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Functional Itō calculus and stochastic integral representation of martingales. (English) Zbl 1272.60031
The authors develop an extended Itō calculus relative to continuous semimartingales, applicable to adapted functionals which may depend on the trajectory (up to the current time). They proceed by specifying some simplified Malliavin calculus. Namely, they consider two pathwise derivatives,
– a horizontal derivative is defined by continuing a \(\mathbb{R}^d\)-valued path \(\omega[0,t]\) into a path \(\omega[0,t+\varepsilon[\), taken to be constant on \([t,t+\varepsilon[\);
– a vertical derivative is defined by varying the path \(\omega[0,t]\) according to \((\omega[0,t]+ 1_{\{t\}}\varepsilon e_j)\).
Such use of the only very particular case of Malliavin derivatives happens to be effective, when considering continuous functionals of both \((X_t,\langle X\rangle_t)\), \(X_t= (X(s),0\leq s\leq t)\) being a continuous semimartingale, and \(\langle X\rangle_t\) denoting its quadratic variation process.
Examples of functionals to which this extended calculus can be applied are \[ \int^t_0 g(s, X_s)\,d\langle X\rangle_s,\quad G(t,X_t,\langle X\rangle_t),\quad \operatorname{E}[G(t, X_t,\langle X\rangle_t)/F_s]. \] Under convenient regularity assumptions on such functionals, the authors establish the Itō formula, an integration by parts formula, and the Clark-Ocone martingale representation formula. Of course, in the usual setting all this reduces to the normal notions and formulas.

MSC:
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G44 Martingales with continuous parameter
60H25 Random operators and equations (aspects of stochastic analysis)
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References:
[1] Ahn, H. (1997). Semimartingale integral representation. Ann. Probab. 25 997-1010. · Zbl 0876.60025 · doi:10.1214/aop/1024404427
[2] Bismut, J.-M. (1981). A generalized formula of Itô and some other properties of stochastic flows. Z. Wahrsch. Verw. Gebiete 55 331-350. · Zbl 0456.60063 · doi:10.1007/BF00532124
[3] Bismut, J.-M. (1983). Calcul des variations stochastique et processus de sauts. Z. Wahrsch. Verw. Gebiete 63 147-235. · Zbl 0494.60082 · doi:10.1007/BF00538963
[4] Clark, J. M. C. (1970). The representation of functionals of Brownian motion by stochastic integrals. Ann. Math. Statist. 41 1282-1295. · Zbl 0213.19402 · doi:10.1214/aoms/1177696903
[5] Cont, R. and Fournie, D. (2010). A functional extension of the Ito formula. C. R. Math. Acad. Sci. Paris 348 57-61. · Zbl 1202.60082 · doi:10.1016/j.crma.2009.11.013
[6] Cont, R. and Fournié, D.-A. (2010). Change of variable formulas for nonanticipative functionals on path space. J. Funct. Anal. 259 1043-1072. · Zbl 1201.60051 · doi:10.1016/j.jfa.2010.04.017
[7] Davis, M. H. A. (1980). Functionals of diffusion processes as stochastic integrals. Math. Proc. Cambridge Philos. Soc. 87 157-166. · Zbl 0424.60063 · doi:10.1017/S0305004100056590
[8] Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies 29 . North-Holland, Amsterdam. · Zbl 0494.60001
[9] Dupire, B. (2009). Functional Itô calculus. Portfolio Research Paper 2009-04, Bloomberg.
