1_ 0 I_ B(X_ s)ds\), \(B\) Borel set from \(R^ 1\), the authors introduce the occupation density of \(\mu\) w.r.t. the Lebesgue measure. Generalizing earlier work of the authors, where \(U\) was assumed to be a process of second Wiener chaos (Imkeller, 1991), and later work, where either \(U\) has to be equal to 1 or \(V\) has to vanish, this paper provides conditions (smoothness of \(U\) and \(V\) in the sense of the Malliavin derivatives including a certain integrability assumption, and \(E\bigl[ \int^ 1_ 0 | U_ t|-P dt\bigr]<+\infty\), for all \(p\geq 2\)) for which the rather general process \(X\) possesses an occupation density which is Hölder continuous of order \(1/2-\varepsilon\), for all \(\varepsilon>0\). The main tool in the author's approach consists in an integration by parts formula with iterated Skorokhod integrals.