Definition
A real valued process ''X'' defined on theAlternative definition
First, the simple predictable processes are defined to be linear combinations of processes of the form ''H''''t'' = ''A''1 for stopping times ''T'' and ''F''''T'' -measurable random variables ''A''. The integral ''H'' · ''X'' for any such simple predictable process ''H'' 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 continuous finite variation processes, and hence semimartingales. *Properties
* The semimartingales form the largest class of processes for which the Itō integral can be defined. * Linear combinations of semimartingales are semimartingales. * Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the Itō integral. * The quadratic variation exists for every semimartingale. * The class of semimartingales is closed under optional stopping, localization, change of time and absolutely continuous change of measure. * If ''X'' is an R''m'' valued semimartingale and ''f'' is a twice continuously differentiable function from R''m'' to R''n'', then ''f''(''X'') is a semimartingale. This is a consequence of Itō's lemma. * The property of being a semimartingale is preserved under shrinking the filtration. More precisely, if ''X'' is a semimartingale with respect to the filtration ''F''t, and is adapted with respect to the subfiltration ''G''t, then ''X'' is a ''G''t-semimartingale. * (Jacod's Countable Expansion) The property of being a semimartingale is preserved under enlarging the filtration by a countable set of disjoint sets. Suppose that ''F''t is a filtration, and ''G''t is the filtration generated by ''F''t and a countable set of disjoint measurable sets. Then, every ''F''t-semimartingale is also a ''G''t-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 d''X''t = σt d''W''t + ''b''t dt, then :Special semimartingales
A special semimartingale is a real valued process '''' with the decomposition , where is a local martingale and 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 ''X''t* ≡ sup''s'' ≤ ''t'' , X''s'', is locally integrable . For example, every continuous semimartingale is a special semimartingale, in which case ''M'' and ''A'' are both continuous processes.Multiplicative decompositions
Recall that denotes thePurely discontinuous semimartingales / quadratic pure-jump semimartingales
A semimartingale is called ''purely discontinuous'' ( Kallenberg 2002) if its quadratic variation 'X''is a finite variation pure-jump process, i.e., :. By this definition, ''time'' is a purely discontinuous semimartingale even though it exhibits no jumps at all. Alternative (and preferred) terminology ''quadratic pure-jump'' semimartingale refers to the fact that the quadratic variation of a purely discontinuous semimartingale is a pure jump process. Every finite variation semimartingale is a quadratic pure-jump semimartingale. An adapted continuous process is a quadratic pure-jump semimartingale if and only if it is of finite variation. For every semimartingale X there is a unique continuous local martingale starting at zero such that is a quadratic pure-jump semimartingale (; ). The local martingale is called the ''continuous martingale part of'' ''X''. Observe that is measure-specific. If '''' and '''' are two equivalent measures then is typically different from , while both and are quadratic pure-jump semimartingales. ByContinuous-time and discrete-time components of a semimartingale
Every semimartingale has a unique decomposition where , the continuous-time component does not jump at predictable times, and the discrete-time component is equal to the sum of its jumps at predictable times in the semimartingale topology. One then has . Typical examples of the continuous-time component are Itô process and Lévy process. The discrete-time component is often taken to be aSemimartingales on a manifold
The concept of semimartingales, and the associated theory of stochastic calculus, extends to processes taking values in a differentiable manifold. A process ''X'' on the manifold ''M'' is a semimartingale if ''f''(''X'') 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.See also
* Sigma-martingaleReferences
* * * * * {{Stochastic processes Martingale theory