Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Fundamental Aspects in Modelling the Constitutive Behaviour of Fibered Soft Tissues Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

Introduction to Stochastic Calculus and to the Resolution of PDEs Using Monte Carlo Simulations

verfasst von : Emmanuel Gobet

Erschienen in: Advances in Numerical Simulation in Physics and Engineering

Verlag: Springer International Publishing

Einloggen

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

search-config
loading …

Abstract

I give a pedagogical introduction to Brownian motion, stochastic calculus introduced by Itô in the fifties, following the elementary (at least not too technical) approach by Föllmer [Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), pp. 143–150. Springer, Berlin, 1981]. Based on this, I develop the connection with linear and semi-linear parabolic PDEs. Then, I provide and analyze some Monte Carlo methods to approximate the solution to these PDEs. This course is aimed at master students, Ph.D. students and researchers interesting in the connection of stochastic processes with PDEs and their numerical counterpart. The reader is supposed to be familiar with basic concepts of probability (say first chapters of the book Probability essentials by Jacod and Protter [Probability Essentials, 2nd edn. Springer, Berlin, 2003]), but no a priori knowledge on martingales and stochastic processes is required.

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 "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

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 Some Remarks on Avalanches Modelling: An Introduction to Shallow Flows Models
Nächstes Kapitel Structure-Preserving Shock-Capturing Methods: Late-Time Asymptotics, Curved Geometry, Small-Scale Dissipation, and Nonconservative Products
Fußnoten
1
A Gaussian random variable X (see [46]) with mean μ and variance σ 2 > 0 (often denoted by \(\mathcal{N}(\mu,\sigma ^{2})\)) is the r.v. with density
$$\displaystyle{ g_{\mu,\sigma ^{2}}(x) = \frac{1} {\sigma \sqrt{2\pi }}\mathrm{exp}[-\frac{(x-\mu )^{2}} {2\sigma ^{2}} ],\quad x \in \mathbb{R}. }$$
If σ 2 = 0, X = μ with probability 1. Moreover, for any \(u \in \mathbb{R}\), \(\mathbb{E}(e^{\mathit{uX}}) = e^{u\mu +\frac{1} {2} u^{2}\sigma ^{2}}\).
 
2
Two random variables X 1 and X 2 are independent if and only if \(\mathbb{E}(f(X_{1})g(X_{2})) = \mathbb{E}(f(X_{1}))\mathbb{E}(g(X_{2}))\) for any bounded functions f and g. This extends similarly to a vector.
 
3
\((X_{1},\ldots,X_{n})\) is a Gaussian vector if and only if for any \((\lambda _{i})_{1\leq i\leq n} \in \mathbb{R}^{n}\), \(\sum _{i=1}^{n}\lambda _{i}X_{i}\) has a Gaussian distribution. Independent Gaussian random variables form a Gaussian vector. A process (X t ) t is Gaussian if \((X_{t_{1}},\ldots,X_{t_{n}})\) is a Gaussian vector for any times \((t_{1},\ldots,t_{n})\) and any n. A Gaussian process is characterized by its mean \(m(t) = \mathbb{E}(X_{t})\) and its covariance function \(K(s,t) = \mathbb{C}\mathrm{ov}(X_{s},X_{t})\).
 
4
We recall that “an event A occurs a.s.” (almost surely) if \(\mathbb{P}(\omega:\omega \in A) = 1\) or equivalently if {w: w∉A} is a set of zero probability measure.
 
5
Here, we use the following standard result: let (X n ) n ≥ 1 be a sequence of random variables, each having the Gaussian distribution with mean μ n and variance σ n 2. If the distribution of X n converges, then (μ n , σ n 2) converge to (μ, σ 2), and the limit distribution is Gaussian with mean μ and variance σ 2. We recall that if X n converges a.s., then it also converges in distribution.
 
6
This growth condition can be relaxed into \(\vert f(x)\vert \leq C\exp \left (\frac{\vert x\vert ^{2}} {2\alpha } \right )\) for any x, for some positive constants C and α: in that case, the smoothness of the function u is satisfied for t < α only.
 
