Journal of Applied Probability

Stochastic integrals and conditional full support

Mikko S. Pakkanen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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.

Article information

Source
J. Appl. Probab. Volume 47, Number 3 (2010), 650-667.

Dates
First available in Project Euclid: 24 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.jap/1285335401

Digital Object Identifier
doi:10.1239/jap/1285335401

Mathematical Reviews number (MathSciNet)
MR2731340

Zentralblatt MATH identifier
1216.60047

Subjects
Primary: 91B28
Secondary: 60H05: Stochastic integrals

Keywords
Conditional full support stochastic integral stochastic volatility stochastic differential equation

Citation

Pakkanen, Mikko S. Stochastic integrals and conditional full support. J. Appl. Probab. 47 (2010), no. 3, 650--667. doi:10.1239/jap/1285335401. https://projecteuclid.org/euclid.jap/1285335401


Export citation

References

  • 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, 167--241.
  • Bender, C., Sottinen, T. and Valkeila, E. (2008). Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch. 12, 441--468.
  • Cheridito, P. (2001). Mixed fractional Brownian motion. Bernoulli 7, 913--934.
  • Cherny, A. (2008). Brownian moving averages have conditional full support. Ann. Appl. Prob. 18, 1825--1830.
  • Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291--323.
  • Delbaen, F. and Schachermayer, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Appl. Prob. 5, 926--945.
  • Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel. Hermann, Paris.
  • Frey, R. (1997). Derivative asset analysis in models with level-dependent and stochastic volatility. CWI Quart. 10, 1--34.
  • 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.
  • Guasoni, P., Rásonyi, M. and Schachermayer, W. (2008). Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Prob. 18, 491--520.
  • Guo, X. (2001). An explicit solution to an optimal stopping problem with regime switching. J. Appl. Prob. 38, 464--481.
  • 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. 332--359.
  • 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. 439--442.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
  • Kallianpur, G. (1971). Abstract Wiener processes and their reproducing kernel Hilbert spaces. Z. Wahrscheinlichkeitsth. 17, 113--123.
  • 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. 36--48.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.
  • Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2. Cambridge University Press.
  • Stroock, D. W. (1971). On the growth of stochastic integrals. Z. Wahrscheinlichkeitsth. 18, 340--344.
  • 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. 333--359.