nach oben

Erschienen in:

2018 | OriginalPaper | Buchkapitel

8. The Girsanov Theorem Without (So Much) Stochastic Analysis

verfasst von : Antoine Lejay

Erschienen in: Séminaire de Probabilités XLIX

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this pedagogical note, we construct the semi-group associated to a stochastic differential equation with a constant diffusion and a Lipschitz drift by composing over small times the semi-groups generated respectively by the Brownian motion and the drift part. Similarly to the interpretation of the Feynman-Kac formula through the Trotter-Kato-Lie formula in which the exponential term appears naturally, we construct by doing so an approximation of the exponential weight of the Girsanov theorem. As this approach only relies on the basic properties of the Gaussian distribution, it provides an alternative explanation of the form of the Girsanov weights without referring to a change of measure nor on stochastic calculus.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Large Deviations for Infectious Diseases Models

Nächstes Kapitel On Drifting Brownian Motion Made Periodic

Nur mit Berechtigung zugänglich

In the original paper [9], the result is stated for 2^−1∕2W and 2^1∕2b.

The norm of a matrix is \(\|a\|=\left (\sum _{i,j=1}^n |a_{i,j}^2|\right )^{1/2}\) while the norm of a vector is \(\|a\|=\left (\sum _{i=1}^n |a_i|{ }^2\right )^{1/2}\).

In the domain of SDE, among others, the Ninomiya-Victoir scheme [41] relies on an astute way to compose the operators.

In [31], R. Léandre gives an interpretation of the Girsanov formula and Malliavin calculus in terms of manipulation on semi-groups.

This is one of the central ideas of Malliavin calculus to express the expectation involving the derivative of a function as the expectation involving the function multiplied by a weight.

This is, X _t(ω) = ω(t) when the probability space Ω is \(\mathrm {C}(\mathbb {R}_+,\mathbb {R})\).

These semi-groups are actually time-homogeneous. We however found it more convenient to keep the time dependence for our purpose.

These semi-groups satisfies the far more finer properties of being Feller, as for (X _t)_t≥0, but we do not use it here.

This condition is stronger than the one given in the original article of M. Kac on that subject.

J. Akahori, T. Amaba, S. Uraguchi, An algebraic approach to the Cameron-Martin-Maruyama-Girsanov formula. Math. J. Okayama Univ. 55, 167–190 (2013)MathSciNetMATH

L. Ambrosio, D. Trevisan, Lecture notes on the DiPerna-Lions theory in abstract measure spaces (2015). arXiv: 1505.05292

R. Bafico, P. Baldi, Small random perturbations of Peano phenomena. Stochastics 6(3–4), 279–292 (1981/1982). https://doi.org/10.1080/17442508208833208 MathSciNetCrossRef

V. Bally, A. Kohatsu-Higa, A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25(6), 3095–3138 (2015). https://doi.org/10.1214/14-AAP1068 MathSciNetCrossRef

V. Bally, C. Rey, Approximation of Markov semigroups in total variation distance. Electron. J. Probab. 21, 44 (2016). Paper No. 12. https://doi.org/10.1214/16-EJP4079

D.R. Bell, The Malliavin Calculus. Pitman Monographs and Surveys in Pure and Applied Mathematics (Longman Scientific & Technical/Wiley, Harlow/New York, 1987)

A. Beskos, G.O. Roberts, Exact simulation of diffusions. Ann. Appl. Probab. 15(4), 2422–2444 (2005). https://doi.org/10.1214/105051605000000485 MathSciNetCrossRef

S. Blanes, F. Casas, A. Murua, Splitting and composition methods in the numerical integration of differential equations. Bol. Soc. Esp. Mat. Apl. S^e MA 45, 89–145 (2008)

R.H. Cameron, W.T. Martin, Transformations of Wiener integrals under translations. Ann. Math. (2) 45, 386–396 (1944)

10.

R.H. Cameron, W.T. Martin, Transformations of Wiener integrals under a general class of linear transformations. Trans. Am. Math. Soc. 58, 184–219 (1945)MathSciNetCrossRef

11.