7
Here again, the boundedness could be relaxed to some exponential growth.
 
8
Meaning that for a deterministic positive constant C, \(\mathbb{P}(U \leq C) = 1\).
 
9
Actually, (19) proves that M is a martingale and the result to be proved is related to the optional sampling theorem.
 
10
Indeed, the result gives uniqueness and not the existence.
 
12
Indeed, we can show that \(\mathbb{E}(\sigma _{M}^{2}) = \mathbb{V}\mathrm{ar}(X)\).
 
13
In fact, it generally depends on the regularity of u.
 
14
This labeling comes from the infinitesimal decomposition of \(\mathbb{E}(f(X_{t}))\) as time is small, \(\partial _{t}\mathbb{E}(f(X_{t}))\vert _{t=0} = L_{b,\sigma ^{2}}^{\mathtt{ABM}}f(x)\), see Proposition 12.
 
15
Integrability is the right assumption.
 
16
That is the sum of \(\sum _{t_{i}\leq t}\vert A_{t_{i+1}} - A_{t_{i}}\vert \) exists and is finite, for instance A is continuously differentiable.
 
17
Leading to the notion of strong solution; the case of non-smooth coefficients is much more delicate and related to weak solutions, see [67].
 
18
Up to a set of zero probability measure.
 
19
To simplify the exposure.
 
20
Uniquely defined if y is close to the boundary.
 
21
Meaning that \(m(O) = \mathbb{E}(R\vert O)\).
 
22
There may be some colinearities within (ϕ j ) 1≤j≤K.
 
23
Also called variance term.
 
24
Squared bias term.
 
Literatur
1.
Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing. SIAM Series. Frontiers in Applied Mathematics, Philadelphia (2005)CrossRefMATH
2.
Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability, vol. 57. Springer, New York (2007)
3.
4.
Bachelier, L.: Théorie de la spéculation. Ph.D. thesis, Ann. Sci. École Norm. Sup. (1900)
5.
Bally, V., Pagès, G.: Error analysis of the optimal quantization algorithm for obstacle problems. Stoch. Process. Appl. 106(1), 1–40 (2003)MATH
6.
Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function. Probab. Theory Relat. Fields 104(1), 43–60 (1996)CrossRefMATHMathSciNet
7.
8.
Bender, C., Steiner, J.: Least-squares Monte Carlo for BSDEs. In: Carmona, R., Del Moral, P., Hu, P., Oudjane, N. (eds.) Numerical Methods in Finance. Series: Springer Proceedings in Mathematics, vol. 12. Springer, Berlin (2012)
9.
Ben Zineb, T., Gobet, E.: Preliminary control variates to improve empirical regression methods. Monte Carlo Methods Appl. 19(4), 331–354 (2013)CrossRefMATHMathSciNet
10.
Bossy, M., Gobet, E., Talay, D.: Symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41(3), 877–889 (2004)CrossRefMATHMathSciNet
11.
Bouchard, B., Touzi, N.: Discrete time approximation and Monte Carlo simulation of backward stochastic differential equations. Stoch. Process. Their Appl. 111, 175–206 (2004)CrossRefMATHMathSciNet
12.
Breiman, L.: Probability. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992) [Corrected reprint of the 1968 original]
13.
Cessenat, M., Dautray, R., Ledanois, G., Lions, P.L., Pardoux, E., Sentis, R.: Méthodes probabilistes pour les équations de la physique. Collection CEA, Eyrolles (1989)
14.
Crisan, D., Manolarakis, K.: Solving backward stochastic differential equations using the cubature method. Preprint (2010)
15.
Durrett, R.: Brownian Motion and Martingales in Analysis. Wadsworth Mathematics Series. Wadsworth International Group, Belmont (1984)MATH
16.
El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.C.: Reflected solutions of backward SDE’s and related obstacle problems for PDE’s. Ann. Probab. 25(2), 702–737 (1997)MATHMathSciNet
17.
El Karoui, N., Peng, S.G., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Financ. 7(1), 1–71 (1997)MATH
18.
