  The process of fluctuations of trajectory observables of stochastic systems
is associated with processes with independent increments from the risk theory.
Boundary problems are considered: first-passage times (times when a certain
value of the variable is first reached), overjumps at the moments of the
first-passage times to reach positive and negative levels. A correspondence
between the expressions of the theory of random processes and thermodynamics of
trajectories is established, as well as deviations from such a correspondence
for the process of fluctuations of trajectory observables. Relations between
the general regularities of the first-passage times in the theory of random
processes and thermodynamics of trajectories are discussed. A more complete use
of the theory of random processes in physical problems is proposed, the
possibilities of combining the approaches of the theory of random processes and
statistical physics are indicated.
