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)

Comments

See also Itô formula and Stochastic integral. Semi-martingales are the most general stochastic processes with respect to which it is possible to integrate predictable processes in a reasonable way.

References

[a1]	K. Bichteler, "The stochastic integral as a vector measure" , Measure Theory (Oberwolfach, 1979) , Lect. notes in math. , 794 , Springer (1980) pp. 348–360
[a2]	C. Dellacherie, "Un survol de la théorie de l'intégrale stochastique" , Measure Theory (Oberwolfach, 1979) , Lect. notes in math. , 794 , Springer (1980) pp. 365–395
[a3]	C. Dellacherie, P.A. Meyer, "Probabilités et potentiels" , 2 , Hermann (1980) pp. Chapts. V-VIII: Théorie des martingales
[a4]	M. Metivier, "Semimartingales" , de Gruyter (1982)
[a5]	L. Schwartz, "Les semi-martingales formelles" , Sem. Probab. XV , Lect. notes in math. , 850 , Springer (1981) pp. 413–489
[a6]	J. Jacod, A.N. Shiryaev, "Limit theorems for stochastic processes" , Springer (1987) (Translated from Russian)