El Karoui, N., Hamadène, S., Matoussi, A.: Backward stochastic differential equations and applications. In: Carmona, R. (ed.) Indifference Pricing: Theory and Applications, Chap. 8, pp. 267–320. Princeton University Press, Princeton (2008)
19.
Föllmer, H.: Calcul d’Itô sans probabilités. In: Seminar on Probability, XV (University of Strasbourg, Strasbourg, 1979/1980) (French), pp. 143–150. Springer, Berlin (1981)
20.
Freidlin, M.I.: Functional Integration and Partial Differential Equations. Annals of Mathematics Studies. Princeton University Press, Princeton (1985)MATH
21.
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Eglewood Cliffs (1964)MATH
22.
Friedman, A.: Stochastic Differential Equations and Applications, vol. 1. Academic, New York (1975) [A Subsidiary of Harcourt Brace Jovanovich, Publishers, XIII]MATH
23.
Friedman, A.: Stochastic Differential Equations and Applications, vol. 2. Academic, New York (1976) [A Subsidiary of Harcourt Brace Jovanovich, Publishers, XIII]MATH
24.
Gavalas, G.R.: Nonlinear Differential Equations of Chemically Reacting Systems. Springer Tracts in Natural Philosophy, vol. 17. Springer, New York (1968)
25.
Geiss, C., Geiss, S., Gobet, E.: Generalized fractional smoothness and L p -variation of BSDEs with non-Lipschitz terminal condition. Stoch. Process. Their Appli. 122(5), 2078–2116 (2012)CrossRefMATHMathSciNet
26.
Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York (2003)CrossRef
27.
28.
Gobet, E.: Euler schemes and half-space approximation for the simulation of diffusions in a domain. ESAIM Probab. Stat. 5, 261–297 (2001)CrossRefMATHMathSciNet
29.
Gobet, E.: Advanced Monte Carlo methods for barrier and related exotic options. In: Ciarlet, P.G., Bensoussan, A., Zhang, Q. (eds.) Handbook of Numerical Analysis, vol. XV. Special Volume: Mathematical Modeling and Numerical Methods in Finance, pp. 497–528. Elsevier/North-Holland, Netherlands (2009)CrossRef
30.
Gobet, E.: Méthodes de Monte-Carlo et processus stochastiques: du linéaire au non-linéaire. Editions de l’École Polytechnique, Palaiseau (2013)
31.
Gobet, E., Labart, C.: Error expansion for the discretization of backward stochastic differential equations. Stoch. Process. Their Appl. 117(7), 803–829 (2007)CrossRefMATHMathSciNet
32.
Gobet, E., Labart, C.: Solving BSDE with adaptive control variate. SIAM Numer. Anal. 48(1), 257–277 (2010)MATHMathSciNet
33.
34.
Gobet, E., Maire, S.: Sequential control variates for functionals of Markov processes. SIAM J. Numer. Anal. 43(3), 1256–1275 (2005)CrossRefMATHMathSciNet
35.
Gobet, E., Maire, S.: Sequential Monte Carlo domain decomposition for the Poisson equation. In: Proceedings of the 17th IMACS World Congress, Scientific Computation, Applied Mathematics and Simulation, Paris, 11–15 July 2005
36.
Gobet, E., Makhlouf, A.: L2-time regularity of BSDEs with irregular terminal functions. Stoch. Proces. Their Appli. 120, 1105–1132 (2010)CrossRefMATHMathSciNet
37.
Gobet, E., Menozzi, S.: Stopped diffusion processes: Boundary corrections and overshoot. Stoch. Process. Their Appl. 120, 130–162 (2010)CrossRefMATHMathSciNet
38.
Gobet, E., Munos, R.: Sensitivity analysis using Itô-Malliavin calculus and martingales. Application to stochastic control problem. SIAM J. Control Optim. 43(5), 1676–1713 (2005)CrossRefMATHMathSciNet
39.
40.
Gobet, E., Lemor, J.P., Warin, X.: A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15(3), 2172–2202 (2005)CrossRefMATHMathSciNet
41.
Golub, G., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Balimore (1996)MATH
42.
Gyorfi, L., Kohler, M., Krzyzak, A., Walk, H.: A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics. Springer, New York (2002)CrossRef
43.
44.
