  Our approach is part of the close link between continuous dissipative
dynamical systems and optimization algorithms. We aim to solve convex
minimization problems by means of stochastic inertial differential equations
which are driven by the gradient of the objective function. This will provide a
general mathematical framework for analyzing fast optimization algorithms with
stochastic gradient input. Our study is a natural extension of our previous
work devoted to the first-order in time stochastic steepest descent. Our goal
is to develop these results further by considering second-order stochastic
differential equations in time, incorporating a viscous time-dependent damping
and a Hessian-driven damping. To develop this program, we rely on stochastic
Lyapunov analysis. Assuming a square-integrability condition on the diffusion
term times a function dependant on the viscous damping, and that the
Hessian-driven damping is a positive constant, our first main result shows that
almost surely, there is convergence of the values, and states fast convergence
of the values in expectation. Besides, in the case where the Hessian-driven
damping is zero, we conclude with the fast convergence of the values in
expectation and in almost sure sense, we also managed to prove almost sure weak
convergence of the trajectory. We provide a comprehensive complexity analysis
by establishing several new pointwise and ergodic convergence rates in
expectation for the convex and strongly convex case.
