The Taylor formula plays a very significant role in numerical analysis. We can obtain the approximation of a sufficiently smooth function in a neighborhood of a given point to any desired order of accuracy with the Taylor formula.

Enlarging the increments of smooth functions of Ito processes, it is beneficial to have a stochastic expansion formula with correspondent specialities to the deterministic Taylor formula. Such a stochastic Taylor formula has some possibilities. One of these possibilities is an Ito-Taylor expansion obtained via Ito’s formula [7].

### Ito-Taylor expansion

First we can obtain an Ito-Taylor expansion for the stochastic case. Consider

$$\begin{aligned} dX(t)=f\bigl(X(t)\bigr)\,dt+g\bigl(X(t)\bigr)\,dW(t), \end{aligned}$$

(3)

where *f* and *g* satisfy a linear growth bound and are sufficiently smooth.

Let *F* be a twice continuously differentiable function of \(X(t)\), then from Ito’s lemma we hav

$$\begin{aligned} dF\bigl[X(t)\bigr]&= \biggl\{ f\bigl[X(t)\bigr]\frac{\partial{{F[X(t)]}}}{\partial{{X}}}+ \frac {1}{2}g^{2}\bigl[X(t)\bigr]\frac{\partial^{2}{{F[X(t)]}}}{\partial{{X^{2}}}} \biggr\} \,dt \\ &\quad{}+g\bigl[X(t)\bigr]\frac{\partial{{F[X(t)]}}}{\partial{{X}}}\,dW(t). \end{aligned}$$

(4)

Defining the following operators:

$$\begin{aligned}& {L^{0}} \equiv f\bigl[X(t)\bigr]\frac{\partial}{\partial{{X}}}+ \frac {1}{2}g^{2}\bigl[X(t)\bigr]\frac{\partial^{2}}{\partial{{X^{2}}}}, \end{aligned}$$

(5)

$$\begin{aligned}& {L^{1}} \equiv g\bigl[X(t)\bigr]\frac{\partial}{\partial{{X}}}, \end{aligned}$$

(6)

(4) becomes

$$\begin{aligned} dF\bigl[X(t)\bigr]={L^{0}} {F\bigl[X(t) \bigr]}\,dt+{L^{1}} {F\bigl[X(t)\bigr]}\,dW(t), \end{aligned}$$

(7)

and integral form of (7) is

$$\begin{aligned} F\bigl[X(t)\bigr]=F\bigl[X(t_{0})\bigr]+ \int_{t_{0}}^{t} { L^{0} F\bigl[X(\tau)\bigr] }\,d \tau+ \int _{t_{0}}^{t}{ L^{1} F\bigl[X(\tau)} \bigr]\,dW(\tau). \end{aligned}$$

(8)

Choosing \(F(x)=x\), \(F(x)=f(x)\) and \(F(x)=g(x)\), (4) becomes respectively

$$\begin{aligned}& X(t) = X(t_{0})+ \int_{t_{0}}^{t} {f\bigl[X(\tau)\bigr]}\,d\tau+ \int_{t_{0}}^{t}{ g\bigl[X(\tau )\bigr]}\,dW(\tau), \end{aligned}$$

(9)

$$\begin{aligned}& f\bigl[X(t)\bigr] = f\bigl[X(t_{0})\bigr]+ \int_{t_{0}}^{t} { L^{0} f\bigl[X(\tau)\bigr] }\,d\tau+ \int _{t_{0}}^{t}{ L^{1} f\bigl[X(\tau)} \bigr]\,dW(\tau), \end{aligned}$$

(10)

$$\begin{aligned}& g\bigl[X(t)\bigr] = g\bigl[X(t_{0})\bigr]+ \int_{t_{0}}^{t} { L^{0} g\bigl[X(\tau)\bigr] }\,ds+ \int_{t_{0}}^{t}{ L^{1}g\bigl[X(\tau)} \bigr]\,dW(\tau). \end{aligned}$$

(11)

