  In this paper, we study the sufficient conditions for the existence of
solutions of first-order Hamiltonian stochastic impulsive differential
equations under Dirichlet boundary value conditions. By using the variational
method, we first obtain the corresponding energy functional. And by using
Legendre transformation, we obtain the conjugation of the functional. Then the
existence of critical point is obtained by mountain pass lemma. Finally, we
assert that the critical point of the energy functional is the mild solution of
the first order Hamiltonian stochastic impulsive differential equation.Finally,
an example are presented to illustrate the feasibility and effectiveness of our
results.
