Difference between revisions of "Stochastic integral"
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− | An integral | + | An integral "∫ H dX" with respect to a [[Semi-martingale|semi-martingale]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901301.png" /> on some stochastic basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901302.png" />, defined for every locally bounded predictable process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901303.png" />. One of the possible constructions of a stochastic integral is as follows. At first a stochastic integral is defined for simple predictable processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901304.png" />, of the form |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901305.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901305.png" /></td> </tr></table> | ||
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given for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013028.png" /> holds for any process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013029.png" />, not only for semi-martingales. The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013030.png" /> with properties a)–c) onto the class of bounded predictable processes is only possible for the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013031.png" /> 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. | given for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013028.png" /> holds for any process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013029.png" />, not only for semi-martingales. The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013030.png" /> with properties a)–c) onto the class of bounded predictable processes is only possible for the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013031.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013032.png" /> is a semi-martingale and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013033.png" /> is a Markov time (stopping time), then the | + | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013032.png" /> is a semi-martingale and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013033.png" /> is a Markov time (stopping time), then the "stopped" process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013034.png" /> is also a semi-martingale and for every predictable bounded process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013035.png" />, |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013036.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013036.png" /></td> </tr></table> | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Jacod, "Calcul stochastique et problèmes de martingales" , ''Lect. notes in math.'' , '''714''' , Springer (1979) {{MR|0542115}} {{ZBL|0414.60053}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Dellacherie, P.A. Meyer, "Probabilities and potential" , '''A-C''' , North-Holland (1978–1988) (Translated from French) {{MR|0939365}} {{MR|0898005}} {{MR|0727641}} {{MR|0745449}} {{MR|0566768}} {{MR|0521810}} {{ZBL|0716.60001}} {{ZBL|0494.60002}} {{ZBL|0494.60001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.Sh. Liptser, A.N. [A.N. Shiryaev] Shiryayev, "Theory of martingales" , Kluwer (1989) (Translated from Russian) {{MR|1022664}} {{ZBL|0728.60048}} </TD></TR></table> |
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013078.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013078.png" /></td> </tr></table> | ||
− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013079.png" /> denotes the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013080.png" />. Say that | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013079.png" /> denotes the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013080.png" />. Say that "X has the property (C)" if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013081.png" /> is continuous for all stopping times. |
The Bichteler–Dellacherie theorem: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013082.png" /> has property (C) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013083.png" /> is a semi-martingale. | The Bichteler–Dellacherie theorem: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013082.png" /> has property (C) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013083.png" /> is a semi-martingale. | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Bichteler, "Stochastic integrators" ''Bull. Amer. Math. Soc.'' , '''1''' (1979) pp. 761–765 {{MR|0537627}} {{ZBL|0416.60066}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Bichteler, "Stochastic integrators and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013088.png" /> theory of semimartingales" ''Ann. Probab.'' , '''9''' (1981) pp. 49–89</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> C. Dellacherie, "Un survol de la théorie de l'intégrale stochastique" ''Stoch. Proc. & Appl.'' , '''10''' (1980) pp. 115–144 {{MR|0587420}} {{MR|0562680}} {{MR|0577985}} {{ZBL|0436.60043}} {{ZBL|0429.60053}} {{ZBL|0427.60055}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> P. Protter, "Stochastic integration and differential equations" , Springer (1990) {{MR|1037262}} {{ZBL|0694.60047}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> K.L. Chung, R.J. Williams, "Introduction to stochastic integration" , Birkhäuser (1990) {{MR|1102676}} {{ZBL|0725.60050}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> R.J. Elliott, "Stochastic calculus and applications" , Springer (1982) {{MR|0678919}} {{ZBL|0503.60062}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988) {{MR|0917065}} {{ZBL|0638.60065}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> L.C.G. Rogers, D. Williams, "Diffusions, Markov processes and martingales" , '''II. Ito calculus''' , Wiley (1987) {{MR|0921238}} {{ZBL|0627.60001}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> H.P. McKean jr., "Stochastic integrals" , Acad. Press (1969)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> M. Metivier, J. Pellaumail, "Stochastic integration" , Acad. Press (1980) {{MR|0578177}} {{ZBL|0463.60004}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> E.J. McShane, "Stochastic calculus and stochastic models" , Acad. Press (1974)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> M.M. Rao, "Stochastic processes and integration" , Sijthoff & Noordhoff (1979) {{MR|0546709}} {{ZBL|0429.60001}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979) {{MR|0532498}} {{ZBL|0426.60069}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> P.E. Kopp, "Martingales and stochastic integrals" , Cambridge Univ. Press (1984) {{MR|0774050}} {{ZBL|0537.60047}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980) {{MR|0569058}} {{ZBL|0422.31007}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, "Nonstandard methods in stochastic analysis and mathematical physics" , Acad. Press (1986) {{MR|0859372}} {{ZBL|0605.60005}} </TD></TR></table> |
Revision as of 10:32, 27 March 2012
An integral "∫ H dX" with respect to a semi-martingale on some stochastic basis
, defined for every locally bounded predictable process
. One of the possible constructions of a stochastic integral is as follows. At first a stochastic integral is defined for simple predictable processes
, of the form
![]() |
where is
-measurable. In this case, by the stochastic integral
(or
, or
) one understands the variable
![]() |
The mapping , where
![]() |
permits an extension (also denoted by ) onto the set of all bounded predictable functions, which possesses the following properties:
a) the process ,
, is continuous from the right and has limits from the left;
b) is linear, i.e.
