Stochastic differential formula and solution of control problem

Exact solution of finite-dimensional linear stochastic differential system with control is derived. Its uniqueness up to stochastic modification is proved. In order to achieve this, detailed grounding of existence for stochastic integral over a process with orthogonal increments is provided. Particularly, density of step-functions set in the set of all integrable functions is proved. Integration by parts formula for stochastic integral is derived. Analogue of Fubini theorem is proved in the case, when one measure is linear and another one is a random orthogonal measure. Stochastic differential formula for finite-dimensional integral functional is established. Formulation of Ito’s stochastic differential formula and its comparison with the result is presented. The solution formula for controlled linear stochastic differential system was conditionally used in applied sciences. Its rigorous proof provides the necessary background for further research.