BichtelerDellacherie theorem
The BichtelerDellacherie 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 DellacherieMeyerMokobodzky theorem. Prior to its discovery, a theory of stochastic integration had been developed for local martingales^{}. As standard LebesgueStieltjes 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 BichtelerDellacherie 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.
We consider a real valued stochastic process^{} $X$ adapted to a filtered probability space $(\mathrm{\Omega},\mathcal{F},{({\mathcal{F}}_{t})}_{t\in {\mathbb{R}}_{+}},\mathbb{P})$. Then, the integral ${\int}_{0}^{t}\xi \mathit{d}X$ can be written out explicitly for any simple predictable process $\xi $.
Theorem (BichtelerDellacherie).
Let $X$ be a cadlag adapted stochastic process. Then, the following are equivalent^{}.

1.
For every $t>0$, the set
$$\{{\int}_{0}^{t}\xi \mathit{d}X:\xi \le 1\mathit{\text{is simple predictable}}\}$$ is bounded in probability.

2.
A decomposition $X=M+V$ exists, where $M$ is a local martingale and $V$ is a finite variation process.

3.
A decomposition $X=M+V$ exists, where $M$ is locally a uniformly bounded martingale^{} and $V$ is a finite variation process.
Condition 1 is equivalent to stating that if ${\xi}^{n}$ is a sequence of simple predictable processes converging uniformly to zero, then the integrals ${\int}_{0}^{t}{\xi}^{n}\mathit{d}X$ tend to zero in probability as $n\to \mathrm{\infty}$, 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 ${\tau}_{n}$, almost surely increasing to infinity^{}, such that the stopped processes ${M}^{{\tau}_{n}}$ are uniformly bounded martingales.
References
 1 Philip E. Protter, Stochastic integration and differential equations^{}. Second edition. Applications of Mathematics, 21. Stochastic Modelling and Applied Probability. SpringerVerlag, 2004.
Title  BichtelerDellacherie theorem 

Canonical name  BichtelerDellacherieTheorem 
Date of creation  20130322 18:36:48 
Last modified on  20130322 18:36:48 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  8 
Author  gel (22282) 
Entry type  Theorem^{} 
Classification  msc 60G48 
Classification  msc 60H05 
Classification  msc 60G07 
Synonym  DellacherieMeyerMokobodzky theorem 