Quasi sure analysis of local times of anticipating smooth semimartingales

Abstract

Listen

Translate article

Let X t = ∫ 0 t f s d W s + ∫ 0 t g s d s be the anticipating smooth semimartingale and L t x be its generalized local time. In this paper, we give some estimates about the quasi sure property of X t and its quadratic variation process 〈 X 〉 t . We also study the fractional smoothness of L t x and prove that the quadratic variation process of L t x can be constructed as the quasi sure limit of the form ∑ Δ n ( L t a i + 1 n − L t a i n ) 2 , where Δ n = ( a i n , a i + 1 n ) is a sequence of subdivisions of [ a , b ] , a i n = i ( b − a ) / 2 n + a , i = 0 , 1 , … , 2 n .