# semimartingale

Given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}},\mathbb{P})$, we consider real-valued stochastic processes $X_{t}$ with time index $t$ ranging over the nonnegative real numbers. Then, semimartingales have historically been defined as follows.

###### Definition.

A semimartingale $X$ is a cadlag adapted process having the decomposition $X=M+V$ for a local martingale $M$ and a finite variation process $V$.

More recently, the following alternative definition has also become common. For simple predictable integrands $\xi$, the stochastic integral $\int\xi\,dX$ is easily defined for any process $X$. The following definition characterizes semimartingales as processes for which this integral is well behaved.

###### Definition.

A semimartingale $X$ is a cadlag adapted process such that

 $\left\{\int_{0}^{t}\xi\,dX:|\xi|\leq 1\textrm{ is simple predictable}\right\}$

Writing $\|\xi\|$ for the supremum norm of a process $\xi$, this definition characterizes semimartingales as processes for which

 $\int_{0}^{t}\xi^{n}\,dX\rightarrow 0$

in probability as $n\rightarrow\infty$ for each $t>0$, where $\xi^{n}$ is any sequence of simple predictable processes satisfying $\|\xi^{n}\|\rightarrow 0$. This property is necessary and, as it turns out, sufficient for the development of a theory of stochastic integration for which results such as bounded convergence holds.

The equivalence of these two definitions of semimartingales is stated by the Bichteler-Dellacherie theorem.

A stochastic process $X_{t}=(X^{1}_{t},X^{2}_{t},\ldots,X^{n}_{t})$ taking values in $\mathbb{R}^{n}$ is said to be a semimartingale if $X^{k}_{t}$ is a semimartingale for each $k=1,2,\ldots,n$.

Title semimartingale Semimartingale 2013-03-22 18:36:38 2013-03-22 18:36:38 gel (22282) gel (22282) 8 gel (22282) Definition msc 60G07 msc 60G48 msc 60H05