Bichteler-Dellacherie theorem

The Bichteler-Dellacherie theorem is an important result in stochastic calculus, and states the equivalence of two very different definitions of semimartingales. The result also goes under other names, such as the Dellacherie-Meyer-Mokobodzky theorem. Prior to its discovery, a theory of stochastic integration had been developed for local martingalesPlanetmathPlanetmath. As standard Lebesgue-Stieltjes integration can be applied to finite variation processes, this allowed an integral to be defined with respect to sums of local martingales and finite variation processes, known as a semimartingales. The Bichteler-Dellacherie theorem then shows that, as long as we require stochastic integration to satisfy boundedPlanetmathPlanetmathPlanetmath convergence, then semimartingales are actually the most general objects which can be used.

We consider a real valued stochastic processMathworldPlanetmath X adapted to a filtered probability space (Ω,,(t)t+,). Then, the integral 0tξ𝑑X can be written out explicitly for any simple predictable process ξ.

Theorem (Bichteler-Dellacherie).

Let X be a cadlag adapted stochastic process. Then, the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

  1. 1.

    For every t>0, the set

    {0tξ𝑑X:|ξ|1 is simple predictable}

    is bounded in probability.

  2. 2.

    A decomposition X=M+V exists, where M is a local martingale and V is a finite variation process.

  3. 3.

    A decomposition X=M+V exists, where M is locally a uniformly bounded martingaleMathworldPlanetmath and V is a finite variation process.

Condition 1 is equivalent to stating that if ξn is a sequence of simple predictable processes converging uniformly to zero, then the integrals 0tξn𝑑X tend to zero in probability as n, which is a weak form of bounded convergence for stochastic integration.

Conditions 1 and 2 are the two definitions often used for the process X to be a semimartingale. However, condition 3 gives a stronger decomposition which is often more useful in practise. The property that M is locally a uniformly bounded martingale means that there exists a sequence of stopping times τn, almost surely increasing to infinityMathworldPlanetmath, such that the stopped processes Mτn are uniformly bounded martingales.


  • 1 Philip E. Protter, Stochastic integration and differential equationsMathworldPlanetmath. Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. Springer-Verlag, 2004.
Title Bichteler-Dellacherie theorem
Canonical name BichtelerDellacherieTheorem
Date of creation 2013-03-22 18:36:48
Last modified on 2013-03-22 18:36:48
Owner gel (22282)
Last modified by gel (22282)
Numerical id 8
Author gel (22282)
Entry type TheoremMathworldPlanetmath
Classification msc 60G48
Classification msc 60H05
Classification msc 60G07
Synonym Dellacherie-Meyer-Mokobodzky theorem