Stochastic integral

2020 Mathematics Subject Classification: Primary: 60H05 [MSN][ZBL]

An integral "∫ H dX" with respect to a semi-martingale $ X $ on some stochastic basis $ ( \Omega , {\mathcal F} , ( {\mathcal F} _ {t} ) _ {t} , {\mathsf P} ) $, defined for every locally bounded predictable process $ H = ( H _ {t} , {\mathcal F} _ {t} ) $. One of the possible constructions of a stochastic integral is as follows. At first a stochastic integral is defined for simple predictable processes $ H $, of the form

$$ H _ {t} = h( \omega ) I _ {( a,b] } ( t),\ a < b, $$

where $ h $ is $ {\mathcal F} _ {a} $- measurable. In this case, by the stochastic integral $ \int _ {0} ^ {t} H _ {s} dX _ {s} $( or $ ( H \cdot X) _ {t} $, or $ \int _ {( t,0] } H _ {s} dX _ {s} $) one understands the variable

$$ h ( \omega ) ( X _ {b\wedge} t - X _ {a\wedge} t ). $$

The mapping $ H \mapsto H \cdot X $, where

$$ H \cdot X = ( H \cdot X) _ {t} ,\ t \geq 0, $$

permits an extension (also denoted by $ H \cdot X $) onto the set of all bounded predictable functions, which possesses the following properties:

a) the process $ ( H \cdot X) _ {t} $, $ t \geq 0 $, is continuous from the right and has limits from the left;

b) $ H \mapsto H \cdot X $ is linear, i.e.

$$ ( cH _ {1} + H _ {2} ) \cdot X = c( H _ {1} \cdot X) + H _ {2} \cdot X; $$

c) If $ \{ H ^ {n} \} $ is a sequence of uniformly-bounded predictable functions, $ H $ is a predictable function and

$$ \sup _ { s\leq } t | H _ {s} ^ {n} - H _ {s} | \mathop \rightarrow \limits ^ {\mathsf P} 0,\ t > 0, $$

then

$$ ( H ^ {n} \cdot X) _ {t} \mathop \rightarrow \limits ^ {\mathsf P} ( H \cdot X) _ {t} ,\ t > 0. $$

The extension $ H \cdot X $ is therefore unique in the sense that if $ H \mapsto \alpha ( H) $ is another mapping with the properties a)–c), then $ H \cdot X $ and $ \alpha ( H) $ are stochastically indistinguishable (cf. Stochastic indistinguishability).

The definition

$$ ( H \cdot X) _ {t} = h( \omega )( X _ {b\wedge} t - X _ {a\wedge} t ), $$

given for functions $ H _ {t} = h( \omega ) I _ {( a,b] } ( t) $ holds for any process $ X $, not only for semi-martingales. The extension $ H \cdot X $ with properties a)–c) onto the class of bounded predictable processes is only possible for the case where $ X $ is a semi-martingale. In this sense, the class of semi-martingales is the maximal class for which a stochastic integral with the natural properties a)–c) is defined.

If $ X $ is a semi-martingale and $ T = T( \omega ) $ is a Markov time (stopping time), then the "stopped" process $ X ^ {T} = ( X _ {t\wedge} T , {\mathcal F} _ {t} ) $ is also a semi-martingale and for every predictable bounded process $ H $,

$$ ( H \cdot X) ^ {T} = H \cdot X ^ {T} = \ ( HI _ {[[ 0,T ]] } ) \cdot X . $$

This property enables one to extend the definition of a stochastic integral to the case of locally-bounded predictable functions $ H $. If $ T _ {n} $ is a localizing (for $ H $) sequence of Markov times, then the $ H ^ {T _ {n} } $ are bounded. Hence, the $ H \cdot I _ {[[ 0,T _ {n} ]] } $ are bounded and

$$ [ ( HI _ {[[ 0, T _ {n+1} ]] } ) \cdot X ] ^ {T _ {n} } $$

is stochastically indistinguishable from $ HI _ {[[ 0,T _ {n} ]] } \cdot X $. A process $ H \cdot X $, again called a stochastic integral, therefore exists, such that

$$ ( H \cdot X) ^ {T _ {n} } = \ HI _ {[[ 0,T _ {n} ]] } \cdot X,\ n \geq 0. $$

The constructed stochastic integral $ H \cdot X $ possesses the following properties: $ H \cdot X $ is a semi-martingale; the mapping $ H \mapsto H \cdot X $ is linear; if $ X $ is a process of locally bounded variation, then so is the integral $ H \cdot X $, and $ H \cdot X $ then coincides with the Stieltjes integral of $ H $ with respect to $ dX $; $ \Delta ( H \cdot X) = H \Delta X $; $ K \cdot ( H \cdot X) = ( KH) \cdot X $.

Depending on extra assumptions concerning $ X $, the stochastic integral $ H \cdot X $ can also be defined for broader classes of functions $ H $. For example, if $ X $ is a locally square-integrable martingale, then a stochastic integral $ H \cdot X $( with the properties a)–c)) can be defined for any predictable process $ H $ that possesses the property that the process

$$ \left ( \int\limits _ { 0 } ^ { t } H _ {s} ^ {2} d\langle X\rangle _ {s} \right ) _ {t \geq 0 } $$

is locally integrable (here $ \langle X\rangle $ is the quadratic variation of $ X $, i.e. the predictable increasing process such that $ X ^ {2} - \langle X\rangle $ is a local martingale).

