On exponential local martingales associated with strong Markov continuous local martingales

We investigate integral functionals Tt=∫RLY(t,a)m(da), t≥0t≥0, where mm is a nonnegative measure on (R,ℬ(R))(R,ℬ(R)) and LYLY is the local time of a Wiener process with drift, i.e., Yt=Wt+tYt=Wt+t, t≥0t≥0, with a standard Wiener process WW. We give conditions for a.s. convergence and divergence of TtTt, t≥0t≥0, and T∞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.