Cost of excursions until first crossing of the origin for random walk and Lévy flights: An exact general formula

We consider a discrete-time random walk on a line starting at x0≥0 where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary jump distribution and cost function (heavy and light tailed alike), provided they are symmetric and continuous. We analyze the formula in different scaling regimes and find a high degree of universality with respect to the details of the jump distribution and the cost function. Applications are given to the motion of an active run-and-tumble particle in one dimension and extensions to multiple cost variables are considered. The analytical results are in perfect agreement with numerical simulations.