Ito, K.: On stochastic differential equations. Mem. Am. Math. Soc. (4), 1–51 (1951)
45.
Itô, K., McKean, H.P.: Diffusion Processes and Their Sample Paths. Springer, Berlin (1965)CrossRefMATH
46.
Jacod, J., Protter, P.: Probability Essentials, 2nd edn. Springer, Berlin (2003)
47.
Jourdain, B., Lelong, J.: Robust adaptive importance sampling for normal random vectors. Ann. Appl. Probab. 19(5), 1687–1718 (2009)CrossRefMATHMathSciNet
48.
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)MATH
49.
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1995)
50.
Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28(2), 558–602 (2000)CrossRefMATHMathSciNet
51.
Kolmogorov, A.N., Petrovsky, I.G., Piskunov, N.S.: Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin Université d’État à Moscou, Série internationale A 1, pp. 1–26 (1937)
52.
Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. In: Ecole d’Eté de Probabilités de St-Flour XII, 1982. Lecture Notes in Mathematics, vol. 1097, pp. 144–305. Springer, Berlin (1984)
53.
Lapeyre, B., Pardoux, E., Sentis, R.: Methodes de Monte Carlo pour les processus de transport et de diffusion. Collection Mathématiques et Applications, vol. 29. Springer, Berlin (1998)
54.
Lemor, J.P., Gobet, E., Warin, X.: Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12(5), 889–916 (2006)CrossRefMATHMathSciNet
55.
Lévy, P.: Sur certains processus stochastiques homogènes. Compos. Math. 7, 283–339 (1939)
56.
Ma, J., Yong, J.: Forward-Backward Stochastic Differential Equations. Lecture Notes in Mathematics, vol. 1702. Springer, Berlin (1999)
57.
Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)MATH
58.
59.
Pardoux, E.: Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order. In: Stochastic Analysis and Related Topics, VI (Geilo, 1996). Progress in Probability, vol. 42, pp. 79–127. Birkhäuser, Boston (1998)
60.
Pardoux, E., Peng, S.G.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)CrossRefMATHMathSciNet
61.
Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications. Proceedings of the IFIP International Conference, Charlotte/NC, 1991. Lecture Notes in Control and Information Sciences, vol. 176, pp. 200–217. Springer, Berlin (1992)
62.
Peirano, É., Talay, D.: Domain decomposition by stochastic methods. In: Domain Decomposition Methods in Science and Engineering, pp. 131–147 (electronic). National Autonomous University of Mexico, México (2003)
63.
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Comprehensive Studies in Mathematics. Springer, Berlin (1999)CrossRefMATH
64.
Sabelfeld, K.K.: Monte Carlo Methods in Boundary Value Problems. Springer Series in Computational Physics. Springer, Berlin (1991) [translated from Russian]MATH
65.
Samuelson, P.A.: Proof that properly anticipated prices fluctuate randomly. Ind. Manag. Rev. 6, 42–49 (1965)
66.
Shigesada, N., Kawasaki, K. (eds.): Biological Invasions: Theory and Practice. Oxford Series in Ecology and Evolution. Oxford University Press, Oxford (1997)
67.
Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes, 2nd edn. Comprehensive Studies in Mathematics. Springer, Berlin (1997)
68.
Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8(4), 94–120 (1990)CrossRefMathSciNet
69.
Metadaten
Titel
Introduction to Stochastic Calculus and to the Resolution of PDEs Using Monte Carlo Simulations
verfasst von
Emmanuel Gobet
Copyright-Jahr
2014
Buch
Advances in Numerical Simulation in Physics and Engineering

Print ISBN: 978-3-319-02838-5

Electronic ISBN: 978-3-319-02839-2

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-02839-2_3

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Zur Marktübersicht
    Bildnachweise