R.H. Cameron, W.T. Martin, The transformation of Wiener integrals by nonlinear transformations. Trans. Am. Math. Soc. 66, 253–283 (1949). https://doi.org/10.109/S0002-9947-1949-0031196-6 MathSciNetCrossRef

12.

P.R. Chernoff, Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators. Memoirs of the American Mathematical Society, vol. 140 (American Mathematical Society, Providence, 1974)

13.

R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)MathSciNetCrossRef

14.

C. Doléans-Dade, Quelques applications de la formule de changement de variables pour les semimartingales. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16, 181–194 (1970)MathSciNetCrossRef

15.

J.L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458 (1957)MathSciNetCrossRef

16.

D. Duffie, P. Protter, From discrete- to continuous-time finance: weak convergence of the financial gain process. J. Math. Finance 2, 1–15 (1992)CrossRef

17.

K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli, R. Schnaubelt (Springer, New York, 2000)

18.

M. Erraoui, E.H. Essaky, Canonical representation for Gaussian processes, in Séminaire de Probabilités XLII. Lecture Notes in Mathematics, vol. 1979 (Springer, Berlin, 2009), pp. 365–381. https://doi.org/10.1007/978-3-642-01763-6_13.

19.

F. Flandoli, Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations. Metrika 69(2–3), 101–123 (2009). https://doi.org/10.1007/s00184-008-0210-7 MathSciNetCrossRef

20.

I.V. Girsanov, On transforming a class of stochastic processes by absolutely continuous substitution of measures. Theor. Probab. Appl. 5, 285–301 (1960). https://doi.org/10.1137/1105027 MathSciNetCrossRef

21.

J.A. Goldstein, Remarks on the Feynman-Kac formula, in Partial Differential Equations and Dynamical Systems. Research Notes in Mathematics, vol. 101 (Pitman, Boston, 1984), pp. 164–180

22.

H. Gzyl, The Feynman-Kac formula and the Hamilton-Jacobi equation. J. Math. Anal. Appl. 142(1), 74–82 (1989). https://doi.org/10.1016/0022-247X(89)90165-0 MathSciNetCrossRef

23.

E. Hairer, C. Lubich, G. Wanner, Structure-preserving algorithms for ordinary differential equations, in Geometric Numerical Integration. Springer Series in Computational Mathematics, 2nd edn., vol. 31 (Springer, Berlin, 2006)

24.

H. Hult, Approximating some Volterra type stochastic integrals with applications to parameter estimation. Stoch. Process. Appl. 105(1), 1–32 (2003). https://doi.org/10.1016/S0304-4149(02)00250-8 MathSciNetCrossRef

25.

M. Jeanblanc, J. Pitman, M. Yor, The Feynman-Kac formula and decomposition of Brownian paths. Mat. Appl. Comput. 16(1), 27–52 (1997)MathSciNetMATH

26.

M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial Markets. Springer Finance (Springer, London, 2009). https://doi.org/10.1007/978-1-84628-737-4 CrossRef

27.

M. Kac, On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)MathSciNetCrossRef

28.

T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in Topics in Functional Analysis (Essays Dedicated to M. G. Kreı̆n on the Occasion of his 70th Birthday). Advances in Mathematics Supplied Studies, vol. 3 (Academic, New York, 1978), pp. 185–195

29.

A. Kohatsu-Higa, A. Lejay, K. Yasuda, Weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with non-regular drift. J. Comput. Appl. Math. 326C, 138–158 (2017). https://doi.org/10.1016/j.cam.2017.05.015 MathSciNetCrossRef

30.

T.G. Kurtz, P.E. Protter, Weak convergence of stochastic integrals and differential equations, in Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Mathematics, vol. 1627 (Springer, Berlin, 1996), pp. 1–41. https://doi.org/10.1007/BFb0093176

31.

R. Leandre, Applications of the Malliavin calculus of Bismut type without probability. WSEAS Trans. Math. 5(11), 1205–1210 (2006)MathSciNetMATH

32.

S. Lie, F. Engel, Theorie der Transformationsgruppen (Teubner, Leipzig, 1888)

33.

