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Working paper

THE MOST PROBABLE PATHS FOR DIFFUSIONS WITH JUMPS

Book of abstracts for 12th International Vilnius Conference on Probability Theory and Mathematical Statistics. e-book. Vilnius University, 2018
We generalize well-known results on the Onsager–Machlup functional for diffusions. The case of L´evy processes with finite number of jumps and diffusions with jumps were considered in our work. The Onsager–Machlup functional of a c´adl´ag process X is defined by the following expression OM( f ,ψ) = lim_{ε→0}P{ω :||X(ω)− f || <ε }/P{ω : ||(ω)−ψ|| <ε }, where || ·|| is a norm in the Skorokhod space D[0,∞]. This expression gives a tool to compare weights of trajectories of the corresponding process, also it is naturally connected with the most probable sample path of the process. This interpretation leads to the number of applications, for example, to the popular Customer Journey Maps problem.