Bulletin of the American Mathematical Society · 1977 · 66 citations · 0 references
EngineeringMathematicsDifferential EquationsProbability TheoryStochastic ProcessesContinuous-time Stochastic ProcessesStochastic CalculusStochastic Differential EquationsItô CalculusStochastic Ordinary Differential Equation ModelsStochastic Process AnalysisStratonovich Calculus
Stochastic differential equations are difficult to solve for arbitrary continuous driving functions, leading to alternative theories such as Ito’s and Fisk–Stratonovich’s. This note announces that the previously described approach can be fully pursued, yielding a simple construction of solutions for arbitrary continuous sample‑path processes. The authors define a solution of the deterministic equation for continuous w and construct the stochastic solution by extending this definition to arbitrary continuous processes W. For a Wiener process, the construction reproduces the Stratonovich solution, confirming its validity.
where ww(/) = Wt(cS). Usually, this is not followed because of the technical difficulties that appear when one tries to solve (det) for arbitrary continuous w. One is therefore forced to use other lines of attack, and to study (stoch) directly. This gives rise to at least two nonequivalent theories, namely, the one due to Ito, and that of Fisk and Stratonovich. The purpose of this note is to announce that the obvious approach described in the beginning of the preceding paragraph can actually be pursued all the way, leading to a simple construction of solutions of (stoch) for arbitrary processes W with continuous sample paths. When W is a Wiener process, our construction gives the same result as the ordinary solution in the sense of Stratonovich. First, let us define what is meant by a solution of (det) when w is only continuous. We say that a curve x: t —• x(t), a C°([g, b], R), such that: