Stochastic Integral Equations Associated with Stratonovich Curveline Integral

An explicit representation formula for a solution is given in Theorem 1, when g is a bounded smooth vector field. The case of a complete vector field g ∈ C1(Rn,Rn) is analyzed in Theorem 2, introducing adequate stopping times. The main support in writing a solution comes from the solution yλ(τ1, τ2) = G (F (τ1, τ2)) [λ], (τ1, τ2) ∈ R, λ ∈ R, satisfying a deterministic gradient system  ∂τ1yλ (τ1, τ2) = g (yλ(τ1, τ2)) ∂τ1F (τ1, τ2), ∂τ2yλ (τ1, τ2) = g (yλ(τ1, τ2)) ∂τ2F (τ1, τ2), (τ1, τ2) ∈ R, yλ(0, 0) = λ ∈ R. Here, {G(τ)[λ] : τ ∈ R, λ ∈ Rn} is the global flow generated by the complete vector field g ∈ C1(Rn,Rn) and F ∈ C2(R2) fulfils a polynomial growth condition.