Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by ${\psi }_p\left(x\right)={exp\left(\right|x|}^p)-1$ with 0 < p ≤ 2) of $ma{x}_{0\le t\le \tau }\left|B_t\right|$ or $|B_{\tau }|$ to be finite, where $B={\left(B_t\right)}_{t\ge 0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}<\infty$ as soon as $E\left({\tau }^k\right)=O\left(C^k{k}^k\right)$ for some constant C > 0 as k → ∞ (or equivalently $\parallel \tau {\parallel }_{{\psi }_1}<\infty$). In particular, if τ ∼ Exp(λ) or $\left|N\right(0,{\sigma }^2\left)\right|$ then the last condition is satisfied, and we obtain $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}\le K\surd E\left(\tau \right)$ with some universal constant K > 0....

We study the pathwise regularity of the map φ↦I(φ)=∫0T〈φ(Xt), dXt〉, where φ is a vector function on ℝd belonging to some Banach space V, X is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A continuous version of this map, seen as a random element of the topological dual of V will be called stochastic current. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture...

We study optimal stopping problems for some functionals of Brownian motion in the case when the decision whether or not to stop before (or at) time t is allowed to be based on the δ-advanced information ${ℱ}_{t+\delta }$, where ${ℱ}_s$ is the σ-algebra generated by Brownian motion up to time s, s ≥ -δ, δ > 0 being a fixed constant. Our approach involves the forward integral and the Malliavin calculus for Brownian motion.