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An approach to the stochastic calculus in the non-Gaussian case. (English) Zbl 0844.60036
Using ideas analogous to those of distributional concepts of modern PDE, the author introduces a generalized stochastic derivative and integral. This may be described in the author’s own words as follows: “Operators of stochastic differentiation $$D$$ and an extended integration $$I = D^*$$ [= the adjoint operator] play an important role in stochastic calculus. In the Gaussian case and for certain special martingales, $$D$$ and $$I$$ can be defined with the aid of an orthogonal expansion. Also $$D$$ and $$I$$ can be defined by means of the usual differentiation with respect to the admissible translation of the probability measure. In all these situations there are some common features. In this article we consider a general scheme in which the operators $$D$$ and $$I$$ are constructed for a non-Gaussian case. Since $$I$$ plays the role of stochastic integration, an analog of the Itô formula is also established”.

MSC:
 60H25 Random operators and equations (aspects of stochastic analysis) 60H05 Stochastic integrals 60G15 Gaussian processes
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