Existencia de soluciones de ecuaciones diferenciales estocásticas

In classic theorems, when we have a stochastic differential equation of the form dX_t =

f(t,X_t)dt + G(t,X_t)dW_t, X_t = ξ,  t_o ≤ t ≤ T <  ∞, where W_t is a Wiener Process and ξ is a random variable independent of W_t-Wt_o for t ≥ t_o 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 [t_o,T], x,y ∈R^d,

|f(t,x)-f(t,y)| + |G(t,x)-G(t,y)| ≤  K|x-y|·

And for all t ∈|t_o,T | and x ∈ R^d,

|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 [t_o,T] such that

|f(t,x)|+|G(t,x)| ≤ m(t)

for alI t ∈ [t_o,T] and x ∈ R^d.