Semi-martingale

A stochastic process that can be represented as the sum of a local martingale and a process of locally bounded variation. For the formal definition of a semi-martingale one starts from a stochastic basis $( \Omega , {\mathcal F} , \mathbf F , {\mathsf P} )$, where $\mathbf F = ( {\mathcal F} _ {t} ) _ {t \geq 0 }$( cf. Stochastic processes, filtering of). A stochastic process $X = ( X _ {t} , {\mathcal F} _ {t} ) _ {t \geq 0 }$ is called a semi-martingale if its trajectories are right-continuous and have left limits, and if it can be represented in the form $X _ {t} = M _ {t} + V _ {t}$, where $M = ( M _ {t} , {\mathcal F} _ {t} )$ is a local martingale and $V = ( V _ {t} , {\mathcal F} _ {t} )$ is a process of locally bounded variation, that is,

$$\int\limits _ { 0 } ^ { t } | dV _ {s} ( \omega ) | < \infty ,\ \ t > 0,\ \ \omega \in \Omega .$$

In general this representation is non-unique. But in the class of representations with predictable processes $V$, the representation is unique (up to stochastic equivalence). The following belong to the class of semi-martingales (apart from local martingales and processes of locally bounded variation): local super-martingales and submartingales, processes $X$ with independent increments for which $f ( t) = {\mathsf E} e ^ {i \lambda X _ {t} }$ is a function of locally bounded variation for any $\lambda \in \mathbf R$( and so all processes with stationary independent increments), Itô processes, diffusion-type processes, and others. The class of semi-martingales is invariant under an equivalent change of measure. If $X$ is a semi-martingale and $f$ is twice continuously differentiable, then $f ( X) = ( f ( X _ {t} ), {\mathcal F} _ {t} )$ is also a semi-martingale. Here (Itô's formula)

$$f ( X _ {t} ) = \ f ( X _ {0} ) + \int\limits _ { 0 } ^ { t } f ^ { \prime } ( X _ {s _ {-} } ) dX _ {s} + { \frac{1}{2} } \int\limits _ { 0 } ^ { t } f ^ { \prime\prime } ( X _ {s _ {-} } ) d [ X, X] _ {s} ^ {c} +$$

$$+ \sum _ {0 < s \leq t } [ f ( X _ {s} ) - f ( X _ {s ^ {-} } ) - f ^ { \prime } ( X _ {s ^ {-} } ) \Delta X _ {s} ]$$

or, equivalently,

$$f ( X _ {t} ) = \ f ( X _ {0} ) + \int\limits _ { 0 } ^ { t } f ^ { \prime } ( X _ {s ^ {-} } ) dX _ {s} + { \frac{1}{2} } \int\limits _ { 0 } ^ { t } f ^ { \prime\prime } ( X _ {s ^ {-} } ) d [ X, X] _ {s} +$$

$$+ \sum _ {0 < s \leq t } \left [ f ( X _ {s} ) - f ( X _ {s ^ {-} } ) - f ^ { \prime } ( X _ {s ^ {-} } ) \Delta X _ {s ^ {-} } - { \frac{1}{2} } f ^ { \prime\prime } ( X _ {s ^ {-} } ) ( \Delta X _ {s} ) ^ {2} \right ] ,$$

where $[ X, X] = ([ X, X] _ {t} , {\mathcal F} _ {t} )$ is the quadratic variation of the semi-martingale $X$, that is,

$$[ X, X] _ {t} = \ X _ {0} ^ {2} + 2 \int\limits _ { 0 } ^ { t } X _ {s ^ {-} } dX _ {s} ,$$

$$[ X, X] _ {t} ^ {c} = [ X, X] _ {t} - \sum _ {0 < x \leq t } ( \Delta X _ {s} ) ^ {2}$$

is the continuous part of the quadratic variation $[ X, X]$, $\Delta X _ {s} = X _ {s} - X _ {s - }$, and the integrals are understood as stochastic integrals with respect to a semi-martingale (cf. Stochastic integral).

If $X$ is a semi-martingale, then the process $X ^ {(\leq 1) } = ( X _ {t} ^ {(\leq 1) } , {\mathcal F} _ {t} )$ with

$$X _ {t} ^ {(\leq 1) } = \ X _ {t} - \sum _ {0 < s \leq t } \Delta X _ {s} I (| \Delta X _ {s} | > 1)$$

has bounded jumps, $| \Delta X _ {t} ^ {(\leq 1) } | \leq 1$, and so can be uniquely represented as

$$X _ {t} ^ {(\leq 1) } = \ X _ {0} + B _ {t} + M _ {t} ,$$

where $B = ( B _ {t} , {\mathcal F} _ {t} )$ is a predictable random process of locally bounded variation and $M = ( M _ {t} , {\mathcal F} _ {t} )$ is a local martingale. This martingale can be uniquely represented as $M = M ^ {c} + M ^ {d}$, where $M ^ {c} = ( M _ {t} ^ {c} , {\mathcal F} _ {t} )$ is a continuous local martingale (a continuous martingale forming the semi-martingale $X$) and $M ^ {d} = ( M _ {t} ^ {d} , {\mathcal F} _ {t} )$ is a purely-discontinuous local martingale that can be written in the form

$$M _ {t} ^ {d} = \ \int\limits _ { 0 } ^ { t } \int\limits _ {| x| \leq 1 } x d ( \mu - \nu ),$$

where $d \mu = \mu ( \omega , dt, dx)$ is the random jump measure of $X$, that is,

$$\mu ( \omega , ( 0, t], \Gamma ) = \ \sum _ {0 < s \leq t } I ( \Delta X _ {s} \in \Gamma ),\ \ \Gamma \in {\mathcal B} ( \mathbf R \setminus \{ 0 \} ),$$

and $d v = \nu ( \omega , dt, dx)$ is its compensator. Since

$$\sum _ {0 < s \leq t } \Delta X _ {s} I (| \Delta X _ {s} | > 1) = \ \int\limits _ { 0 } ^ { t } \int\limits _ {| x | > 1 } x d \mu ,$$

each semi-martingale $X$ admits a representation

$$X _ {t} = \ X _ {0} + B _ {t} + M _ {t} ^ {c} + \int\limits _ { 0 } ^ { t } \int\limits _ {| x | \leq 1 } x d ( \mu - \nu ) + \int\limits _ { 0 } ^ { t } \int\limits _ {| x | > 1 } x d \mu ,$$

called the canonical representation (decomposition).

The set of (predictable) characteristics $( B, \langle M ^ {c} \rangle , \nu )$, where $\langle M ^ {c} \rangle$ is the quadratic characteristic of $M ^ {c}$, that is, a predictable increasing process such that $( M ^ {c} ) ^ {2} - \langle M ^ {c} \rangle$ is a local martingale, is called a triplet of local (predictable) characteristics of $X$.

References

 [1] J. Jacod, "Calcul stochastique et problèmes de martingales" , Lect. notes in math. , 714 , Springer (1979) [2] R.Sh. Liptser, A.N. [A.N. Shiryaev] Shiryayev, "Theory of martingales" , Kluwer (1989) (Translated from Russian)