We consider inverse problems for differential systems in the presence of random perturbations. By the method of additional variables, we establish sufficient conditions for the representation of stochastic differential equations of the second order in the form of stochastic Lagrange equations, as well as for the representation of stochastic differential equations of the first order in the form of stochastic canonical equations. The obtained results are illustrated by specific examples.
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Tleubergenov, M.I., Azhymbaev, D.T. On the Solvability of Stochastic Helmholtz Problem. J Math Sci 253, 297–305 (2021). https://doi.org/10.1007/s10958-021-05229-1