Almost Sure

28 March 11

The Bichteler-Dellacherie Theorem

In this post, I will give a statement and proof of the Bichteler-Dellacherie theorem describing the space of semimartingales. A semimartingale, as defined in these notes, is a cadlag adapted stochastic process X such that the stochastic integral {\int\xi\,dX} is well-defined for all bounded predictable integrands {\xi}. More precisely, an integral should exist which agrees with the explicit expression for elementary integrands, and satisfies bounded convergence in the following sense. If {\{\xi^n\}_{n=1,2,\ldots}} is a uniformly bounded sequence of predictable processes tending to a limit {\xi}, then {\int_0^t\xi^n\,dX\rightarrow\int_0^t\xi\,dX} in probability as n goes to infinity. If such an integral exists, then it is uniquely defined up to zero probability sets.

An immediate consequence of bounded convergence is that the set of integrals {\int_0^t\xi\,dX} for a fixed time t and bounded elementary integrands {\vert\xi\vert\le1} is bounded in probability. That is,

\displaystyle  \left\{\int_0^t\xi\,dX\colon\xi{\rm\ is\ elementary},\ \vert\xi\vert\le1\right\} (1)

is bounded in probability, for each {t\ge0}. For cadlag adapted processes, it was shown in a previous post that this is both a necessary and sufficient condition to be a semimartingale. Some authors use the property that (1) is bounded in probability as the definition of semimartingales (e.g., Protter, Stochastic Calculus and Differential Equations). The existence of the stochastic integral for arbitrary predictable integrands does not follow particularly easily from this definition, at least, not without using results on extensions of vector valued measures. On the other hand, if you are content to restrict to integrands which are left-continuous with right limits, the integral can be constructed very efficiently and, furthermore, such integrands are sufficient for many uses (integration by parts, Ito’s formula, a large class of stochastic differential equations, etc).

It was previously shown in these notes that, if X can be decomposed as {X=M+V} for a local martingale M and FV process V then it is possible to construct the stochastic integral, so X is a semimartingale. The importance of the Bichteler-Dellacherie theorem is that it tells us that a process is a semimartingale if and only if it is the sum of a local martingale and an FV process. In fact this was the historical definition used of semimartingales, and is still probably the most common definition.

Throughout, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}, and all processes are real-valued.

Theorem 1 (Bichteler-Dellacherie) For a cadlag adapted process X, the following are equivalent.

  1. X is a semimartingale.
  2. For each {t\ge0}, the set given by (1) is bounded in probability.
  3. X is the sum of a local martingale and an FV process.

Furthermore, the local martingale term in 3 can be taken to be locally bounded.

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