A real valued process X defined on the filtered probability space is called a semimartingale if it can be decomposed as where M is a local martingale and A is a càdlàgadapted process of locally bounded variation. An Rn-valued process X = is a semimartingale if each of its components Xi is a semimartingale.
Alternative definition
First, the simple predictable processes are defined to be linear combinations of processes of the form Ht = A1 for stopping times T and FT -measurable random variablesA. The integral H · X for any such simple predictable processH and real valued process X is This is extended to all simple predictable processes by the linearity of H · X in H. A real valued process X is a semimartingale if it is càdlàg, adapted, and for every t ≥ 0, is bounded in probability. The Bichteler-Dellacherie Theorem states that these two definitions are equivalent.
Examples
Adapted and continuously differentiable processes are finite variation processes, and hence are semimartingales.
The property of being a semimartingale is preserved under shrinking the filtration. More precisely, if X is a semimartingale with respect to the filtration Ft, and is adapted with respect to the subfiltration Gt, then X is a Gt-semimartingale.
The property of being a semimartingale is preserved under enlarging the filtration by a countable set of disjoint sets. Suppose that Ft is a filtration, and Gt is the filtration generated by Ft and a countable set of disjoint measurable sets. Then, every Ft-semimartingale is also a Gt-semimartingale.
Semimartingale decompositions
By definition, every semimartingale is a sum of a local martingale and a finite variation process. However, this decomposition is not unique.
Continuous semimartingales
A continuous semimartingale uniquely decomposes as X = M + A where M is a continuous local martingale and A is a continuous finite variation process starting at zero. For example, if X is an Itō process satisfying the stochastic differential equation dXt = σt dWt + bt dt, then
Special semimartingales
A special semimartingale is a real valued process X with the decomposition X = M + A, where M is a local martingale and A is a predictable finite variation process starting at zero. If this decomposition exists, then it is unique up to a P-null set. Every special semimartingale is a semimartingale. Conversely, a semimartingale is a special semimartingale if and only if the process Xt* ≡ sups ≤ t |Xs| is locally integrable. For example, every continuous semimartingale is a special semimartingale, in which case M and A are both continuous processes.
Purely discontinuous semimartingales
A semimartingale is called purely discontinuous if its quadratic variation is a pure jump process, Every adapted finite variation process is a purely discontinuous semimartingale. A continuous process is a purely discontinuous semimartingale if and only if it is an adapted finite variation process. Then, every semimartingale has the unique decomposition X = M + A where M is a continuous local martingale and A is a purely discontinuous semimartingale starting at zero. The local martingale M - M0 is called the continuous martingale part of X, and written as Xc. In particular, if X is continuous, then M and A are continuous.
Semimartingales on a manifold
The concept of semimartingales, and the associated theory ofstochastic calculus, extends to processes taking values in a differentiable manifold. A process X on the manifold M is a semimartingale if f is a semimartingale for every smooth function f from M to R. Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.
