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 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.

We consider a real valued stochastic process $X$ adapted to a filtered probability space $(\mathrm{\Omega },ℱ,{({ℱ}_t)}_{t\in {ℝ}_{+}},\mathbb{P})$. Then, the integral ${\int }_0^t\xi 𝑑X$ can be written out explicitly for any simple predictable process $\xi$.

Theorem (Bichteler-Dellacherie).

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

1. For every $t>0$, the set

   $$\{{\int }_0^t\xi 𝑑X:|\xi |\le 1\mathit{\text{ is simple predictable}}\}$$

   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 ${\int }_0^t{\xi }^n𝑑X$ 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.