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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s0904501.png" /> be continuous semi-martingales (cf. [[Semi-martingale|Semi-martingale]]) defined on a filtered [[Probability space|probability space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s0904502.png" />. The Stratonovich integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s0904503.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s0904504.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s0904505.png" /> is defined as
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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/s090450/s0904506.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where the integral on the right-hand side is the Itô stochastic integral and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s0904507.png" /> denotes the quadratic cross-variation process of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s0904508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s0904509.png" />. There is no universally agreed notation for the integral, but the above is the most common (see [[#References|[a1]]], [[#References|[a2]]], for example). It is also known as the Fisk, Fisk–Stratonovich or symmetrized stochastic integral, the latter in view of the property that
+
Let  $  ( X, Y)=( X( t), Y( t)) _ {t \geq  0 }  $
 +
be continuous semi-martingales (cf. [[Semi-martingale|Semi-martingale]]) defined on a filtered [[Probability space|probability space]] $  ( \Omega , {\mathcal F} , {\mathcal F} _ {t} , P) $.  
 +
The Stratonovich integral of $  Y $
 +
with respect to  $  X $
 +
on the interval  $  [ 0, t] $
 +
is defined as
  
<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/s090450/s09045010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a1 }
 +
\int\limits _ { 0 } ^ { t }  Y( s) \circ  dX( s)  = \int\limits Y( s)  dX( s)+
  
<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/s090450/s09045011.png" /></td> </tr></table>
+
\frac{1 }{2 }
 +
\langle  X, Y \rangle _ {t} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045012.png" /> denotes a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045014.png" />, and the limit is in probability. Indeed, this property was the original definition of the integral, [[#References|[a3]]], [[#References|[a4]]]. As an immediate application of (a2) one sees that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045015.png" />, and this points to the main feature of the Stratonovich integral, namely that the [[Itô formula|Itô formula]] expressed in terms of Stratonovich integrals coincides with the "ordinary"  (Newton–Leibniz) formula. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045016.png" /> be continuous semi-martingales, defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045017.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045018.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045019.png" />-function. The Itô formula is
+
where the integral on the right-hand side is the Itô stochastic integral and  $  \langle  X, Y \rangle _ {t} $
 +
denotes the quadratic cross-variation process of $  X $
 +
and  $  Y $.
 +
There is no universally agreed notation for the integral, but the above is the most common (see [[#References|[a1]]], [[#References|[a2]]], for example). It is also known as the Fisk, Fisk–Stratonovich or symmetrized stochastic integral, the latter in view of the property 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/s090450/s09045020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
$$ \tag{a2 }
 +
\int\limits _ { 0 } ^ { t }  Y( s) \circ  dX( s) =
 +
$$
  
<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/s090450/s09045021.png" /></td> </tr></table>
+
$$
 +
= \
 +
\lim\limits _ {| \Delta | \rightarrow 0 }  \sum _ {i= 1 } ^ { n }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045022.png" />. If one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045023.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045024.png" /> it is readily verified that
+
\frac{Y( t _ {i} )+ Y( t _ {i- 1 }  ) }{2}
 +
( X( t _ {i} )- X( t _ {i- 1 }  )),
 +
$$
  
<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/s090450/s09045025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
where  $  \Delta $
 +
denotes a partition  $  0= t _ {0} < t _ {1} < {} \dots < t _ {n} = t $,
 +
$  | \Delta | = \max _ {i} ( t _ {i} - t _ {i- 1 }  ) $,
 +
and the limit is in probability. Indeed, this property was the original definition of the integral, [[#References|[a3]]], [[#References|[a4]]]. As an immediate application of (a2) one sees that  $  \int _ {[ 0,t] }  X \circ  dX =( 1/2) X  ^ {2} ( t) $,
 +
and this points to the main feature of the Stratonovich integral, namely that the [[Itô formula|Itô formula]] expressed in terms of Stratonovich integrals coincides with the  "ordinary" (Newton–Leibniz) formula. Let  $  X  ^ {1} ( t) \dots X  ^ {d} ( t) $
 +
be continuous semi-martingales, defining  $  X( t)=( X  ^ {1} ( t) \dots X  ^ {d} ( t)) $,
 +
and let  $  f: \mathbf R  ^ {d} \rightarrow \mathbf R $
 +
be a  $  C  ^ {2} $-
 +
function. The Itô formula is
 +
 
