  Stochastic differential equations and the associated partial differential
equations are the cornerstone formalism in stochastic control problems. The
universality of bilinear stochastic systems can be found in autonomous systems,
non-linear dynamic circuits, and mathematical finance. Consensus on the Ito
versus Stratonovich suggests stochastic systems embedded with Stratonovich
differential to describe the stochastic evolution of the state vector of real
physical systems. The mathematical theory of a scalar time-varying bilinear
Stratonovich stochastic differential equation is available in current texts.
The theory of scalar Stratonovich systems was developed by deriving their
closed-form solutions and then conditional moments. Practical problems obeying
vector time-varying bilinear Stratonovich stochastic differential equations are
ubiquitous. However, their formal and systematic estimation theory is not
available. In this paper, we develop the results for non-homogeneous Markov
processes obeying the vector Stratonovich bilinear stochastic differential
equation. Then, the theory of the paper is applied to a stochastic three-phase
rectifier circuit. The stochastic evolution of the rectifier state vector is
constructed by utilizing the Euler-Lagrange theory and embedding Stratonovich
differential. Since the theory of the paper accounts for the
multidimensionality as well as respects Stratonovich stochasticity of the
ubiquitous bilinear system, the estimation Theorems will be remarkably useful
to estimation and control of nonlinear real physical systems. Finally, this
paper will be of interest to control and computational practitioners aspiring
for the formal theory of bilinear stochastic systems arising from their
application as well as applied mathematicians looking for applications of
formal bilinear stochastic estimation theory.
