Stochastic-processes Defined From a Lagrangian

Let us consider a physical particle with movement described by a Lagrangian function. Then its classical deterministic trajectory x(t) (t(a) less-than-or-equal-to t less-than-or-equal-to t(b)) between to fixed time-instants t(a) and t(b) can be replaced by a stochastic path X(t) (t(a) less-than-or-equal-to t less-than-or-equal-to t(b)) such that X(t) = EX(t). Process X(t) defined in this why can be used to construct models for several actuarial situations. For instance, the rather deterministic analysis of underwriting cycles by Taylor (1991) can be probabilized. Movements, oscillary on the average, with damping effects or not, with exterior perturbative forces or not, all time-dependent or not, can be introduced. In this paper, we present the general theory with the two particular cases, one of which is the Brownian motion. Specific actuarial applications shall be treated in forthcoming publications