 +
$$ \tag{a3 }
 +
f( X( t))- f( X( 0)) =
 +
$$
 +
 
 +
$$
 +
= \
 +
\sum _ {i= 1 } ^ { d }  \int\limits _ { 0 } ^ { t }  D _ {i} f( X( s))  dX  ^ {i} ( s) +
 +
\frac{1}{2}
 +
\sum _ {i,j= 1 } ^ { d }  \int\limits _ { 0 } ^ { t }  D _ {i} D _ {j} f( X( s))  d\langle  X  ^ {i} , X  ^ {j} \rangle _ {s} ,
 +
$$
 +
 
 +
where  $  D _ {i} = \partial  / \partial  x  ^ {i} $.  
 +
If one writes  $  Y( t)= D _ {i} f( X( t)) $,
 +
then for  $  f \in C  ^ {3} $
 +
it is readily verified that
 +
 
 +
$$ \tag{a4 }
 +
\langle  Y, X  ^ {i} \rangle _ {s= \sum _ {j= 1 } ^ { d }  \int\limits _ { 0 } ^ { t }  D _ {i} D _ {j} f( X( s)) d\langle  X  ^ {i} , X  ^ {j} \rangle _ {s} ,
 +
$$
  
 
and hence (a3) becomes simply
 
and hence (a3) becomes simply
  
<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/s090450/s09045026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
$$ \tag{a5 }
 +
f( X( t))- f( X( 0))  = \sum _ {i= 1 } ^ { d }  \int\limits _ { 0 } ^ { t }
 +
D _ {i} f( X( s)) \circ  dX  ^ {i} ( s)
 +
$$
  
— the formula of  "ordinary calculus" . Equations (a4), (a5) remain valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045027.png" /> [[#References|[a2]]], Thm. V20.
+
— the formula of  "ordinary calculus" . Equations (a4), (a5) remain valid for $  f \in C  ^ {2} $[[#References|[a2]]], Thm. V20.
  
 
At first sight the Stratonovich integral appears to be simply a notational trick to obtain (a5). However, it plays an important role in several areas of stochastic analysis, including:
 
At first sight the Stratonovich integral appears to be simply a notational trick to obtain (a5). However, it plays an important role in several areas of stochastic analysis, including:
  
a) Approximation of stochastic differential equations, [[#References|[a1]]], Sect. V1.7. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045028.png" /> be independent Brownian motions (cf. [[Brownian motion|Brownian motion]]) and consider the [[Stochastic differential equation|stochastic differential equation]]
+
a) Approximation of stochastic differential equations, [[#References|[a1]]], Sect. V1.7. Let $  W  ^ {1} ( t) \dots W  ^ {p} ( t) $
 +
be independent Brownian motions (cf. [[Brownian motion|Brownian motion]]) and consider the [[Stochastic differential equation|stochastic differential equation]]
 +
 
 +
$$ \tag{a6 }
 +
dX( t)  =  b( X( t))  dt+ \sigma ( X( t))  dW( t),
 +
$$
  
<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/s090450/s09045029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
where  $  b: \mathbf R  ^ {d} \rightarrow \mathbf R  ^ {d} $
 +
and  $  \sigma : \mathbf R  ^ {d} \rightarrow \mathbf R ^ {d \times p } $
 +
are bounded functions with  $  b _ {i} \in C  ^ {1} ( \mathbf R  ^ {d} ) $
 +
and  $  \sigma _ {ij }  \in C  ^ {2} ( \mathbf R  ^ {d} ) $.
 +
Written in terms of Stratonovich integrals this becomes
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045031.png" /> are bounded functions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045033.png" />. Written in terms of Stratonovich integrals this becomes
+
$$ \tag{a7 }
 +
dX( t)  = \widetilde{b}  ( X( t))  dt+ \sigma ( X( t)) \circ  dW( t),
 +
$$
  
