The Annals of Probability

Nondifferentiable functions of one-dimensional semimartingales

George Lowther

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We consider decompositions of processes of the form Y=f(t, Xt) where X is a semimartingale. The function f is not required to be differentiable, so Itô’s lemma does not apply.

In the case where f(t, x) is independent of t, it is shown that requiring f to be locally Lipschitz continuous in x is enough for an Itô-style decomposition to exist. In particular, Y will be a Dirichlet process. We also look at the case where f(t, x) can depend on t, possibly discontinuously. It is shown, under some additional mild constraints on f, that the same decomposition still holds. Both these results follow as special cases of a more general decomposition which we prove, and which applies to nondifferentiable functions of Dirichlet processes.

Possible applications of these results to the theory of one-dimensional diffusions are briefly discussed.

Article information

Ann. Probab. Volume 38, Number 1 (2010), 76-101.

First available in Project Euclid: 25 January 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals 60G48: Generalizations of martingales 60G44: Martingales with continuous parameter 60G20: Generalized stochastic processes

Semimartingale Itô’s lemma quadratic variation covariation Dirichlet process


Lowther, George. Nondifferentiable functions of one-dimensional semimartingales. Ann. Probab. 38 (2010), no. 1, 76--101. doi:10.1214/09-AOP476.

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  • [1] Bass, R. F. and Chen, Z.-Q. (2001). Stochastic differential equations for Dirichlet processes. Probab. Theory Related Fields 121 422–446.
  • [2] Bertoin, J. (1986). Les processus de Dirichlet en tant qu’espace de Banach. Stochastics 18 155–168.
  • [3] Coquet, F., Mémin, J. and Słomiński, L. (2003). On noncontinuous Dirichlet processes. J. Theoret. Probab. 16 197–216.
  • [4] Coquet, F. and Słomiński, L. (1999). On the convergence of Dirichlet processes. Bernoulli 5 615–639.
  • [5] Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel. Hermann, Paris. Chapitres I à IV, Édition entièrement refondue, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XV, Actualités Scientifiques et Industrielles, No. 1372.
  • [6] Elworthy, K. D., Truman, A. and Zhao, H. (2007). Generalized Itô formulae and space–time Lebesgue–Stieltjes integrals of local times. In Séminaire de Probabilités XL. Lecture Notes in Math. 1899 117–136. Springer, Berlin.
  • [7] Errami, M., Russo, F. and Vallois, P. (2002). Itô’s formula for C1,λ-functions of a càdlàg process and related calculus. Probab. Theory Related Fields 122 191–221.
  • [8] Flandoli, F., Russo, F. and Wolf, J. (2003). Some SDEs with distributional drift. I. General calculus. Osaka J. Math. 40 493–542.
  • [9] Flandoli, F., Russo, F. and Wolf, J. (2004). Some SDEs with distributional drift. II. Lyons–Zheng structure, Itô’s formula and semimartingale characterization. Random Oper. Stoch. Equ. 12 145–184.
  • [10] Föllmer, H. (1981). Dirichlet processes. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Math. 851 476–478. Springer, Berlin.
  • [11] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • [12] He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Kexue Chubanshe (Science Press), Beijing.
  • [13] Heinonen, J. (2005). Lectures on Lipschitz Analysis. Report. 100. Univ. Jyväskylä, Jyväskylä.
  • [14] Hobson, D. G. (1998). Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Probab. 8 193–205.
  • [15] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [16] Lowther, G. (2008). Properties of expectations of functions of martingale diffusions. Preprint. Available at arXiv:0801.0330v1.
  • [17] Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics 21. Springer, Berlin.
  • [18] Stricker, C. (1988). Variation conditionnelle des processus stochastiques. Ann. Inst. H. Poincaré Probab. Statist. 24 295–305.