The unified Itô formula has the the pseudo-Poisson structure \(df(x)={[}f(x+b)-f(x)]^{\mu}_{\nu}da^{\nu}_{\mu}\). (English) Zbl 0773.60040
Summary: A universal functional Itô formula, recently discovered within the *- Itô noncommutative algebraic approach, is described herein. The formula unifies the classical Itô formulas for Wiener, Poisson, and quantum cases, extends the chain rule \(df(x_ 1,\dots,x_ l)=\sum f_ i'(x)dx_ i\) to the case of noncommuting stochastically differentiable operators \(X_ i(t)\), and also admits a nonadapted generalization.
References:
| [1] | DOI: 10.3792/pia/1195572786 · Zbl 0060.29105 · doi:10.3792/pia/1195572786 |
| [2] | Meyer P. A., J. Math. 6 pp 193– (1962) |
| [3] | DOI: 10.1007/BF01258530 · Zbl 0546.60058 · doi:10.1007/BF01258530 |
| [4] | Belavkin V. P., Mat. Zamietki 49 pp 135– (1991) |
| [5] | DOI: 10.1016/0022-1236(91)90129-S · doi:10.1016/0022-1236(91)90129-S |
| [6] | DOI: 10.1063/1.528945 · Zbl 0718.47034 · doi:10.1063/1.528945 |
| [7] | DOI: 10.1063/1.1705306 · Zbl 0173.29604 · doi:10.1063/1.1705306 |
| [8] | DOI: 10.3792/pja/1195519933 · Zbl 0256.60029 · doi:10.3792/pja/1195519933 |
| [9] | Kannan D., Ann. Inst. H. Poincaré 8 pp 9– (1972) |
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