Chapter 6 Stochastic Representations For Nonlinear Parabolic PDEs

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We discuss several different representations of nonlinear parabolic partial differential equations in terms of Markov processes. After a brief introduction of the linear case, different representations for nonlinear equations are discussed. One class of representations is in terms of stochastic control and differential games. An extension to geometric equations is also discussed. All of these representations are through the appropriate expected values of the data. Different type of representations are also available through backward stochastic differential equations. A recent extension to second-order backward stochastic differential equations allow us to represent all fully nonlinear scalar parabolic equations.

Original languageEnglish (US)
Title of host publicationHandbook of Differential Equations
Subtitle of host publicationEvolutionary Equations
Number of pages50
ISBN (Print)9780444528483
StatePublished - 2007
Externally publishedYes

Publication series

NameHandbook of Differential Equations: Evolutionary Equations
ISSN (Print)1874-5717

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Applied Mathematics


  • 2BSDE
  • 35K55
  • 60H10
  • 60H30
  • 60H35
  • BSDE
  • Feynman-Kac formula
  • Fully nonlinear parabolic partial differential equations
  • Second-order backward stochastic differential equations
  • Viscosity solutions, Superdiffusions


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