On exponential local martingales associated with strong Markov continuous local martingales

• Published in: Stochastic processes and their applications Vol. 119; no. 9; pp. 2859 - 2880
• Main Authors: Blei, Stefan, Engelbert, Hans-Jürgen
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.2009; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: 0–1-laws; Brownian motion; Brownian motion with drift; Continuous exponential local martingales; Continuous local martingales; Continuous local martingales Continuous strong Markov processes Stochastic differential equations Brownian motion Brownian motion with drift Integral functionals 0-1-laws Continuous exponential local martingales Stochastic exponentials Martingale property of stochastic exponentials; Continuous strong Markov processes; Exact sciences and technology; Inference from stochastic processes; time series analysis; Integral functionals; Markov processes; Martingale property of stochastic exponentials; Mathematics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Stochastic analysis; Stochastic differential equations; Stochastic exponentials; Stochastic processes; 60J65; Martingale property of stochastic exponentials; 60G44; 60H10; Stochastic differential equations; Continuous local martingales; Continuous exponential local martingales; 60G30; Brownian motion; 0–1-laws; Stochastic exponentials; 60J25; Brownian motion with drift; Integral functionals; Continuous strong Markov processes; 60G48; Markov process; Divergence; Necessary and sufficient condition; Wiener process; Stochastic process; Local time; Convergence; 0-1-laws; Integral; Speed; Functional; Application
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2009.03.003

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.