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We leverage commutative hypercomplex analysis to find closed-form solutionsof some systems of stochastic differential equations. Specifically, we obtainnecessary and sufficient conditions under which a system of stochasticdifferential equations can be transformed into a scalar one involving processesvalued in a commutative hypercomplex. In the event the targeted scalarstochastic differential equation is solved by quadratures, we recover thesolution of the original system by projecting the solution of the scalarstochastic differential equation along the units of the underlying commutativehypercomplex. The conversion of a system of stochastic differential equationsinvolving real-valued processes into a scalar one written in terms ofhypercomplex-valued processes is termed hypercomplexification. Bothhypercomplexification and its reverse are mediated by the analyticity ofstochastic differential equations data. They may be iterated in order togenerate higher-dimensional integrable systems of stochastic differentialequations and solve them. We showcase the utility of hypercomplexification bytreating several examples including linear, and linearizable systems ofstochastic differential equations and stochastic Lotka-Volterra systems.Although we consider only random systems driven by white noises,hypercomplexification is fundamentally algebraic, and it readily extends tostochastic systems involving other types of disturbances.

Hypercomplexification of SDEs Exact integration of some systems of stochastic differential equations through hypercomplexification