Conditionally Gaussian stochastic integrals (Q5965100)

The authors prove the conditional Brownian identity NEWLINE\[NEWLINE E\Big[\text{exp} \Big( \sqrt{-1} \int^T_0 u_t \, dB_t \Big| \int^T_0 |u_t|^2 \, dt \Big] = \text{exp} \Big( -\frac12 \int^T_0 |u_t|^2 \, dt \Big),NEWLINE\]NEWLINE for adapted processes \((u_t )\) having their Malliavin derivative \(Du \in \underset{p>1} \bigcap L^p\) such that \(\langle u, (Du)^k u\rangle_ { L^2} = 0\) for any \(k \geq 1\), and also satisfying \(\text{exp} \Big(\frac12 \int^T_0 |u_t|^2 \, dt \Big) \in \, L^1\).NEWLINENEWLINEThis applies in particular to the quadratic Brownian integral \(\int^T_0 AB_t\, dB_t\) considered by \textit{M. Yor} [Lect. Notes Math. 721, 427--440 (1979; Zbl 0418.60057)] under his condition \(^tA \times A^2 = 0\) on the deterministic matrix \(A\). The authors also notice that the latter can be replaced by the condition that \(^tA \times A^2\) is skew-symmetric and \(^tA \times A\) is proportional to a projector, up to replacing the condition \(\int^T_0 |AB_t |^2 \,dt\) by \(|AB_t |\) for \(0\leq t\leq T\).