  Large deviation theory and instanton calculus for stochastic systems are
widely used to gain insight into the evolution and probability of rare events.
At its core lies the realization that rare events are, under the right
circumstances, dominated by their least unlikely realization. Their computation
through a saddle-point approximation of the path integral for the corresponding
stochastic field theory then reduces an inefficient stochastic sampling problem
into a deterministic optimization problem: finding the path of smallest action,
the instanton. In the presence of heavy tails, though, standard algorithms to
compute the instanton critically fail to converge. The reason for this failure
is the divergence of the scaled cumulant generating function (CGF) due to a
non-convex large deviation rate function. We propose a solution to this problem
by "convexifying" the rate function through nonlinear reparametrization of the
observable, which allows us to compute instantons even in the presence of
super-exponential or algebraic tail decay. The approach is generalizable to
other situations where the existence of the CGF is required, such as
exponential tilting in importance sampling for Monte-Carlo algorithms. We
demonstrate the proposed formalism by applying it to rare events in several
stochastic systems with heavy tails, including extreme power spikes in fiber
optics induced by soliton formation.
