6.4.2 Stratonovich-Taylor expansion

The Stratonovich-Taylor expansion of a Stratonovich process of the form (

) is given by [40]

$\displaystyle X(t)$	$\textstyle =$	$\displaystyle X(t_0) + \bar a J_{(0)} + b J_{(1)} + + \bar a \bar a' J_{(0,0)} + \bar a b' J_{(0,1)} + b \bar a' J_{(1,0)}$
		$\displaystyle + b b' J_{(1,1)} + \bar a(\bar a\bar a''+\bar a'^2)J_{(0,0,0)} + \bar a(\bar ab''+\bar a'b')J_{(0,0,1)}$
		$\displaystyle + \bar a(\bar a''b+\bar a'b')J_{(0,1,0)} + b(\bar a\bar a''+\bar a'^2)J_{(1,0,0)}$
		$\displaystyle + \bar a(bb''+b'^2)J_{(0,1,1)} + b(\bar ab''+\bar a'b')J_{(1,0,1)}$
		$\displaystyle + b(\bar a''b+\bar a'b')J_{(1,1,0)} + b(bb''+b'^2)J_{(1,1,1)} + R$	(6.21)

$\displaystyle J_{(0)}$	$\textstyle =$	$\displaystyle I_{(0)}$
$\displaystyle J_{(1)}$	$\textstyle =$	$\displaystyle I_{(1)}$
$\displaystyle J_{(0,0)}$	$\textstyle =$	$\displaystyle I_{(0,0)}$
$\displaystyle J_{(0,1)}$	$\textstyle =$	$\displaystyle I_{(0,1)}$
$\displaystyle J_{(1,1)}$	$\textstyle =$	$\displaystyle I_{(1,1)} + \frac{1}{2}I_{(1,1)}I_{(0)}$
$\displaystyle J_{(j_1, j_2, j_3)}$	$\textstyle =$	$\displaystyle I_{(j_1, j_2, \ldots)} + \frac{1}{2} ( I_{\{j_1=j_2 \not= 0\}} I_{(0,j_3)}+ I_{\{j_2=j_3 \not= 0\}} I_{(j_1,0)} )$
	$\textstyle \vdots$