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Stochastic Integrals and Conditional Full Support

Published online by Cambridge University Press:  14 July 2016

Mikko S. Pakkanen*
Affiliation:
University of Helsinki
*
Postal address: Department of Mathematics and Statistics, University of Helsinki, PO Box 68, FI-00014 Helsingin yliopisto, Finland. Email address: msp@iki.fi
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Abstract

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We present conditions that imply the conditional full support (CFS) property, introduced in Guasoni, Rásonyi and Schachermayer (2008), for processes Z := H + ∫K dW, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case, under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.

MSC classification

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. Ser. B 63, 167241.CrossRefGoogle Scholar
[2] Bender, C., Sottinen, T. and Valkeila, E. (2008). Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch. 12, 441468.CrossRefGoogle Scholar
[3] Cheridito, P. (2001). Mixed fractional Brownian motion. Bernoulli 7, 913934.CrossRefGoogle Scholar
[4] Cherny, A. (2008). Brownian moving averages have conditional full support. Ann. Appl. Prob. 18, 18251830.CrossRefGoogle Scholar
[5] Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291323.CrossRefGoogle Scholar
[6] Delbaen, F. and Schachermayer, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Appl. Prob. 5, 926945.CrossRefGoogle Scholar
[7] Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel. Hermann, Paris.Google Scholar
[8] Frey, R. (1997). Derivative asset analysis in models with level-dependent and stochastic volatility. CWI Quart. 10, 134.Google Scholar
[9] Gasbarra, D., Sottinen, T. and van Zanten, H. (2008). Conditional full support of Gaussian processes with stationary increments. Preprint 487, Department of Mathematics and Statistics, University of Helsinki.Google Scholar
[10] Guasoni, P., Rásonyi, M. and Schachermayer, W. (2008). Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Prob. 18, 491520.CrossRefGoogle Scholar
[11] Guo, X. (2001). An explicit solution to an optimal stopping problem with regime switching. J. Appl. Prob. 38, 464481.CrossRefGoogle Scholar
[12] Jeulin, T. and Yor, M. (1979). Inégalité de Hardy, semimartingales, et faux-amis. In Séminaire de Probabilités XIII (Lecture Notes Math. 721), Springer, Berlin, pp. 332359.CrossRefGoogle Scholar
[13] Kabanov, Y. and Stricker, C. (2008). On martingale selectors of cone-valued processes. In Séminaire de Probabilités XLI (Lecture Notes Math. 1934), Springer, Berlin, pp. 439442.CrossRefGoogle Scholar
[14] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
[15] Kallianpur, G. (1971). Abstract Wiener processes and their reproducing kernel Hilbert spaces. Z. Wahrscheinlichkeitsth. 17, 113123.CrossRefGoogle Scholar
[16] Millet, A. and Sanz-Solé, M. (1994). A simple proof of the support theorem for diffusion processes. In Séminaire de Probabilités XXVIII (Lecture Notes Math. 1583), Springer, Berlin, pp. 3648.CrossRefGoogle Scholar
[17] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
[18] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2. Cambridge University Press.Google Scholar
[19] Stroock, D. W. (1971). On the growth of stochastic integrals. Z. Wahrscheinlichkeitsth. 18, 340344.CrossRefGoogle Scholar
[20] Stroock, D. W. and Varadhan, S. R. S. (1972). On the support of diffusion processes with applications to the strong maximum principle. In Proc. 6th Berkeley Symp. Mathematical Statistics and Probability, Vol. III, California Press, Berkeley, CA, pp. 333359.Google Scholar
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