<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/s090450/s09045034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
where  $  \widetilde{b}  ( x) = b( x)- \widehat{b}  ( x) $
 +
and
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045035.png" /> and
+
$$
 +
\widehat{b}  _ {i} ( x)  =
 +
\frac{1}{2}
 +
\sum _ {i,j } \sigma _ {jk }  ( x) D _ {k} \sigma _ {ij} ( x) .
 +
$$
  
<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/s090450/s09045036.png" /></td> </tr></table>
+
Now, let  $  W  ^ {(} n) ( s) $
 +
be a piecewise-linear approximation to the Brownian path, i.e.
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045037.png" /> be a piecewise-linear approximation to the Brownian path, i.e.
+
$$
 +
W  ^ {(} n) ( s)  =
 +
\frac{( t _ {i} - s) W( t _ {i-} 1 ) +( s- t _ {i-} 1 ) W( t _ {i} ) }{t _ {i} - t _ {i-} 1 }
 +
,
 +
$$
  
<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/s090450/s09045038.png" /></td> </tr></table>
+
$  s \in [ t _ {i-} 1 , t _ {i} ] $,
 +
for a partition  $  ( t _ {i} ) $
 +
of  $  [ 0, t] $
 +
as above, and consider the approximating sequence of random ordinary differential equations
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045039.png" />, for a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045041.png" /> as above, and consider the approximating sequence of random ordinary differential equations
+
$$ \tag{a8 }
 +
\dot{X}  ^ {(} n) ( s)  = \widetilde{b}  ( X  ^ {(} n) ( s))+ \sigma
 +
( X  ^ {(} n) ( s)) \dot{W}  ^ {(} n) ( s),
 +
$$
  
<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/s090450/s09045042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
where the dot denotes  $  d/ds $.  
 +
Then
  
where the dot denotes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045043.png" />. Then
+
$$
 +
\lim\limits _ {| \Delta | \rightarrow 0 }  {\mathsf E} \left ( \sup _ {0 \leq
 +
s \leq  t }  | X( s)- X  ^ {(} n) ( s) |  ^ {2} \right )  = 0.
 +
$$
  
<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/s090450/s09045044.png" /></td> </tr></table>
+
Thus, the approximation scheme (a8) derives naturally from the Stratonovich equation (a7), and not from the Itô equation (a6). In general, the limit obtained depends on the specific approximation to the Brownian path chosen, unless the  $  p $
 +
columns of  $  \sigma $,
 +
considered as vector fields (cf. [[Vector field|Vector field]]), commute, i.e.  $  \sum _ {mk} ( \sigma _ {ml} D _ {m} \sigma _ {kj} - \sigma _ {mj} D _ {m} \sigma _ {kl} ) = 0 $,
 +
$  l, j= 1 \dots p $,
 +
in which case the limit (a6) or (a7) is obtained for any  "reasonable" approximation to the Brownian path; in particular, this is true when  $  p= 1 $.
 +
Questions of this sort were first investigated by E. Wong and M. Zakai [[#References|[a5]]], and  $  \widehat{b}  ( x) $
 +
is sometimes known as the Wong–Zakai correction term.
  
Thus, the approximation scheme (a8) derives naturally from the Stratonovich equation (a7), and not from the Itô equation (a6). In general, the limit obtained depends on the specific approximation to the Brownian path chosen, unless the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045045.png" /> columns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045046.png" />, considered as vector fields (cf. [[Vector field|Vector field]]), commute, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045048.png" />, in which case the limit (a6) or (a7) is obtained for any "reasonable" approximation to the Brownian path; in particular, this is true when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045049.png" />. Questions of this sort were first investigated by E. Wong and M. Zakai [[#References|[a5]]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045050.png" /> is sometimes known as the Wong–Zakai correction term.
+
b) Support of diffusion processes, [[#References|[a1]]], Sect. V1.8, [[#References|[a6]]]. Consider the solution of (a6) or (a7) starting from a fixed point  $  x \in \mathbf R  ^ {d} $.
 +
It defines a measure $ {\mathcal P} $
 +
on the sample space  $  {\mathcal C}  ^ {d} = C([ 0, \infty ) ;  \mathbf R  ^ {d} ) $.  
 +
Let  $  {\mathcal S} $
 +
be the support of this measure, i.e. the smallest closed set in  $  {\mathcal C}  ^ {d} $
 +
with  $  {\mathcal P} $-
 +
measure  $  1 $.  
 +
Let  $  \Phi $
 +
be the set of $  C  ^  \infty  $-
 +
functions in  $  {\mathcal C}  ^ {p} $,
 +
and, for  $  \phi \in \Phi $,
 +
let  $  \zeta  ^  \phi  \in {\mathcal C}  ^ {d} $
 +
be the solution of the ordinary differential equation
  