R.S. Liptser, A.N. Shiryaev, Statistics of Random Processes. I. General Theory. Applications of Mathematics (New York), 2nd edn., vol. 5 (Springer, Berlin, 2001). Translated from the 1974 Russian original by A.B. Aries. https://doi.org/10.1007/978-3-662-13043-8

34.

R.S. Liptser, A.N. Shiryaev, Statistics of Random Processes. II. Applications. Applications of Mathematics (New York), 2nd edn., vol. 6 (Springer, Berlin, 2001). Translated from the 1974 Russian original by A. B. Aries. https://doi.org/10.1007/978-3-662-10028-8

35.

B. Maisonneuve, Quelques martingales remarquables associées à une martingale continue. Publ. Inst. Statist. Univ. Paris 17.fasc. 3, 13–27 (1968)

36.

P. Malliavin, Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 313 (Springer, Berlin, 1997). https://doi.org/10.1007/978-3-642-15074-6

37.

G. Maruyama, On the transition probability functions of the Markov process. Nat. Sci. Rep. Ochanomizu Univ. 5, 10–20 (1954)MathSciNetMATH

38.

R. Mikulevičius, E. Platen, Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr. 151, 233–239 (1991). https://doi.org/10.1002/mana.19911510114 MathSciNetCrossRef

39.

R. Mikulevičius, C. Zhang, Weak Euler approximation for Itô diffusion and jump processes. Stoch. Anal. Appl. 33(3), 549–571 (2015). https://doi.org/10.1080/07362994.2015.1014102 MathSciNetCrossRef

40.

E. Nelson, Feynman integrals and the Schrödinger equation. J. Math. Phys. 5, 332–343 (1964)CrossRef

41.

S. Ninomiya, N. Victoir, Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15(1–2), 107–121 (2008). https://doi.org/1.108/13504860701413958 MathSciNetCrossRef

42.

S. Orey, Conditions for the absolute continuity of two diffusions. Trans. Am. Math. Soc. 193, 413–426 (1974)MathSciNetCrossRef

43.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44 (Springer, New York, 1983). https://doi.org/10.1007/978-1-4612-5561-1

44.

B. Roynette, M. Yor, Penalising Brownian Paths. Lecture Notes in Mathematics, vol. 1969 (Springer, Berlin, 2009). https://doi.org/10.1007/978-3-540-89699-9

45.

B. Simon, Schrödinger semigroups. Bull. Am. Math. Soc. (N.S.) 7(3), 447–526 (1982). https://doi.org/10.1090/S0273-0979-1982-15041-8

46.

A.V. Skorohod, On the densities of probability measures in functional spaces, in Proceedings of. Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA, 1965/1966) (University of California Press, Berkeley, 1967); vol. II: Contributions to Probability Theory, Part 1

47.

D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes. Classics in Mathematics (Springer, Berlin, 2006). Reprint of the 1997 edn.

48.

S. Takanobu, On the error estimate of the integral kernel for the Trotter product formula for Schrödinger operators. Ann. Probab. 25(4), 1895–1952 (1997). https://doi.org/10.1214/aop/1023481116 MathSciNetCrossRef

49.

D. Talay, Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution. RAIRO Modél. Math. Anal. Numér. 20(1), 141–179 (1986)MathSciNetCrossRef

50.

H.F. Trotter, On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)MathSciNetCrossRef

51.

J. H. Van Schuppen, E. Wong, Transformation of local martingales under a change of law. Ann. Probab. 2, 879–888 (1974)MathSciNetCrossRef

52.

A.J. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb. 39(3), 387–403 (1981). https://doi.org/10.1070/SM1981v039n03ABEH001522 CrossRef

53.

A. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift. Math. USSR Sb. 22, 129–149 (1975). https://doi.org/10.1070/SM1974v022n01ABEH001689 CrossRef

Titel: The Girsanov Theorem Without (So Much) Stochastic Analysis
verfasst von: Antoine Lejay
Verlag: Springer International Publishing
Buch: Séminaire de Probabilités XLIX
Print ISBN: 978-3-319-92419-9

Electronic ISBN: 978-3-319-92420-5

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-92420-5_8