6.4.1 Itô-Taylor expansion

The Itô-Taylor expansion of an Itô process of the form () is given by [40]

$$X(t) = X(t_0) + a I_{(0)} + b I_{(1)} + \left( a a' + \frac{1}{2} b^2 a'' \right) I_{(0,0)}$$

$$+ \left( a b' + \frac{1}{2} b^2 b'' \right) I_{(0,1)} + b a' I_{(1,0)} + b b' I_{(1,1)}$$

$$+ \Biggl[ a \left( a a'' + a'^2 + bb'a'' + \frac{1}{2} b^2 a''' \right)$$

$$\quad +\frac{1}{2} b^2 (a a''' + 3 a' a'' + (b'^2 + b b'')a'' + 2 b b' a''') +\frac{1}{4} b^4 a^{(4)} \Biggr] I_{(0,0,0)}$$

$$+ \Biggl[ a \left( a' b' + ab'' + bb'b'' + \frac{1}{2} b^2 b''' \right)$$

$$\quad +\frac{1}{2} b^2 ( a'' b' + 2 a' b'' + a b''' + (b'^2 + b b'')b'' + 2 b b' b''' ) +\frac{1}{2} b^2 b^{(4)} \Biggr] I_{(0,0,1)}$$

$$+ \left[ a \left( b' a' + ba'' \right) +\frac{1}{2} b^2 ( b'' a' + 2 b' a'' + b a''' \right] I_{(0,1,0)}$$

$$+ \left[ a \left( b'^2 + b b'' \right) +\frac{1}{2} b^2 ( b'' b' + 2 b b'' + b b''' \right] I_{(0,1,1)}$$

$$+ b \left( a a'' + a'^2 + b b' a'' +\frac{1}{2} b^2 a''' \right) I_{(1,0,0)}$$

$$+ b \left( a b'' + a'b' + b b' b'' +\frac{1}{2} b^2 b''' \right) I_{(1,0,1)}$$

$$+ b (a' b' + a'' b) I_{(1,1,0)} + b (b'^2 + b b'') I_{(1,1,1)} + R \quad.$$ (6.20)

The Itô integrals $I_{(j_1, j_2, \ldots)}$ are defined as

$$I_{(0)} = \int_{t_0}^{t} \,d{t'}\,$$

$$I_{(1)} = \int_{t_0}^{t} \,d{W(t')}\,$$

$$I_{(0,0)} = \int_{t_0}^{t}\int_{t_0}^{s} \,d{t'}\, \,d{s}\,$$

$$I_{(0,1)} = \int_{t_0}^{t}\int_{t_0}^{s} \,d{t'}\, \,d{W(s)}\,$$

$$I_{(1,1)} = \int_{t_0}^{t}\int_{t_0}^{s} \,d{W(t')}\, \,d{W(s)}\,$$

$$\vdots$$