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 martingales. 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 bounded convergence, then semimartingales are actually the most general objects which can be used.
Let be a cadlag adapted stochastic process. Then, the following are equivalent.
Condition 1 is equivalent to stating that if is a sequence of simple predictable processes converging uniformly to zero, then the integrals tend to zero in probability as , which is a weak form of bounded convergence for stochastic integration.
Conditions 1 and 2 are the two definitions often used for the process to be a semimartingale. However, condition 3 gives a stronger decomposition which is often more useful in practise. The property that is locally a uniformly bounded martingale means that there exists a sequence of stopping times , almost surely increasing to infinity, such that the stopped processes are uniformly bounded martingales.
- 1 Philip E. Protter, Stochastic integration and differential equations. Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. Springer-Verlag, 2004.
|Date of creation||2013-03-22 18:36:48|
|Last modified on||2013-03-22 18:36:48|
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