Sage Journals: Discover world-class research

Abstract

We consider stochastic differential equations with a drift term of gradient type and driven by Gaussian white noise on $R^{d}$ . Particular attention is given to the kernel $p_{t}$ , $t > 0$ of the transition semigroup associated with the solution process.

Under some rather strong regularity and growth assumptions on the coefficients, we adapt previous work by Thierry Hargé on Schrödinger operators and prove that the small time asymptotic expansion of $p_{t}$ , $t > 0$ is Borel summable.

We also briefly indicate some extensions and applications.

Keywords

SDEs asymptotic expansions Borel summability

Get full access to this article

View all access options for this article.

References

Albeverio, Theory of Dirichlet forms and applications, in: Lectures on Probability Theory and Statistics. St. Flour, 2000,

Albeverio,

Schachermayer and

Talagrand, eds, Springer, 2003, pp. 3–106. doi:10.1007/3-540-44922-1.

Albeverio,

Brzezniak and

Mazzuchi, Probabilistic integrals: Mathematical aspects, Scholarpedia 12(5) (2017), 10429. doi:10.4249/scholarpedia.10429.

Albeverio,

Cordoni,

Di Persio and Pellegrini , Asymptotic expansions for some locality models arising in finance, 2017, Preprint.

Albeverio,

Di Persio and

Mastrogiacomo, Small noise asymptotic expansions for stochastic PDE’s I. The case of a dissipative polynomially bounded nonlinearity, Tohoku. Math. J. 63 (2011), 877–898. doi:10.2748/tmj/1325886292.

Albeverio,

Di Persio,

Mastrogiacomo and

Smii, A class of Lévy driven SDEs and their explicit invariant measures, Pot. Anal. 45(2) (2016), 229–259. doi:10.1007/s11118-016-9544-3.

Albeverio,

Di Persio,

Mastrogiacomo and

Smii, Explicit invariant measures for infinite dimensional SDE driven by Lévy noise with dissipative nonlinear drift I, Comm. Math. Sci. 15(4) (2017), 957–983. doi:10.4310/CMS.2017.v15.n4.a3.

Albeverio,

Gottschalk and

J.-L.

Wu, Convoluted generalized white noise, Schwinger functions and their analytic continuation to wightman functions, Rev. Math. Phys 8(6) (1996), 763–817. doi:10.1142/S0129055X96000287.

Albeverio,

Gottschalk and

M.W.

Yoshida, Systems of classical particles in the grand canonical ensemble, scaling limits and quantum field theory, Rev. Math. Phys 17(02) (2005), 175–226. doi:10.1142/S0129055X05002327.

Albeverio,

Hilbert and

Kolokoltsov, Uniform asymptotic bounds for the heat kernel and the trace of a stochastic geodesic flow, Stochastics. 84(2–3) (2012), 315–333. doi:10.1080/17442508.2010.519029.

10.

Albeverio,

Hoegh-Krohn and

Streit, Energy forms, Hamiltonians and distorted Brownian paths, J. Math. Phy. 18 (1977), 907. doi:10.1063/1.523359.

11.

Albeverio,

Kondratiev,

Kozitsky and

Röckner, The Statistical Mechanics of Quantum Lattice Systems. A Path Integral Approach, EMS, 2009.

12.

Albeverio,

Lytvynov and

Mahnig, A model of the term structure of interest rates based on Lévy fields, Stoch. Proc. Appl. 114(2) (2004), 251–263. doi:10.1016/j.spa.2004.06.006.

13.

Albeverio and

Marinelli, Reconstructing the drift of a diffusion from partially observed transition probabilities, Stoch. Proc. Appl. 115 (2005), 1487–1502. doi:10.1016/j.spa.2005.04.008.

14.

Albeverio,

Mastrogiacomo and

Smii, Small noise asymptotic expansions for stochastic PDE’s driven by dissipative nonlinearity and Lévy noise, Stoch. Proc. Appl. 123 (2013), 2084–2109. doi:10.1016/j.spa.2013.01.013.

15.

Albeverio,

Röckle and

Steblovskaya, Asymptotic expansions for Ornstein-Uhlenbeck semigroups perturbed by potentials over Banach spaces, Stoch and Stoch. Rep. 69 (2000), 195–238. doi:10.1080/17442500008834240.

16.

Albeverio and

Smii, Asymptotic expansions for SDE’s with small multiplicative noise, Stoch. Proc. Appl. 125(3) (2015), 1009–1031. doi:10.1016/j.spa.2014.09.009.

17.

