Existencia de soluciones de ecuaciones diferenciales estocásticas

Abstract

In classic theorems, when we have a stochastic differential equation of the form dXt = f(t,Xt)dt + G(t,Xt)dWt, Xt = ξ, to ≤ t ≤ T and lt; ∞, where Wt is a Wiener Process and ξ is a random variable independent of Wt-Wto for t ≥ to in order to have existence and uniqueness of solutions it is supposed the existence of a constant K such that: (Lipschitz condition) for all t ϵ [to,T], x,y ∈Rd, |f(t,x)-f(t,y)| + |G(t,x)-G(t,y)| ≤ K|x-y|· And for all t ∈|to,T | and x ∈ Rd, |f(t,x)|2+|G(t,x)|2≤ K2(1+|x|2). In this article we prove an existence theorem under weaker hypothesis: we require only that f and G be continuous in the second variable and the existence of a function m ϵ L2 [to,T] such that |f(t,x)|+|G(t,x)| ≤ m(t) for alI t ∈ [to,T] and x ∈ Rd.