The Annals of Applied Probability

Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions

Stefano De Marco

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Abstract

We study smoothness of densities for the solutions of SDEs whose coefficients are smooth and nondegenerate only on an open domain D. We prove that a smooth density exists on D and give upper bounds for this density. Under some additional conditions (mainly dealing with the growth of the coefficients and their derivatives), we formulate upper bounds that are suitable to obtain asymptotic estimates of the density for large values of the state variable (“tail” estimates). These results specify and extend some results by Kusuoka and Stroock [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1–76], but our approach is substantially different and based on a technique to estimate the Fourier transform inspired from Fournier [Electron. J. Probab. 13 (2008) 135–156] and Bally [Integration by parts formula for locally smooth laws and applications to equations with jumps I (2007) The Royal Swedish Academy of Sciences]. This study is motivated by existing models for financial securities which rely on SDEs with non-Lipschitz coefficients. Indeed, we apply our results to a square root-type diffusion (CIR or CEV) with coefficients depending on the state variable, that is, a situation where standard techniques for density estimation based on Malliavin calculus do not apply. We establish the existence of a smooth density, for which we give exponential estimates and study the behavior at the origin (the singular point).

Article information

Source
Ann. Appl. Probab., Volume 21, Number 4 (2011), 1282-1321.

Dates
First available in Project Euclid: 8 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1312818837

Digital Object Identifier
doi:10.1214/10-AAP717

Mathematical Reviews number (MathSciNet)
MR2857449

Zentralblatt MATH identifier
1246.60082

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H07: Stochastic calculus of variations and the Malliavin calculus 60J60: Diffusion processes [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Keywords
Smoothness of densities stochastic differential equations locally smooth coefficients tail estimates Malliavin calculus square-root process

Citation

De Marco, Stefano. Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions. Ann. Appl. Probab. 21 (2011), no. 4, 1282--1321. doi:10.1214/10-AAP717. https://projecteuclid.org/euclid.aoap/1312818837


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References

  • [1] Alòs, E. and Ewald, C.-O. (2008). Malliavin differentiability of the Heston volatility and applications to option pricing. Adv. in Appl. Probab. 40 144–162.
  • [2] Bally, V. (2007). Integration by parts formula for locally smooth laws and applications to equations with jumps I. Preprint, Institut Mittag-Leffler, The Royal Swedish Academy of Sciences, Stockholm.
  • [3] Berkaoui, A., Bossy, M. and Diop, A. (2008). Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence. ESAIM Probab. Stat. 12 1–11 (electronic).
  • [4] Bossy, M. and Diop, A. (2004). An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form |x|α, α ∈ [1/2, 1). Technical Report INRIA, preprint RR-5396.
  • [5] Cox, J. C., Ingersoll, J. E. Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53 385–407.
  • [6] Dalang, R. C. and Nualart, E. (2004). Potential theory for hyperbolic SPDEs. Ann. Probab. 32 2099–2148.
  • [7] Forde, M. (2008). Tail asymptotics for diffusion processes, with applications to local volatility and CEV-Heston models. Available at arXiv:math/0608634v6.
  • [8] Fournier, N. (2008). Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump. Electron. J. Probab. 13 135–156.
  • [9] Hagan, P., Kumar, D., Lesniewski, A. and Woodward, D. (2002). Managing smile risk. Wilmott Magazine 3 84–108.
  • [10] Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies 6 327–343.
  • [11] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [12] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [13] Kusuoka, S. and Stroock, D. (1985). Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 1–76.
  • [14] Lamberton, D. and Lapeyre, B. (1997). Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall/CRC Press, Boca Raton, FL.
  • [15] Malliavin, P. (1978). Stochastic calculus of variation and hypoelliptic operators. In Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) 195–263. Wiley, New York.
  • [16] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [17] Viens, F. G. and Vizcarra, A. B. (2007). Supremum concentration inequality and modulus of continuity for sub-nth chaos processes. J. Funct. Anal. 248 1–26.