BESOV REGULARITY OF TRAJECTORIES OF THE I NTEGRAL SKOROKHOD PROCESS

Let {W-t : 0 less than or equal to t less than or equal to 1} be a lin ear Brownian motion, starting from 0, defined on the canonical probabi lity space (Omega, F, P). Consider a process {u(t) : 0 less than or eq ual to t less than or equal to 1} belonging to the space L(2,1) (see D efinition II.2). The Skorokhod integral U-t = integral(0)(t) u delta W is then well defined, for every t is an element of [0, 1]. In this pa per, we study the Besov regularity of the Skorokhod integral process t --> U-t. More precisely, we prove the following THEOREM III.1. (1) If 0 < alpha < 1/2 and u is an element of L(p,1) with 1/alpha < p < infi nity, then a.s.t --> U-t is an element of B-p,q(alpha) for all q is an element of [1, infinity], and t --> U-t is an element of B-p,infinity (alpha 0). (2) For every even integer p greater than or equal to 4, if there exists delta > 2(p + 1) such that u is an element of L(delta,2) boolean AND L(infinity)([0, 1] X Omega), then a.s.t --> U-t is an ele ment of B-p,infinity(1/2). (For the definition of the Besov spaces B-p ,q(alpha) and B-p,infinity(alpha,0), see Section I; for the definition of the spaces L(p,1) and L(p,2), p greater than or equal to 2, see De finition II.2.) An analogous result for the classical Ito integral pro cess has been obtained by B. Roynette in [R]. Let us finally observe t hat D. Nualart and E. Pardoux [NP] showed that the Skorokhod integral process t --> U-t admits an a.s. continuous modification, under smooth ness conditions on the integrand similar to those stated in Theorem II .1 (cf. Theorems 5.2 and 5.3 of [NP]).