b) Support of diffusion processes, [[#References|[a1]]], Sect. V1.8, [[#References|[a6]]]. Consider the solution of (a6) or (a7) starting from a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045051.png" />. It defines a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045052.png" /> on the sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045053.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045054.png" /> be the support of this measure, i.e. the smallest closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045055.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045056.png" />-measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045057.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045058.png" /> be the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045059.png" />-functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045060.png" />, and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045061.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045062.png" /> be the solution of the ordinary differential equation
+
$$
  
<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/s090450/s09045063.png" /></td> </tr></table>
+
\frac{d}{dt}
 +
\zeta _  \phi  ( t)  = \widetilde{b}  ( \zeta _  \phi  ( t))+ \sigma ( \zeta _  \phi  ( t)) \dot \phi  ( t) ,\  \zeta _  \phi  ( 0= x.
 +
$$
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045064.png" />. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045065.png" /> is just the closure of the set of  "outputs"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045066.png" /> of equation (a7) when the  "input"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045067.png" /> is replaced by smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045068.png" />. As in a), the Stratonovich formulation is required to preserve consistency between systems with  "smooth"  and  "Brownian"  inputs.
+
Then $  {\mathcal S}= \mathop{\rm cl} \{ {\zeta _  \phi  } : {\phi \in \Phi } \} $.  
 +
Thus $  {\mathcal S} $
 +
is just the closure of the set of  "outputs"   $ X( \cdot ) $
 +
of equation (a7) when the  "input"   $ W $
 +
is replaced by smooth functions $  \phi $.  
 +
As in a), the Stratonovich formulation is required to preserve consistency between systems with  "smooth"  and  "Brownian"  inputs.
  
c) Diffusion processes on manifolds. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045069.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045070.png" />-manifold (cf. [[Manifold|Manifold]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045071.png" /> be smooth vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045072.png" /> (cf. [[Vector field on a manifold|Vector field on a manifold]]), and fix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045073.png" />. Then there is a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045074.png" />-valued process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045075.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045076.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045077.png" />,
+
c) Diffusion processes on manifolds. Let $  M $
 +
be a $  C  ^  \infty  $-
 +
manifold (cf. [[Manifold|Manifold]]), let $  A _ {0} \dots A _ {p} $
 +
be smooth vector fields on $  M $(
 +
cf. [[Vector field on a manifold|Vector field on a manifold]]), and fix $  x _ {0} \in M $.  
 +
Then there is a unique $  M $-
 +
valued process $  X( t) $
 +
such that $  X( 0)= x _ {0} $
 +
and for $  f \in C  ^  \infty  ( M) $,
  
<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/s090450/s09045078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
+
$$ \tag{a9 }
 +
df( X( t)) =
 +
$$
  
<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/s090450/s09045079.png" /></td> </tr></table>
+
$$
 +
= \
 +
A _ {0} f( X( t))  dt+ \sum _ { j= } 1 ^ { p }  A _ {j} f( X( t)) \circ  dW  ^ {j} ( t).
 +
$$
  
It is essential to use the Stratonovich integral here; the same equation written with Itô integrals fails to provide a coordinate-independent description, since the rules of Itô calculus conflict with the ordinary rules of calculus relating different coordinate systems. It is immediate from (a9) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045080.png" /> and hence that the Itô form of (a9) is
+
It is essential to use the Stratonovich integral here; the same equation written with Itô integrals fails to provide a coordinate-independent description, since the rules of Itô calculus conflict with the ordinary rules of calculus relating different coordinate systems. It is immediate from (a9) that $  d\langle  A _ {j} f( X( \cdot )), W  ^ {j} \rangle _ {t} = A _ {j}  ^ {2} f( X( t)) $
 +
and hence that the Itô form of (a9) is
  
<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/s090450/s09045081.png" /></td> </tr></table>
+
$$
 +
df( X( t)) =
 +
$$
  
