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Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions. (English) Zbl 1246.60082
This paper deals with existence of smooth densities for solutions of Stochastic Differential Equations (SDEs) whose coefficients are smooth (i.e., admit bounded partial derivatives) and nondegenerate only on an open domain \(D\). More precisely, it is proved in Theorem 2.1 that under these conditions, a smooth density exists for the solution of the SDE. If in addition, the domain \(D\) is the complementary of a compact ball and under additional assumptions on the coefficients, asymptotics (meaning for large values of the state variable) upper bounds for the density are given in Theorem 2.2. In addition, the dependence between the bounding constants and the coefficients of the SDE is made explicit. These results find applications in Finance where for instance stochastic volatility models like the Heston model are defined via such non-classical (non-Lipschitz) SDEs. These results are presented in Section 3. The proofs of the main results are presented in Section 2 and make use of the Malliavin calculus and of a Fourier transform argument.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
60H07 Stochastic calculus of variations and the Malliavin calculus
60J60 Diffusion processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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[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. · Zbl 1137.91422 · doi:10.1239/aap/1208358890
[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). · Zbl 1183.65004 · doi:10.1051/ps:2007030 · eudml:246056
[4] Bossy, M. and Diop, A. (2004). An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form | x | \alpha , \alpha \in [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. · Zbl 1274.91447
[6] Dalang, R. C. and Nualart, E. (2004). Potential theory for hyperbolic SPDEs. Ann. Probab. 32 2099-2148. · Zbl 1054.60066 · doi:10.1214/009117904000000685
[7] Forde, M. (2008). Tail asymptotics for diffusion processes, with applications to local volatility and CEV-Heston models. Available at . · arxiv.org
[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. · Zbl 1191.60072 · emis:journals/EJP-ECP/_ejpecp/viewarticle408f.html · eudml:229126
[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. · Zbl 1384.35131
[11] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes , 2nd ed. North-Holland Mathematical Library 24 . North-Holland, Amsterdam. · Zbl 0684.60040
[12] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus , 2nd ed. Graduate Texts in Mathematics 113 . Springer, New York. · Zbl 0734.60060
[13] Kusuoka, S. and Stroock, D. (1985). Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 1-76. · Zbl 0568.60059
[14] Lamberton, D. and Lapeyre, B. (1997). Introduction to Stochastic Calculus Applied to Finance . Chapman & Hall/CRC Press, Boca Raton, FL. · Zbl 0949.60005
[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. · Zbl 0411.60060
[16] Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050
[17] Viens, F. G. and Vizcarra, A. B. (2007). Supremum concentration inequality and modulus of continuity for sub- n th chaos processes. J. Funct. Anal. 248 1-26. · Zbl 1126.60041 · doi:10.1016/j.jfa.2007.03.019
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