An integral representation of the local time of the Brownian motion via the Clark–Ocone formula

Let ( L B ( t , x ) , t ≥ 0 , x ∈ R ) be the local time of ( B t , t ≥ 0 ) , the real-valued one-dimensional Brownian motion. In this paper, in case of g , a strictly increasing and bijective function, we propose some integral representations of L g ( B ) ( t , x ) , of the form: R ( t , x ) + ∫ 0 t K ( t , x , B s ) d B s , where R ( t , x ) is a deterministic function and K ( t , x , B s ) is a random function depending on t and F , the cumulative distribution function of the standard normal distribution N ( 0 , 1 ) and some Brownian functionals with no Malliavin derivative. Our study is based on the case L B ( t , x ) . An exact formula of the expectation E [ L B ( t , x ) ] is given in this paper.