Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction Read chapter Read first chapter

2013 | OriginalPaper | Chapter

2. Conditional Expectation and Linear Parabolic PDEs

Author : Nizar Touzi

Published in: Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

Publisher: Springer New York

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Throughout this chapter, \((\Omega,\mathcal{F}, \mathbb{F},P)\) is a filtered probability space with filtration \(\mathbb{F} =\{ {\mathcal{F}}_{t},\) t ≥ 0} satisfying the usual conditions. Let W = {W t ,t ≥ 0} be a Brownian motion valued in \({\mathbb{R}}^{d}\), defined on \((\Omega,\mathcal{F}, \mathbb{F},P)\).

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

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!

inform now
previous chapter Introduction
next chapter Stochastic Control and Dynamic Programming
Literature
1.
P. Bank, D. Baum, Hedging and portfolio optimization in financial markets with a large trader. Math. Finance. 14, 1–18 (2004)MathSciNetMATHCrossRef
2.
G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi. Mathématiques & Applications (Springer, New York, 1994)
3.
G. Barles, E.R. Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Math. Comput. 76, 1861–1893 (2007)MathSciNetMATHCrossRef
4.
G. Barles, P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4, 271–283 (1991)MathSciNetMATH
5.
G. Barles, C. Daher, M. Romano, Convergence of numerical schemes for parabolic equations arising in finance theory. Math. Model Meth. Appl. Sci. 5, 125–143 (1995)MathSciNetMATHCrossRef
6.
7.
F. Bonnans, H. Zidani, Consistency of generalized finite difference schemes for the stochastic HJB equation. Siam J. Numer. Anal. 41, 1008–1021 (2003)MathSciNetMATHCrossRef
8.
9.
10.
U. Cetin, R. Jarrow, P. Protter, M. Warachka, Pricing options in an extended black-scholes economy with illiquidity: theory and empirical evidence. Rev. Financ. Stud. 19, 493–529 (2006).CrossRef
11.
P. Cheridito, M. Soner, N. Touzi, Small time path behavior of double stochastic integrals, and application to stochastic control. Ann. Appl. Probab. 15, 2472–2495 (2005)MathSciNetMATHCrossRef
12.
P. Cheridito, M. Soner, N. Touzi, The multi-dimensional super-replication problem under gamma constraints. Annales de l’Institut Henri Poincaré, Série C: Analyse Non-Linéaire. 22, 633–666 (2005)
13.
P. Cheridito, M. Soner, N. Touzi, N. Victoir, Second order backward stochastic differential equations and fully non-linear parabolic pdes. Comm. Pure Appl. Math. 60, (2007)
14.
M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)MathSciNetMATHCrossRef
15.
J. Dugundji, Topology (Allyn and Bacon series in Advanced Mathematics, Allyn and Bacon edt.) (1966)
16.
N. ElKaroui, M. Jeanblanc-Picqué, Contrôle de processus de markov, Séminaire de Probabilités Springer Lecture Notes in Math. vol. 1321 (1988), pp. 508–541
17.
18.
N. ElKaroui, S. Peng, M.-C. Quenez, Backward stochastic differential equations in fiannce. Math. Finance. 7, 1–71 (1997)MathSciNetCrossRef
19.
A. Fahim, N. Touzi, X. Warin, A probabilistic numerical method for fully nonlinear parabolic PDEs. 21(4), 1322–1364 (2011)
20.
W.H. Fleming, H.M. Soner, Controlled Markov Processes and Viscosity Solutions (Springer, New York, 1993)MATH
21.
22.
23.
I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)MATHCrossRef
24.
I. Karatzas, S. Shreve, Methods of Mathematical Finance, 2nd edn. (Springer, New York, 1999)
25.
N. Kazamaki, Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics (Springer, Berlin 1579)
26.
M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Appl. Probab. 28, 558–602 (2000)MathSciNetMATH
27.
N.V. Krylov, Controlled Diffusion Processes (Springer, New York, 1977)MATH
28.
H. J. Kushner, P.G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time (Springer, New York, 1992)MATHCrossRef
29.
R.C. Merton, Lifetime portfolio selection under uncertainty: the continuous-time model. Rev. Econ. Stat. 51, 247–257 (1969)CrossRef
30.
R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theor. 3, 373–413 (1971)MathSciNetCrossRef
31.
M.-A. Morlais, Equations Différentielles Stochastiques Rétrogrades à Croissance Quadratique et Applications. Ph.D. thesis, Université de Rennes, 2007
32.
33.
D. Pooley, P. Forsyth, K. Vetzal, Numerical convergence properties of option pricing pdes with uncertain volatility. IMA J. Numer. Anal. 23, 241–267 (2003)MathSciNetMATHCrossRef
34.
P.J. Reny, On the existence of pure and mixed strategy nash equilibria in discontinuous games. Econometrica. 67, 1029–1056 (1999)MathSciNetMATHCrossRef
35.
R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)MATH
36.
H.M. Soner, N. Touzi, Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201–236 (2002)MathSciNetMATHCrossRef
37.
M. Soner, N. Touzi, The dynamic programming equation for second order stochastic target problems. SIAM J. Contr. Optim. 48(4), 2344–2365 (2009)MathSciNetMATHCrossRef
38.
H.M. Soner, N. Touzi, J. Zhang, Wellposedness of second order backward sdes. Probability Theory and Related Fields 153, 149–190 (2012)MathSciNetMATHCrossRef
39.
R. Tevzadze, Solvability of backward stochastic differential equation with quadratic growth. Stochast. Process. Appl. 118, 503–515 (2008)MathSciNetMATHCrossRef
40.
J. Thomas, Numerical Partial Differential Equations: Finite Difference methods (Texts in Applied Mathematics) (Springer, New York, 1998)
41.
J. Wang, P. Forsyth, Maximal use of central differencing for hamilton-jacobi-bellman pdes in finance. Siam J. Numer. Anal. 46, 1580–1601 (2008)MathSciNetMATHCrossRef
42.
43.
R. Zvan, K.R. Vetzal, P.A. Forsyth PDE methods for pricing barrier options. J. Econom. Dynam. Control 24, 1563–1590 (2000)
Metadata
Title
Conditional Expectation and Linear Parabolic PDEs
Author
Nizar Touzi
Copyright Year
2013
Publisher
Springer New York
Book
Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE

Print ISBN: 978-1-4614-4285-1

Electronic ISBN: 978-1-4614-4286-8

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-4286-8_2