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The process of fluctuations of trajectory observables of stochastic systemsis associated with processes with independent increments from the risk theory.Boundary problems are considered: first-passage times (times when a certainvalue of the variable is first reached), overjumps at the moments of thefirst-passage times to reach positive and negative levels. A correspondencebetween the expressions of the theory of random processes and thermodynamics oftrajectories is established, as well as deviations from such a correspondencefor the process of fluctuations of trajectory observables. Relations betweenthe general regularities of the first-passage times in the theory of randomprocesses and thermodynamics of trajectories are discussed. A more complete useof the theory of random processes in physical problems is proposed, thepossibilities of combining the approaches of the theory of random processes andstatistical physics are indicated.

First passage time of a given level and value of overjump for fluctuations of trajectory observables