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An Itô formula for the fermionic Brownian motion. (Une formule d’Itô pour le mouvement brownien fermionique.) (French) Zbl 0766.60060

Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 575-578 (1992).
[For the entire collection see Zbl 0754.00008.]
The so-called quantum Itô formula is in cases of bosons and for fermions only for creation and annihilation operators. It is actually an integration by parts formula. The author considers the following problem: when \(B_ t\) is a fermionic Brownian motion (autoadjoint), what is the formula for \(f(B_ t)\) for a general function \(f\)? (Previous result is for the case \(f(x)=x^ 2\).) He finds that \(f(B_ t)=f(B_ 0)+\int^ t_ 0\int^ 1_ 0f'(B_ s+u)du dB_ s+\int^ r_ 0\int_{0 < \theta < 1}f''(B_ s+\theta ) ( 1-\theta)d\theta ds\) which resemble very much to an Azema’s formula.
Reviewer: Y.-Z.Hu (Wuhan)

MSC:

60H05 Stochastic integrals
81S25 Quantum stochastic calculus

Citations:

Zbl 0754.00008
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