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

Ogawa, Shigeyoshi

Tôhoku Math. J., II. Ser. 36, 41-48 (1984).

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

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).