![]() |
c) If is a sequence of uniformly-bounded predictable functions,
is a predictable function and
![]() |
then
![]() |
The extension is therefore unique in the sense that if
is another mapping with the properties a)–c), then
and
are stochastically indistinguishable (cf. Stochastic indistinguishability).
The definition
![]() |
given for functions holds for any process
, not only for semi-martingales. The extension
with properties a)–c) onto the class of bounded predictable processes is only possible for the case where
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 is a semi-martingale and
is a Markov time (stopping time), then the "stopped" process
is also a semi-martingale and for every predictable bounded process
,
![]() |
This property enables one to extend the definition of a stochastic integral to the case of locally-bounded predictable functions . If
is a localizing (for
) sequence of Markov times, then the
are bounded. Hence, the
are bounded and
![]() |
is stochastically indistinguishable from . A process
, again called a stochastic integral, therefore exists, such that
![]() |
The constructed stochastic integral possesses the following properties:
is a semi-martingale; the mapping
is linear; if
is a process of locally bounded variation, then so is the integral
, and
then coincides with the Stieltjes integral of
with respect to
;
;
.
Depending on extra assumptions concerning , the stochastic integral
can also be defined for broader classes of functions
. For example, if
is a locally square-integrable martingale, then a stochastic integral
(with the properties a)–c)) can be defined for any predictable process
that possesses the property that the process
![]() |
is locally integrable (here is the quadratic variation of
, i.e. the predictable increasing process such that
is a local martingale).
References
[1] | J. Jacod, "Calcul stochastique et problèmes de martingales" , Lect. notes in math. , 714 , Springer (1979) MR0542115 Zbl 0414.60053 |
[2] | 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 |
[3] | R.Sh. Liptser, A.N. [A.N. Shiryaev] 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 [a1]–[a3], and can be formulated as follows [a4], Thm. III.22. Call a process elementary predictable if it has a representation
![]() |
where are stopping times and
is
-measurable with
a.s.,
. Let
be the set of elementary predictable processes, topologized by uniform convergence in
. Let
be the set of finite-valued random variables, topologized by convergence in probability. Fix a stochastic process
and for each stopping time
define a mapping
by
![]() |
where denotes the process
. Say that "X has the property (C)" if
is continuous for all stopping times.
The Bichteler–Dellacherie theorem: has property (C) if and only if
is a semi-martingale.
Since the topology on is very strong and that on
very weak, property (C) is a minimal requirement if the definition of
is to be extended beyond
.
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 [a4]. There are many excellent textbook expositions of stochastic integration from the conventional point of view; see, e.g., [a5]–[a8].
References
[a1] | K. Bichteler, "Stochastic integrators" Bull. Amer. Math. Soc. , 1 (1979) pp. 761–765 MR0537627 Zbl 0416.60066 |
[a2] | K. Bichteler, "Stochastic integrators and the ![]() |
[a3] | 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 |
[a4] | P. Protter, "Stochastic integration and differential equations" , Springer (1990) MR1037262 Zbl 0694.60047 |
[a5] | K.L. Chung, R.J. Williams, "Introduction to stochastic integration" , Birkhäuser (1990) MR1102676 Zbl 0725.60050 |
[a6] | R.J. Elliott, "Stochastic calculus and applications" , Springer (1982) MR0678919 Zbl 0503.60062 |
[a7] | I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988) MR0917065 Zbl 0638.60065 |
[a8] | L.C.G. Rogers, D. Williams, "Diffusions, Markov processes and martingales" , II. Ito calculus , Wiley (1987) MR0921238 Zbl 0627.60001 |
[a9] | H.P. McKean jr., "Stochastic integrals" , Acad. Press (1969) |
[a10] | M. Metivier, J. Pellaumail, "Stochastic integration" , Acad. Press (1980) MR0578177 Zbl 0463.60004 |
[a11] | E.J. McShane, "Stochastic calculus and stochastic models" , Acad. Press (1974) |
[a12] | M.M. Rao, "Stochastic processes and integration" , Sijthoff & Noordhoff (1979) MR0546709 Zbl 0429.60001 |
[a13] | D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979) MR0532498 Zbl 0426.60069 |
[a14] | P.E. Kopp, "Martingales and stochastic integrals" , Cambridge Univ. Press (1984) MR0774050 Zbl 0537.60047 |
[a15] | M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980) MR0569058 Zbl 0422.31007 |
[a16] | 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 |
Stochastic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_integral&oldid=15041