## Nondifferentiable functions of one-dimensional semimartingales.(English)Zbl 1193.60072

Let $$X$$ be a real-valued aemimartingale and $$f:\mathbb{R}_+\times\mathbb{R}\to \mathbb{R}$$. if $$f$$ is twice continuously differentiable, it follows from Itô’s lemma that $f(t, X_t)= \int^t_0 D_x f(s, X_{s-})\,dX_s+ V_t,\tag{$$*$$}$ where $$V_t$$ is a finite variation process. A real-valued process $$X$$ is called a Dirichiet process if it has a decomposition $$X,Y+ V$$, where $$Y$$ is a semimartingale and is a càdlàg adapted zero continuous quadratic variation (z.c.q.v.) process, i.e., the quadratic variation $$[V]$$ of $$V$$ exists, and $$[V]^c= 0$$. Let $${\mathcal D}_0$$ denote the set of $$f: \mathbb{R}_+\times \mathbb{R}\to\mathbb{R}$$ such that
(i) $$f(t,x)$$ is locally Lipschitz continuuus in $$x$$ and càdlàg in $$t$$;
(ii) for every $$K_0< K_1\in \mathbb{R}$$ and $$T\in \mathbb{R}_+$$, $\int^{K_1}_{K_0} \int^T_0 |d_t f(t,x)|\,dx< \infty.$ If, furthermore, the left and right derivatives of $$f(t,x)$$ with respect to $$x$$ exist everywhere, this is written as $$f\in{\mathcal D}$$. If $$f\in{\mathcal D}_0$$, $$D_x f(t,x)$$ is used to denote $$\limsup_{n\to 0}(f(t, x+ h)- f(t, x))/h$$ which is locally bounded for any $$f\in{\mathcal D}_0$$. Then the following theorem holds: Let $$X$$ be a semimartingale and $$f\in{\mathcal D}$$. Then $$(*)$$ holds where $$V$$ has z.c.q.v. In particular, $$f(t,X_t)$$ is a Dirichlet process. This result is a consequence of a more general decomposition result which generalizes the just mentioned theorem to arbitrary Dirichlet processes.

### MSC:

 60H05 Stochastic integrals 60G48 Generalizations of martingales 60G44 Martingales with continuous parameter 60G20 Generalized stochastic processes 60J60 Diffusion processes 60H20 Stochastic integral equations
Full Text:

### References:

 [1] Bass, R. F. and Chen, Z.-Q. (2001). Stochastic differential equations for Dirichlet processes. Probab. Theory Related Fields 121 422-446. · Zbl 0995.60053 [2] Bertoin, J. (1986). Les processus de Dirichlet en tant qu’espace de Banach. Stochastics 18 155-168. · Zbl 0602.60069 [3] Coquet, F., Mémin, J. and Słomiński, L. (2003). On noncontinuous Dirichlet processes. J. Theoret. Probab. 16 197-216. · Zbl 1026.60072 [4] Coquet, F. and Słomiński, L. (1999). On the convergence of Dirichlet processes. Bernoulli 5 615-639. · Zbl 0953.60001 [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. · Zbl 0323.60039 [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. · Zbl 1126.60044 [7] Errami, M., Russo, F. and Vallois, P. (2002). Itô’s formula for C 1, \lambda -functions of a càdlàg process and related calculus. Probab. Theory Related Fields 122 191-221. · Zbl 0999.60048 [8] Flandoli, F., Russo, F. and Wolf, J. (2003). Some SDEs with distributional drift. I. General calculus. Osaka J. Math. 40 493-542. · Zbl 1054.60069 [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. · Zbl 1088.60058 [10] Föllmer, H. (1981). Dirichlet processes. In Stochastic Integrals ( Proc. Sympos. , Univ. Durham , Durham , 1980). Lecture Notes in Math. 851 476-478. Springer, Berlin. · Zbl 0462.60046 [11] Friedman, A. (1964). Partial Differential Equations of Parabolic Type . Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0144.34903 [12] He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochastic Calculus . Kexue Chubanshe (Science Press), Beijing. · Zbl 0781.60002 [13] Heinonen, J. (2005). Lectures on Lipschitz Analysis. Report . 100 . Univ. Jyväskylä, Jyväskylä. · Zbl 1086.30003 [14] Hobson, D. G. (1998). Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Probab. 8 193-205. · Zbl 0933.91012 [15] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd ed. Springer, New York. · Zbl 0996.60001 [16] Lowther, G. (2008). Properties of expectations of functions of martingale diffusions. Preprint. Available at · Zbl 1153.78301 [17] Protter, P. E. (2004). Stochastic Integration and Differential Equations , 2nd ed. Applications of Mathematics 21 . Springer, Berlin. · Zbl 1041.60005 [18] Stricker, C. (1988). Variation conditionnelle des processus stochastiques. Ann. Inst. H. Poincaré Probab. Statist. 24 295-305. · Zbl 0647.60053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.