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A law of iterated logarithm (LIL) in small time and an asymptotic estimate of modulus of continuity are proved for Brownian motion on the loop group ℒ( G ) over a compact connected Lie group G . Upper bounds are obtained via infinite-dimensional deviation inequalities for functionals on the path space ℙ(ℒ( G )) on ℒ( G ), such as the supremum of Brownian motion on ℒ( G ), which are proved from the Clark–Ocone formula on ℙ(ℒ( G )). The lower bounds rely on analog finite-dimensional results that are proved separately on Riemannian path space. [ABSTRACT FROM AUTHOR]

We derive convex comparison inequalities for stochastic integrals of the formand, whereare adapted processes with respect to the filtration generated by a standard Brownian motion, andis an independent Brownian motion. Our method uses forward–backward stochastic integration and the Malliavin calculus, and is also applied to jump–diffusion processes. [ABSTRACT FROM AUTHOR]