[10] Elliott, R. J. and Kohlmann, M. (1988). A short proof of a martingale representation result. Statist. Probab. Lett. 6 327-329. · Zbl 0645.60053 · doi:10.1016/0167-7152(88)90008-9
[11] Fitzsimmons, P. J. and Rajeev, B. (2009). A new approach to the martingale representation theorem. Stochastics 81 467-476. · Zbl 1186.60046 · doi:10.1080/17442500802343417
[12] Föllmer, H. (1981). Calcul d’Itô sans probabilités. In Seminar on Probability , XV ( Univ. Strasbourg , Strasbourg , 1979 / 1980) ( French ). Lecture Notes in Math. 850 143-150. Springer, Berlin. · Zbl 0461.60074 · numdam:SPS_1981__15__143_0 · eudml:113318
[13] Haussmann, U. G. (1978). Functionals of Itô processes as stochastic integrals. SIAM J. Control Optim. 16 252-269. · Zbl 0375.60070 · doi:10.1137/0316016
[14] Haussmann, U. G. (1979). On the integral representation of functionals of Itô processes. Stochastics 3 17-27. · Zbl 0427.60056 · doi:10.1080/17442507908833134
[15] Itô, K. (1944). Stochastic integral. Proc. Imp. Acad. Tokyo 20 519-524. · Zbl 0060.29105 · doi:10.3792/pia/1195572786
[16] Itô, K. (1946). On stochastic differential equations. Proc. Imp. Acad. Tokyo 22 32-35. · Zbl 0063.02992 · doi:10.3792/pja/1195572371
[17] Jacod, J., Méléard, S. and Protter, P. (2000). Explicit form and robustness of martingale representations. Ann. Probab. 28 1747-1780. · Zbl 1044.60042 · doi:10.1214/aop/1019160506
[18] Karatzas, I., Ocone, D. L. and Li, J. (1991). An extension of Clark’s formula. Stochastics Stochastics Rep. 37 127-131. · Zbl 0745.60056
[19] Kunita, H. and Watanabe, S. (1967). On square integrable martingales. Nagoya Math. J. 30 209-245. · Zbl 0167.46602
[20] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215-310. · Zbl 0923.34056 · doi:10.4171/RMI/240 · eudml:39555
[21] Malliavin, P. (1978). Stochastic calculus of variation and hypoelliptic operators. In Proceedings of the International Symposium on Stochastic Differential Equations ( Res. Inst. Math. Sci. , Kyoto Univ. , Kyoto , 1976) 195-263. Wiley, New York. · Zbl 0411.60060
[22] Malliavin, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 313 . Springer, Berlin. · Zbl 0878.60001
[23] Meyer, P. A. (1976). Un cours sur les intégrales stochastiques. In Séminaire de Probabilités , X ( Seconde Partie : Théorie des Intégrales Stochastiques , Univ. Strasbourg , Strasbourg , Année Universitaire 1974 / 1975). Lecture Notes in Math. 511 245-400. Springer, Berlin. · Zbl 0374.60070 · numdam:SPS_1976__10__245_0 · eudml:113083
[24] Nualart, D. (2009). Malliavin Calculus and Its Applications. CBMS Regional Conference Series in Mathematics 110 . CBMS, Washington, DC. · Zbl 1198.60006
[25] Ocone, D. (1984). Malliavin’s calculus and stochastic integral representations of functionals of diffusion processes. Stochastics 12 161-185. · Zbl 0542.60055 · doi:10.1080/17442508408833299
[26] Pardoux, É. and Peng, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic Partial Differential Equations and Their Applications ( Charlotte , NC , 1991). Lecture Notes in Control and Inform. Sci. 176 200-217. Springer, Berlin. · Zbl 0766.60079 · doi:10.1007/BFb0007334
[27] Picard, J. (2006). Brownian excursions, stochastic integrals, and representation of Wiener functionals. Electron. J. Probab. 11 199-248 (electronic). · Zbl 1112.60043 · doi:10.1214/EJP.v11-310 · eudml:127424
[28] Protter, P. E. (2005). Stochastic Integration and Differential Equations , 2nd ed. Stochastic Modelling and Applied Probability 21 . Springer, Berlin.
[29] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Springer, Berlin. · Zbl 0917.60006
[30] Shigekawa, I. (1980). Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20 263-289. · Zbl 0476.28008
[31] Stroock, D. W. (1981). The Malliavin calculus, a functional analytic approach. J. Funct. Anal. 44 212-257. · Zbl 0475.60060 · doi:10.1016/0022-1236(81)90011-2
[32] Watanabe, S. (1987). Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15 1-39. · Zbl 0633.60077 · doi:10.1214/aop/1176992255
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