References

[J]	J. Jacod, "Calcul stochastique et problèmes de martingales" , Lect. notes in math. , 714 , Springer (1979) MR0542115 Zbl 0414.60053
[DM]	C. Dellacherie, P.A. Meyer, "Probabilities and potential" , A-C , North-Holland (1978–1988) (Translated from French) MR0939365 MR0898005 MR0727641 MR0745449 MR0566768 MR0521810 Zbl 0716.60001 Zbl 0494.60002 Zbl 0494.60001
[LS]	R.Sh. Liptser, A.N. Shiryayev, "Theory of martingales" , Kluwer (1989) (Translated from Russian) MR1022664 Zbl 0728.60048

Comments

The result alluded to above, that semi-martingales constitute the widest viable class of stochastic integrators, is the Bichteler–Dellacherie theorem [B]–[D], and can be formulated as follows [P], Thm. III.22. Call a process elementary predictable if it has a representation

$$ H _ {t} = H _ {0} I _ {\{ 0 \} } ( t)+ \sum _ { i=1} ^ { n } H _ {i} I _ {( T _ {i} , T _ {i+1} ] } ( t) , $$

where $ 0 = T _ {0} \leq T _ {1} \leq \dots \leq T _ {n+1} < \infty $ are stopping times and $ H _ {i} $ is $ {\mathcal F} _ {T _ {i} } $- measurable with $ | H _ {i} | < \infty $ a.s., $ 0< i< n $. Let $ E $ be the set of elementary predictable processes, topologized by uniform convergence in $ ( t, \omega ) $. Let $ L ^ {0} $ be the set of finite-valued random variables, topologized by convergence in probability. Fix a stochastic process $ X $ and for each stopping time $ T $ define a mapping $ I _ {X} ^ {T} : E \rightarrow L ^ {0} $ by

$$ I _ {X} ^ {T} ( H) = H _ {0} X _ {0} ^ {T} + \sum _ { i=1} ^ { n } H _ {i} ( X _ {T _ {i+1} } ^ {T} - X _ {T _ {i} } ^ {T} ), $$

where $ X ^ {T} $ denotes the process $ X _ {t} ^ {T} = X _ {t\wedge T } $. Say that "X has the property (C)" if $ I _ {X} ^ {T} $ is continuous for all stopping times.

The Bichteler–Dellacherie theorem: $ X $ has property (C) if and only if $ X $ is a semi-martingale.

Since the topology on $ E $ is very strong and that on $ L ^ {0} $ very weak, property (C) is a minimal requirement if the definition of $ I _ {X} ^ {T} $ is to be extended beyond $ E $.

It is possible to use property (C) as the definition of a semi-martingale, and to develop the theory of stochastic integration from this point of view [P]. There are many excellent textbook expositions of stochastic integration from the conventional point of view; see, e.g., [CW]–[RW].

References

[B]	K. Bichteler, "Stochastic integrators" Bull. Amer. Math. Soc. , 1 (1979) pp. 761–765 MR0537627 Zbl 0416.60066
[B2]	K. Bichteler, "Stochastic integrators and the theory of semimartingales" Ann. Probab. , 9 (1981) pp. 49–89
[D]	C. Dellacherie, "Un survol de la théorie de l'intégrale stochastique" Stoch. Proc. & Appl. , 10 (1980) pp. 115–144 MR0587420 MR0562680 MR0577985 Zbl 0436.60043 Zbl 0429.60053 Zbl 0427.60055
[P]	P. Protter, "Stochastic integration and differential equations" , Springer (1990) MR1037262 Zbl 0694.60047
[CW]	K.L. Chung, R.J. Williams, "Introduction to stochastic integration" , Birkhäuser (1990) MR1102676 Zbl 0725.60050
[E]	R.J. Elliott, "Stochastic calculus and applications" , Springer (1982) MR0678919 Zbl 0503.60062
[KS]	I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988) MR0917065 Zbl 0638.60065
[RW]	L.C.G. Rogers, D. Williams, "Diffusions, Markov processes and martingales" , II. Ito calculus , Wiley (1987) MR0921238 Zbl 0627.60001
[McK]	H.P. McKean jr., "Stochastic integrals" , Acad. Press (1969)
[MP]	M. Metivier, J. Pellaumail, "Stochastic integration" , Acad. Press (1980) MR0578177 Zbl 0463.60004
[McSh]	E.J. McShane, "Stochastic calculus and stochastic models" , Acad. Press (1974)
[R]	M.M. Rao, "Stochastic processes and integration" , Sijthoff & Noordhoff (1979) MR0546709 Zbl 0429.60001
[SV]	D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979) MR0532498 Zbl 0426.60069
[K]	P.E. Kopp, "Martingales and stochastic integrals" , Cambridge Univ. Press (1984) MR0774050 Zbl 0537.60047
[F]	M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980) MR0569058 Zbl 0422.31007
[AFHL]	S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, "Nonstandard methods in stochastic analysis and mathematical physics" , Acad. Press (1986) MR0859372 Zbl 0605.60005