View source for Stochastic integral

{{MSC|60H05}}

[[Category:Stochastic analysis]]

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>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901306.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901307.png" />-measurable. In this case, by the stochastic integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901308.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s0901309.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013010.png" />) one understands the variable

<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/s09013011.png" /></td> </tr></table>

The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013012.png" />, where

<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/s09013013.png" /></td> </tr></table>

permits an extension (also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013014.png" />) onto the set of all bounded predictable functions, which possesses the following properties:

a) the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013016.png" />, is continuous from the right and has limits from the left;

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013017.png" /> is linear, i.e.

<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/s09013018.png" /></td> </tr></table>

c) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013019.png" /> is a sequence of uniformly-bounded predictable functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013020.png" /> is a predictable function and

<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/s09013021.png" /></td> </tr></table>

then

<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/s09013022.png" /></td> </tr></table>

The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013023.png" /> is therefore unique in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013024.png" /> is another mapping with the properties a)–c), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013026.png" /> are stochastically indistinguishable (cf. [[Stochastic indistinguishability|Stochastic indistinguishability]]).

The definition

<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/s09013027.png" /></td> </tr></table>

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 "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>

This property enables one to extend the definition of a stochastic integral to the case of locally-bounded predictable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013037.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013038.png" /> is a localizing (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013039.png" />) sequence of Markov times, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013040.png" /> are bounded. Hence, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013041.png" /> are bounded and

<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/s09013042.png" /></td> </tr></table>

is stochastically indistinguishable from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013043.png" />. A process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013044.png" />, again called a stochastic integral, therefore exists, such that

<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/s09013045.png" /></td> </tr></table>

The constructed stochastic integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013046.png" /> possesses the following properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013047.png" /> is a semi-martingale; the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013048.png" /> is linear; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013049.png" /> is a process of locally bounded variation, then so is the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013050.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013051.png" /> then coincides with the Stieltjes integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013052.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013053.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013054.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013055.png" />.

Depending on extra assumptions concerning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013056.png" />, the stochastic integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013057.png" /> can also be defined for broader classes of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013058.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013059.png" /> is a locally square-integrable martingale, then a stochastic integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013060.png" /> (with the properties a)–c)) can be defined for any predictable process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013061.png" /> that possesses the property that the process

<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/s09013062.png" /></td> </tr></table>

is locally integrable (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013063.png" /> is the quadratic variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013064.png" />, i.e. the predictable increasing process such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013065.png" /> is a local martingale).

====References====
{|
|valign="top"|{{Ref|J}}|| J. Jacod, "Calcul stochastique et problèmes de martingales" , ''Lect. notes in math.'' , '''714''' , Springer (1979) {{MR|0542115}} {{ZBL|0414.60053}}
|-
|valign="top"|{{Ref|DM}}|| 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}}
|-
|valign="top"|{{Ref|LS}}|| R.Sh. Liptser, A.N. Shiryayev, "Theory of martingales" , Kluwer (1989) (Translated from Russian) {{MR|1022664}} {{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 {{Cite|B}}–{{Cite|D}}, and can be formulated as follows {{Cite|P}}, Thm. III.22. Call a process elementary predictable if it has a representation

<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/s09013066.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013067.png" /> are stopping times and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013068.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013069.png" />-measurable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013070.png" /> a.s., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013071.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013072.png" /> be the set of elementary predictable processes, topologized by uniform convergence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013073.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013074.png" /> be the set of finite-valued random variables, topologized by convergence in probability. Fix a stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013075.png" /> and for each stopping time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013076.png" /> define a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013077.png" /> by

<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 "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.

Since the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013084.png" /> is very strong and that on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013085.png" /> very weak, property (C) is a minimal requirement if the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013086.png" /> is to be extended beyond <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090130/s09013087.png" />.

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 {{Cite|P}}. There are many excellent textbook expositions of stochastic integration from the conventional point of view; see, e.g., {{Cite|CW}}–{{Cite|RW}}.

====References====
{|
|valign="top"|{{Ref|B}}|| K. Bichteler, "Stochastic integrators" ''Bull. Amer. Math. Soc.'' , '''1''' (1979) pp. 761–765 {{MR|0537627}} {{ZBL|0416.60066}}
|-
|valign="top"|{{Ref|B2}}|| 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
|-
|valign="top"|{{Ref|D}}|| C. Dellacherie, "Un survol de la théorie de l'intégrale stochastique" ''Stoch. Proc. &amp; Appl.'' , '''10''' (1980) pp. 115–144 {{MR|0587420}} {{MR|0562680}} {{MR|0577985}} {{ZBL|0436.60043}} {{ZBL|0429.60053}} {{ZBL|0427.60055}}
|-
|valign="top"|{{Ref|P}}|| P. Protter, "Stochastic integration and differential equations" , Springer (1990) {{MR|1037262}} {{ZBL|0694.60047}}
|-
|valign="top"|{{Ref|CW}}|| K.L. Chung, R.J. Williams, "Introduction to stochastic integration" , Birkhäuser (1990) {{MR|1102676}} {{ZBL|0725.60050}}
|-
|valign="top"|{{Ref|E}}|| R.J. Elliott, "Stochastic calculus and applications" , Springer (1982) {{MR|0678919}} {{ZBL|0503.60062}}
|-
|valign="top"|{{Ref|KS}}|| I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988) {{MR|0917065}} {{ZBL|0638.60065}}
|-
|valign="top"|{{Ref|RW}}|| L.C.G. Rogers, D. Williams, "Diffusions, Markov processes and martingales" , '''II. Ito calculus''' , Wiley (1987) {{MR|0921238}} {{ZBL|0627.60001}}
|-
|valign="top"|{{Ref|McK}}|| H.P. McKean jr., "Stochastic integrals" , Acad. Press (1969)
|-
|valign="top"|{{Ref|MP}}|| M. Metivier, J. Pellaumail, "Stochastic integration" , Acad. Press (1980) {{MR|0578177}} {{ZBL|0463.60004}}
|-
|valign="top"|{{Ref|McSh}}|| E.J. McShane, "Stochastic calculus and stochastic models" , Acad. Press (1974)
|-
|valign="top"|{{Ref|R}}|| M.M. Rao, "Stochastic processes and integration" , Sijthoff &amp; Noordhoff (1979) {{MR|0546709}} {{ZBL|0429.60001}}
|-
|valign="top"|{{Ref|SV}}|| D.W. Stroock, S.R.S. Varadhan, "Multidimensional diffusion processes" , Springer (1979) {{MR|0532498}} {{ZBL|0426.60069}}
|-
|valign="top"|{{Ref|K}}|| P.E. Kopp, "Martingales and stochastic integrals" , Cambridge Univ. Press (1984) {{MR|0774050}} {{ZBL|0537.60047}}
|-
|valign="top"|{{Ref|F}}|| M. Fukushima, "Dirichlet forms and Markov processes" , North-Holland (1980) {{MR|0569058}} {{ZBL|0422.31007}}
|-
|valign="top"|{{Ref|AFHL}}|| 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}}
|}