<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/s090450/s09045082.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left ( A _ {0} +
 +
\frac{1}{2}
 +
\sum _ { j= } 1 ^ { p }  A _ {j}  ^ {2} \right ) f( X( t))  dt + \sum _ { j= } 1 ^ { p }  A _ {j} f( X( t))  dW  ^ {j} ( t).
 +
$$
  
It follows readily that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090450/s09045083.png" /> is a [[Diffusion process|diffusion process]] with differential generator
+
It follows readily that $  X( t) $
 +
is a [[Diffusion process|diffusion process]] with differential generator
  
<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/s090450/s09045084.png" /></td> </tr></table>
+
$$
 +
{\mathcal A}  = A _ {0} +
 +
\frac{1}{2}
 +
\sum _ {1}  ^ {p} A _ {j}  ^ {2} .
 +
$$
  
 
A detailed account of the Stratonovich integral and its properties is contained in [[#References|[a2]]], Sect. V.5, including an extension of the definition to possibly discontinuous semi-martingales.
 
A detailed account of the Stratonovich integral and its properties is contained in [[#References|[a2]]], Sect. V.5, including an extension of the definition to possibly discontinuous semi-martingales.

Latest revision as of 08:23, 6 June 2020


Let $ ( X, Y)=( X( t), Y( t)) _ {t \geq 0 } $ be continuous semi-martingales (cf. Semi-martingale) defined on a filtered probability space $ ( \Omega , {\mathcal F} , {\mathcal F} _ {t} , P) $. The Stratonovich integral of $ Y $ with respect to $ X $ on the interval $ [ 0, t] $ is defined as

$$ \tag{a1 } \int\limits _ { 0 } ^ { t } Y( s) \circ dX( s) = \int\limits Y( s) dX( s)+ \frac{1 }{2 } \langle X, Y \rangle _ {t} , $$

where the integral on the right-hand side is the Itô stochastic integral and $ \langle X, Y \rangle _ {t} $ denotes the quadratic cross-variation process of $ X $ and $ Y $. There is no universally agreed notation for the integral, but the above is the most common (see [a1], [a2], for example). It is also known as the Fisk, Fisk–Stratonovich or symmetrized stochastic integral, the latter in view of the property that

$$ \tag{a2 } \int\limits _ { 0 } ^ { t } Y( s) \circ dX( s) = $$

$$ = \ \lim\limits _ {| \Delta | \rightarrow 0 } \sum _ {i= 1 } ^ { n } \frac{Y( t _ {i} )+ Y( t _ {i- 1 } ) }{2} ( X( t _ {i} )- X( t _ {i- 1 } )), $$

where $ \Delta $ denotes a partition $ 0= t _ {0} < t _ {1} < {} \dots < t _ {n} = t $, $ | \Delta | = \max _ {i} ( t _ {i} - t _ {i- 1 } ) $, and the limit is in probability. Indeed, this property was the original definition of the integral, [a3], [a4]. As an immediate application of (a2) one sees that $ \int _ {[ 0,t] } X \circ dX =( 1/2) X ^ {2} ( t) $, and this points to the main feature of the Stratonovich integral, namely that the Itô formula expressed in terms of Stratonovich integrals coincides with the "ordinary" (Newton–Leibniz) formula. Let $ X ^ {1} ( t) \dots X ^ {d} ( t) $ be continuous semi-martingales, defining $ X( t)=( X ^ {1} ( t) \dots X ^ {d} ( t)) $, and let $ f: \mathbf R ^ {d} \rightarrow \mathbf R $ be a $ C ^ {2} $- function. The Itô formula is

$$ \tag{a3 } f( X( t))- f( X( 0)) = $$

$$ = \ \sum _ {i= 1 } ^ { d } \int\limits _ { 0 } ^ { t } D _ {i} f( X( s)) dX ^ {i} ( s) + \frac{1}{2} \sum _ {i,j= 1 } ^ { d } \int\limits _ { 0 } ^ { t } D _ {i} D _ {j} f( X( s)) d\langle X ^ {i} , X ^ {j} \rangle _ {s} , $$

