## Une remarque sur l’approximation de l’integrale stochastique du type noncausal par une suite des integrales de Stieltjes.(French)Zbl 0551.60058

If $$f(t,\omega)$$, (t$$\in [0,1]$$, $$\omega\in \Omega ({\mathcal F},P))$$, is a measurable process with $$P\{\int^{1}_{0}f^ 2(t,\omega)dt<\infty \}=1$$, $$B(t,\omega)$$ is a Brownian motion and $$\{\phi_ n\}$$ is an orthonormal basis of $$L^ 2([0,1])$$, then the series $$\sum <f,\phi_ n><\phi_ n,\dot B>$$ with convergence in probability is used to define a stochastic integral $$\int^{1}_{0}f(t,\omega)d^*B(t,\omega)$$. While this definition avoids the customary nonanticipatory condition, it may depend on the choice of the base. The author addresses to this difficulty and proves the following result: If the above process f is integrable in the $$L^ 1(\Omega)$$ sense w.r.t. a trigonometric system in $$L^ 2[0,1]$$, then it is also integrable w.r.t. the Haar system, and the two integrals are equal.
Reviewer: D.Kannan

### MSC:

 60H05 Stochastic integrals

### Keywords:

approximation; Stieltjes integrals
Full Text:

### References:

 [1] S. OGAWA, Sur le produit direct du bruit blanc par luimeme. C. R. Acad. Sc. Paris, 288 (1979), Serie A, 359-362. · Zbl 0397.60047 [2] S. OGAWA, Quelques proprietes de integrate stochastique du type noncausal. (1981, a paraitre).
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