Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 2752-2775.

Sticky processes, local and true martingales

Miklós Rásonyi and Hasanjan Sayit

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Abstract

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^{p}(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choose $Q$ arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present an application in mathematical finance.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2752-2775.

Dates
Received: September 2015
Revised: March 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051224

Digital Object Identifier
doi:10.3150/17-BEJ944

Mathematical Reviews number (MathSciNet)
MR3779701

Zentralblatt MATH identifier
06853264

Keywords
consistent price systems illiquid markets martingales processes with jumps sticky processes

Citation

Rásonyi, Miklós; Sayit, Hasanjan. Sticky processes, local and true martingales. Bernoulli 24 (2018), no. 4A, 2752--2775. doi:10.3150/17-BEJ944. https://projecteuclid.org/euclid.bj/1522051224


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References

  • [1] Aurzada, F. and Dereich, S. (2009). Small deviations of general Lévy processes. Ann. Probab. 37 2066–2092.
  • [2] Bender, C., Pakkanen, M.S. and Sayit, H. (2015). Sticky continuous processes have consistent price systems. J. Appl. Probab. 52 586–594.
  • [3] Cherny, A. (2008). Brownian moving averages have conditional full support. Ann. Appl. Probab. 18 1825–1830.
  • [4] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series. Boca Raton, FL: Chapman & Hall/CRC.
  • [5] Dalang, R.C., Morton, A. and Willinger, W. (1990). Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep. 29 185–201.
  • [6] Delbaen, F. and Schachermayer, W. (2006). The Mathematics of Arbitrage. Springer Finance. Berlin: Springer.
  • [7] Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies 29. Amsterdam-New York: North-Holland.
  • [8] Elworthy, K.D., Li, X.-M. and Yor, M. (1999). The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes. Probab. Theory Related Fields 115 325–355.
  • [9] Gasbarra, D., Sottinen, T. and van Zanten, H. (2011). Conditional full support of Gaussian processes with stationary increments. J. Appl. Probab. 48 561–568.
  • [10] Guasoni, P. (2006). No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16 569–582.
  • [11] Guasoni, P. and Rásonyi, M. (2015). Fragility of arbitrage and bubbles in local martingale diffusion models. Finance Stoch. 19 215–231.
  • [12] Guasoni, P. and Rásonyi, M. (2015). Hedging, arbitrage and optimality with superlinear frictions. Ann. Appl. Probab. 25 2066–2095.
  • [13] Guasoni, P., Rásonyi, M. and Schachermayer, W. (2008). Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18 491–520.
  • [14] Harrison, J.M. and Shepp, L.A. (1981). On skew Brownian motion. Ann. Probab. 9 309–313.
  • [15] Herczegh, A., Prokaj, V. and Rásonyi, M. (2014). Diversity and no arbitrage. Stoch. Anal. Appl. 32 876–888.
  • [16] Jacod, J. and Shiryaev, A.N. (1998). Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch. 2 259–273.
  • [17] Kabanov, Y. and Safarian, M. (2009). Markets with Transaction Costs. Springer Finance. Berlin: Springer.
  • [18] Kabanov, Y. and Stricker, C. (2001). On equivalent martingale measures with bounded densities. In Séminaire de Probabilités, XXXV. Lecture Notes in Math. 1755 139–148. Springer, Berlin.
  • [19] Kabanov, Y. and Stricker, C. (2008). On martingale selectors of cone-valued processes. In Séminaire de Probabilités XLI. Lecture Notes in Math. 1934 439–442. Springer, Berlin.
  • [20] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. New York: Springer.
  • [21] Pakkanen, M.S. (2010). Stochastic integrals and conditional full support. J. Appl. Probab. 47 650–667.
  • [22] Protter, P. (2013). A mathematical theory of financial bubbles. In Paris–Princeton Lectures on Mathematical Finance 2013. Lecture Notes in Math. 2081 1–108. Springer, Cham.
  • [23] Protter, P.E. (2005). Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability 21. Berlin: Springer.
  • [24] Sayit, H. and Viens, F. (2011). Arbitrage-free models in markets with transaction costs. Electron. Commun. Probab. 16 614–622.
  • [25] Simon, T. (2001). Sur les petites déviations d’un processus de Lévy. Potential Anal. 14 155–173.
  • [26] Stroock, D.W. and Varadhan, S.R.S. (1972). On the support of diffusion processes with applications to the strong maximum principle. 333–359.