where $ D _ {i} = \partial / \partial x ^ {i} $. If one writes $ Y( t)= D _ {i} f( X( t)) $, then for $ f \in C ^ {3} $ it is readily verified that

$$ \tag{a4 } \langle Y, X ^ {i} \rangle _ {s} = \sum _ {j= 1 } ^ { d } \int\limits _ { 0 } ^ { t } D _ {i} D _ {j} f( X( s)) d\langle X ^ {i} , X ^ {j} \rangle _ {s} , $$

and hence (a3) becomes simply

$$ \tag{a5 } f( X( t))- f( X( 0)) = \sum _ {i= 1 } ^ { d } \int\limits _ { 0 } ^ { t } D _ {i} f( X( s)) \circ dX ^ {i} ( s) $$

— the formula of "ordinary calculus" . Equations (a4), (a5) remain valid for $ f \in C ^ {2} $[a2], Thm. V20.

At first sight the Stratonovich integral appears to be simply a notational trick to obtain (a5). However, it plays an important role in several areas of stochastic analysis, including:

a) Approximation of stochastic differential equations, [a1], Sect. V1.7. Let $ W ^ {1} ( t) \dots W ^ {p} ( t) $ be independent Brownian motions (cf. Brownian motion) and consider the stochastic differential equation

$$ \tag{a6 } dX( t) = b( X( t)) dt+ \sigma ( X( t)) dW( t), $$

where $ b: \mathbf R ^ {d} \rightarrow \mathbf R ^ {d} $ and $ \sigma : \mathbf R ^ {d} \rightarrow \mathbf R ^ {d \times p } $ are bounded functions with $ b _ {i} \in C ^ {1} ( \mathbf R ^ {d} ) $ and $ \sigma _ {ij } \in C ^ {2} ( \mathbf R ^ {d} ) $. Written in terms of Stratonovich integrals this becomes

$$ \tag{a7 } dX( t) = \widetilde{b} ( X( t)) dt+ \sigma ( X( t)) \circ dW( t), $$

where $ \widetilde{b} ( x) = b( x)- \widehat{b} ( x) $ and

$$ \widehat{b} _ {i} ( x) = \frac{1}{2} \sum _ {i,j } \sigma _ {jk } ( x) D _ {k} \sigma _ {ij} ( x) . $$

Now, let $ W ^ {(} n) ( s) $ be a piecewise-linear approximation to the Brownian path, i.e.

$$ W ^ {(} n) ( s) = \frac{( t _ {i} - s) W( t _ {i-} 1 ) +( s- t _ {i-} 1 ) W( t _ {i} ) }{t _ {i} - t _ {i-} 1 } , $$

$ s \in [ t _ {i-} 1 , t _ {i} ] $, for a partition $ ( t _ {i} ) $ of $ [ 0, t] $ as above, and consider the approximating sequence of random ordinary differential equations

$$ \tag{a8 } \dot{X} ^ {(} n) ( s) = \widetilde{b} ( X ^ {(} n) ( s))+ \sigma ( X ^ {(} n) ( s)) \dot{W} ^ {(} n) ( s), $$

where the dot denotes $ d/ds $. Then

$$ \lim\limits _ {| \Delta | \rightarrow 0 } {\mathsf E} \left ( \sup _ {0 \leq s \leq t } | X( s)- X ^ {(} n) ( s) | ^ {2} \right ) = 0. $$

Thus, the approximation scheme (a8) derives naturally from the Stratonovich equation (a7), and not from the Itô equation (a6). In general, the limit obtained depends on the specific approximation to the Brownian path chosen, unless the $ p $ columns of $ \sigma $, considered as vector fields (cf. Vector field), commute, i.e. $ \sum _ {mk} ( \sigma _ {ml} D _ {m} \sigma _ {kj} - \sigma _ {mj} D _ {m} \sigma _ {kl} ) = 0 $, $ l, j= 1 \dots p $, in which case the limit (a6) or (a7) is obtained for any "reasonable" approximation to the Brownian path; in particular, this is true when $ p= 1 $. Questions of this sort were first investigated by E. Wong and M. Zakai [a5], and $ \widehat{b} ( x) $ is sometimes known as the Wong–Zakai correction term.