Albeverio and

Steblovskaya, A model of financial market with several interacting assets. Complete market case, Finance and Stochastics 6(3) (2002), 383–396. doi:10.1007/s007800100066.

18.

Azencott, Densité des diffusions en temps petit, développement asymptotique, Sém. Prob. XVIII, LN. Math. 1059 (1984), 402–494.

19.

Baldi and

Carmellino, Asymptotic of hitting probability for general one-dimensional pinned diffusions, Ann. Appl. Prob. 12 (2002), 1071–1095. doi:10.1214/aoap/1031863181.

20.

Ben Arous, Méthodes de Laplace et de la phase stationnaire sur l’espace de Wiener, Stochastics. 25 (1988), 125–153. doi:10.1080/17442508808833536.

21.

Berestycki,

Busca and

Florent, Asymptotics and calibration of local volatility models, Quant. Finan. 2 (2002), 61–69. doi:10.1088/1469-7688/2/1/305.

22.

Capriotti, The exponent expansion: An effective approximation of transition probabilities of diffusion processes and pricing kernels of financial derivatives, Int. J. Theor. Appl. Fin. 9(7) (2006), 1179–1199. doi:10.1142/S0219024906003925.

23.

C.Y.

de Verdière, Spectre du Laplacien et longueurs des géodésiques périodiques. I, Composio Mathematica 27(1) (1973), 83–106.

24.

C.Y.

de Verdière, Spectre du Laplacien et longueurs des géodésiques périodiques. II, Composio Mathematica 27(1) (1973), 83–106.

25.

Doss, Sur une résolution stochastique de l’équation de Schrödinger à coefficients analytiques, Comm. Math. Phys. V 73(3) (1980), 247–264. doi:10.1007/BF01197701.

26.

J.L.

Forman and

Sørensen, The Pearson diffusions: A class of statistically tractable diffusion processes, Scandi. J. Stat 35 (2008), 438–465. doi:10.1111/j.1467-9469.2007.00592.x.

27.

Freidlin, Markov Processes and Differential Equations, Asymptotic Problems, Birkhäuser, Basel, 1996.

28.

P.K.

Friz,

Gatheral,

Gulisashvili,

Jacquier and

Teichmann, Large Deviations and Asymptotic Methods in Finance, Proceedings in Mathematics and Statistics, Vol. 110, Springer, 2015.

29.

Fukushima,

Oshima and

Takeda, Dirichlet Forms and Symmetric Markov Processes, Vol. 19, De Gruyter, 2010. doi:10.1515/9783110218091.

30.

Garibaldi and

Scalas, Finitary Probabilistic Methods in Econophysics, Cambridge University Press, Cambridge, UK, 2010.

31.

Gatheral,

E.P.

Hsu,

Laurence,

Ouyang and

T.-H.

Wang, Asymptotics of implied volatility in local volatility models, Math. Fin. 22 (2012), 591–620. doi:10.1111/j.1467-9965.2010.00472.x.

32.

I.I.

Gihman and

A.V.

Skorohod, Stochastic Differential Equations, Springer-Verlag, Berlin-Heidelberg, New York, 1972.

33.

Gottschalk and

Smii, How to determine the law of the solution to a SPDE driven by a Lévy space-time noise, Journal of Mathematical Physics 43 (2007), 1–22.

34.

Gottschalk,

Smii and

Ouerdiane, Convolution Calculus on White Noise Spaces and Feynman Graph Representation of Generalized Renormalization Flows, 2005, pp. 101–111.

35.

Gottschalk,

Smii and

Thaler, The Feynman graph representation of general convolution semigroups and its applications to Lévy statistics, Journal of Bernoulli Society 14(2) (2008), 322–351. doi:10.3150/07-BEJ106.

36.

Gulisashvili, Analytically Tractable Stochastic Stock Price Models, Lect. Notes. Maths, Springer, 2012.

37.

Hargé, Borel summation of the small time expansion of the heat kernel. The scalar potential case, 2013, arXiv:1301.7742v1.

38.

Hargé, Borel summation of the semi-classical expansion of the partition function associated to the Schrödinger equation, 2013, https://hal.archives-ouvertes.fr/hal-00823084/file/ha8.pdf.

39.

Hargé, Some remarks on the complex heat kernel on

C^{ν}

in the scalar potential case, 2013, arXiv:1302.3512v1 [math-ph].

40.

Hargé, Borel summation of the heat kernel with a vector potential, 2013, preprint.

41.

Hargé, A deformation formula for the heat kernel, 2013, https://pdfs.semanticscholar.org/cd43/d7b34a353296541cf1d53d267cc61428d087.pdf.

42.

