Summary
A general one dimensional change of variables formula is established for continuous semimartingales which extends the famous Meyer-Tanaka formula. The inspiration comes from an application arising in stochastic finance theory. For functions mapping ℝn to ℝ, a general change of variables formula is established for arbitrary semimartingales, where the usualC 2 hypothesis is relaxed.
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Brosamler, G.: Quadratic variation of potentials and harmonic functions. Trans. Am. Math. Soc.149, 243–257 (1970)
Carlen, E., Protter, P.: On semimartingale decompositions of convex functions of semimartingales. Ill. J. Math.36, 420–427 (1992)
Çinlar, E., Jacod, J., Protter, P., Sharpe, M.: Semimartingales and Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.54, 161–220 (1980)
Kunita, H.: Some extensions of Itô's formula. In: Azéma, J., Yor, M. (eds.) Semin. de Probab. XV. (Lect. Notes Math., vol. 850, pp. 118–141) Berlin Heidelberg New York: Springer 1981
Krylov, N. V.: Controlled diffusion processes. Berlin Heidelberg New York: Springer 1980
Kubo, I.: Itô formula for generalized Brownian functionals. Lect. Notes Control Inf. Sci.49, 156–166 (1983)
Kuo, H. H., Shieh, N. R.: A generalized Itô's formula for multidimensional Brownian motion and its applications. Chinese J. Math.15, 163–174 (1987)
Meyer, P. A.: Un cours sur les intégrales stochastiques. In: Meyer, P. A. (ed.) Sémin. Probab. X. (Lect. Notes Math., vol. 511, pp. 246–400) Berlin Heidelberg New York: Springer 1976
Meyer, P. A.: La formule d'Itô pour le mouvement Brownien d'après G. Brosamler. In: Dellocherie, C. et al (eds.) Sémin. Probab. XII. (Lect. Notes Math., vol. 649, pp. 763–769) Berlin Heidelberg New York: Springer 1978
Myneni, R.: Personal communications (1990, 1991)
Protter, P.: Stochastic integration and differential equations. A new approach. Berlin Heidelberg New York: Springer 1990
Ouknine, Y., Rutkowski, M.: Local times of functions of continuous semimartingales. (Preprint)
Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Berlin Heidelberg New York: Springer 1990
Rosen, J.: A representation for the intersection local time of Brownian motion in space. Ann. Probab.13, 145–153 (1985)
San Martin, J.: Stochastic differential equations. Ph.D. Thesis, Purdue University 1990
San Martin, J.: One dimensional stochastic differential equations. Ann. Probab.21, 509–553 (1993)
Sznitman, A. S., Varadhan, S. R. S.: A multidimensional process involving local time. Probab. Theory Relat. Fields71, 553–579 (1986)
Yor, M.: Compléments aux formules de Tanaka-Rosen. In: Azéma, J., Yor, M. (eds.) Sémin. de Probab. XIX. (Lect. Notes Math., vol. 1123, pp. 332–349) Berlin Heidelberg New York: Springer 1985
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Supported in part by NSF grant No. DMS-9103454
Supported in part by John D. and Catherine T. MacArthur Foundation award for US-Chile Scientific Cooperation
Supported in part by FONDECYT, grant 92-0881
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Protter, P., San Martin, J. General change of variable formulas for semimartingales in one and finite dimensions. Probab. Th. Rel. Fields 97, 363–381 (1993). https://doi.org/10.1007/BF01195071
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DOI: https://doi.org/10.1007/BF01195071