b) Support of diffusion processes, [a1], Sect. V1.8, [a6]. Consider the solution of (a6) or (a7) starting from a fixed point $ x \in \mathbf R ^ {d} $. It defines a measure $ {\mathcal P} $ on the sample space $ {\mathcal C} ^ {d} = C([ 0, \infty ) ; \mathbf R ^ {d} ) $. Let $ {\mathcal S} $ be the support of this measure, i.e. the smallest closed set in $ {\mathcal C} ^ {d} $ with $ {\mathcal P} $- measure $ 1 $. Let $ \Phi $ be the set of $ C ^ \infty $- functions in $ {\mathcal C} ^ {p} $, and, for $ \phi \in \Phi $, let $ \zeta ^ \phi \in {\mathcal C} ^ {d} $ be the solution of the ordinary differential equation

$$ \frac{d}{dt} \zeta _ \phi ( t) = \widetilde{b} ( \zeta _ \phi ( t))+ \sigma ( \zeta _ \phi ( t)) \dot \phi ( t) ,\ \zeta _ \phi ( 0) = x. $$

Then $ {\mathcal S}= \mathop{\rm cl} \{ {\zeta _ \phi } : {\phi \in \Phi } \} $. Thus $ {\mathcal S} $ is just the closure of the set of "outputs" $ X( \cdot ) $ of equation (a7) when the "input" $ W $ is replaced by smooth functions $ \phi $. As in a), the Stratonovich formulation is required to preserve consistency between systems with "smooth" and "Brownian" inputs.

c) Diffusion processes on manifolds. Let $ M $ be a $ C ^ \infty $- manifold (cf. Manifold), let $ A _ {0} \dots A _ {p} $ be smooth vector fields on $ M $( cf. Vector field on a manifold), and fix $ x _ {0} \in M $. Then there is a unique $ M $- valued process $ X( t) $ such that $ X( 0)= x _ {0} $ and for $ f \in C ^ \infty ( M) $,

$$ \tag{a9 } df( X( t)) = $$

$$ = \ A _ {0} f( X( t)) dt+ \sum _ { j= } 1 ^ { p } A _ {j} f( X( t)) \circ dW ^ {j} ( t). $$

It is essential to use the Stratonovich integral here; the same equation written with Itô integrals fails to provide a coordinate-independent description, since the rules of Itô calculus conflict with the ordinary rules of calculus relating different coordinate systems. It is immediate from (a9) that $ d\langle A _ {j} f( X( \cdot )), W ^ {j} \rangle _ {t} = A _ {j} ^ {2} f( X( t)) $ and hence that the Itô form of (a9) is

$$ df( X( t)) = $$

$$ = \ \left ( A _ {0} + \frac{1}{2} \sum _ { j= } 1 ^ { p } A _ {j} ^ {2} \right ) f( X( t)) dt + \sum _ { j= } 1 ^ { p } A _ {j} f( X( t)) dW ^ {j} ( t). $$

It follows readily that $ X( t) $ is a diffusion process with differential generator

$$ {\mathcal A} = A _ {0} + \frac{1}{2} \sum _ {1} ^ {p} A _ {j} ^ {2} . $$

A detailed account of the Stratonovich integral and its properties is contained in [a2], Sect. V.5, including an extension of the definition to possibly discontinuous semi-martingales.

References

[a1] N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland (1989)
[a2] P. Protter, "Stochastic integrals and differential equations" , Springer (1990)
[a3] R.L. Stratonovich, "A new representation for stochastic integrals and equations" SIAM J. Control , 4 (1966) pp. 362–371
[a4] D.L. Fisk, "Quasi-martingales and stochastic integrals" Techn. Report Dept. Math. Michigan State Univ. , 1 (1963)
[a5] E. Wong, M. Zakai, "On the relation between ordinary and stochastic differential equations" Int. J. Engin. Sci. , 3 (1965) pp. 213–229
[a6] D.W. Stroock, S.R.S. Varadhan, "On the support of diffusion processes with applications to the strong maximum principle" , Proc. VI Berkeley Symp. Math. Statist. Probab. , III , Univ. California Press (1972) pp. 333–359
How to Cite This Entry:
Stratonovich integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stratonovich_integral&oldid=48870
This article was adapted from an original article by M.H.A. Davis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article