On exponential local martingales associated with strong Markov continuous local martingales

Abstract

We investigate integral functionals T t = ∫ R L Y ( t , a ) m ( d a ) , t ≥ 0 , where m is a nonnegative measure on ( R , ℬ ( R ) ) and L Y is the local time of a Wiener process with drift, i.e., Y t = W t + t , t ≥ 0 , with a standard Wiener process W . We give conditions for a.s. convergence and divergence of T t , t ≥ 0 , and T ∞ . In the second part of the present note we apply these results to exponential local martingales associated with strong Markov continuous local martingales. In terms of the speed measure of a strong Markov continuous local martingale, we state a necessary and sufficient condition for the exponential local martingale associated with a strong Markov continuous local martingale to be a martingale.