Hida,

H.-H.

Kuo,

Potthoff and

Streit, White Noise an Infinite Dimensional Calculus, Springer, Netherlands, 1993.

43.

Jordan and

Tier, Asymptotic approximation to deterministic and stochastic volatilty models, SIAM, J. Fin. Math. 2 (2011), 935–964. doi:10.1137/100791890.

44.

Kanai, Short time asymptotics for fundamental solutions of partial differential equations, Comm. Part. Diff. Eq. 2 (1977), 781–830. doi:10.1080/03605307708820048.

45.

Karatzas and

S.E.

Shreve, Methods of Mathematical Finance, Vol. 39, Springer, Verlag, 1998.

46.

Y.I.

Kifer, Transition density of diffusion processes with small diffusion, Theor. Prob. Appl. 21 (1976), 513–522. doi:10.1137/1121063.

47.

Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes, Lect. Notes. Math., Springer, 2000.

48.

Kusuoka and

Osajima, A remark on the asymptotic expansion of density function of Wiener functionals, JFA 255 (2008), 2545–2562. doi:10.1016/j.jfa.2008.03.019.

49.

Lamperti, A simple construction of certain diffusion processes, J. Math. Kyoto Univ. 4 (1964), 161–170. doi:10.1215/kjm/1250524711.

50.

Lütkebohmert, An asymptotic expansion for a Black-Scholes type model, Bull. Sci. Math. 128 (2004), 661–685. doi:10.1016/j.bulsci.2004.02.008.

51.

B.B.

Mandelbrot, Fractals and Scaling in Finance: Discontinuity and Concentration, Springer, Verlag 1997.

52.

Milan and

Miroslav, Stochastic equations for simple discrete models of epitaxial growth, Phys. Rev. E. 54(4) (1996), 3933–3942. doi:10.1103/PhysRevE.54.3933.

53.

S.A.

Molchanov, Diffusion processes and Riemannian geometry, Math. Surv. 30 (1975), 1–63. doi:10.1070/RM1975v030n01ABEH001400.

54.

J.K.

Møller and

Madsen, From State Dependent Diffusion to Constant Diffusion in Stochastic Differential Equations by the Lamperti Transform, Tech. Univ. of Denmark, DTU Informatics 25, 2010.

55.

Øksendal, Stochastic Differential Equations: An Introduction with Applications, 6th edn, Springer-Verlag, Berlin Heidelberg, 2010.

56.

Reed and

Simon, IV Analysis of Operators, Vol. 4, Elsevier Science Publishing Co Inc., 1978.

57.

Rezende, The method of stationary phase for oscillatory integrals on Hilbert spaces, Comm. Math. Phy. V 101(2) (1985), 187–206. doi:10.1007/BF01218758.

58.

A.N.

Sengupta, Pricing Derivatives, Ed. Mcgraw-Hill Professi, 2005.

59.

A.D.

Sokal, An improvement of Watson’s theorem on Borel summability, J. Math. Phys. 21(2) (1980), 261–263. doi:10.1063/1.524408.

60.

Takahashi, Asymptotic Expansion Approach in Finance. Large Deviations and Asymptotic Methods in Finance, Springer Proceeding in Mathematics and Statistics, Vol. 110, 2015. doi:10.1007/978-3-319-11605-1.

61.

H.C.

Tuckwell, Stochastic Processes in the Neurosciences, Vol. 56, Society for industrial and applied mathematics, 1989.

62.

H.C.

Tuckwell, Analytical and simulation results for the stochastic spatial Fitz Hugh-Nagumo model neuron, Neural computation 20(12) (2008), 3003–3033. doi:10.1162/neco.2008.08-07-585.

63.

Uchida and

Nakahiro, Information criteria for small diffusions via the theory of Malliavin–Watanabe, Statistical Inference for Stochastic Processes 7(1) (2004), 35–67. doi:10.1023/B:SISP.0000016462.43348.8f.

64.

Uemura, On a short time expansions of the fundamental solution of heat equations by the method of Wiener functional, J. Math. Kyoto Univ. 27 (1987), 417–431. doi:10.1215/kjm/1250520656.

65.

Vasicek, An equilibrium characterization of the term structure, J. Financ. Ec. 5 (1977), 177–188. doi:10.1016/0304-405X(77)90016-2.

66.

T.-H.

Wang and

Gatheral, Implied Volatility from Local Volatility: A Path Integral Approach, Proceedings in Mathematics and Statistics, Springer, 2015.

Borel summation of the small time expansion of some SDE’s driven by Gaussian white noise

Abstract

Keywords

Get full access to this article

References