Substituting Eqs. (10) and (11) into (9), we obtain the following equation:

$$\begin{aligned} \begin{aligned}[b] X(t)&=X(t_{0}) + \int_{t_{0}}^{t} \biggl( f\bigl[X(t_{0}) \bigr]+ \int_{t_{0}}^{\tau_{1}}{ L^{0} f\bigl[X( \tau_{2})\bigr] }\,d\tau_{2} \\ &\quad{}+ \int_{t_{0}}^{\tau_{1}}{ L^{1} f\bigl[X( \tau_{2})\bigr]}\,dW(\tau_{2}) \biggr) \,d \tau_{1} \\ &\quad{}+ \int_{t_{0}}^{t} \biggl(g\bigl[X(t_{0})\bigr]+ \int_{t_{0}}^{\tau_{1}}{ L^{0} g\bigl[X( \tau_{2})\bigr] }\,d\tau_{2} \\ &\quad{}+ \int_{t_{0}}^{\tau_{1}}{ L^{1} g\bigl[X( \tau_{2})\bigr]}\,dW(\tau_{2}) \biggr) \,dW( \tau_{1} ); \end{aligned} \end{aligned}$$

(12)

and therefore,

$$\begin{aligned} X(t) =&X(t_{0})+f\bigl[X(t_{0})\bigr] \int_{t_{0}}^{t}{d\tau_{1}}+g \bigl[X(t_{0})\bigr] \int _{t_{0}}^{t}{dW( \tau_{1} )}+\mathcal{R,} \end{aligned}$$

(13)

where \(\mathcal{R}\) is the remaining terms which include the double integral terms:

$$ \begin{aligned}[b]\mathcal{R} &\equiv \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} {L^{0} f\bigl[X(\tau _{2})\bigr] }\,d\tau_{2} \,d \tau_{1} + \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} {L^{1} f\bigl[X( \tau_{2})\bigr] }\,dW(\tau_{2}) \,d \tau_{1} \\ &\quad {}+ \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} {L^{0} g\bigl[X( \tau_{2})\bigr] }\,d\tau_{2} \,d W(\tau_{1}) + \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} {L^{1} g\bigl[X( \tau_{2})\bigr] }\,dW(\tau_{2}) \,dW( \tau_{1}). \end{aligned} $$

(14)

Selecting \(F=L^{1} g\) in (8), we obtain

$$ \begin{aligned}[b] &\int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} {L^{1} g\bigl[X( \tau_{2})\bigr] }\,dW(\tau_{2}) \,dW( \tau_{1})\\ &\quad = \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} \biggl( {L^{1} g \bigl[X(t_{0})\bigr] } \\ &\qquad{}+ \int_{t_{0}}^{\tau_{2}} {L^{0} L^{1} g \bigl[X(\tau_{3})\bigr] }\,d\tau_{3}+ \int _{t_{0}}^{\tau_{2}} {L^{1} L^{1} g \bigl[X(\tau_{3})\bigr] }\,dW(\tau_{3}) \biggr)\,dW(\tau _{2})\,dW(\tau_{1}), \end{aligned} $$

(15)

and using \(L^{1} g =g g'\), we have

$$\begin{aligned} \begin{aligned}[b] X(t)&=X(t_{0})+f \bigl[X(t_{0})\bigr] \int_{t_{0}}^{t}{d\tau_{1}} \\ &\quad{}+g\bigl[X(t_{0})\bigr] \int_{t_{0}}^{t}{dW( \tau_{1} )}+g \bigl[X(t_{0})\bigr]g'\bigl[X(t_{0})\bigr] \int _{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}}\,dW(\tau_{2})\,dW( \tau_{1})+\mathcal{\widetilde{R,}} \end{aligned} \end{aligned}$$

(16)

where our new remainder *R̃* is

$$\begin{aligned} \begin{aligned}[b] \mathcal{\widetilde{R}} &\equiv \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} {L^{0} f\bigl[X( \tau_{2})\bigr] }\,d\tau_{2} \,d \tau_{1} + \int_{t_{0}}^{t} \int_{t_{0}}^{\tau _{1}} {L^{1} f\bigl[X( \tau_{2})\bigr] }\,dW(\tau_{2}) \,d \tau_{1} \\ &\quad{}+ \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} {L^{0} g\bigl[X( \tau_{2})\bigr] }\,d\tau _{2} \,d W(\tau_{1}) \\ &\quad{}+ \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} \int_{t_{0}}^{\tau_{2}}{L^{0} L^{1} g \bigl[X(\tau_{3})\bigr] }\,d(\tau_{3})\,dW( \tau_{2}) \,dW( \tau_{1}) \\ &\quad{}+ \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} \int_{t_{0}}^{\tau_{2}}{L^{1} L^{1} g \bigl[X(\tau_{3})\bigr] }\,dW(\tau_{3})\,dW( \tau_{2}) \,dW( \tau_{1}). \end{aligned} \end{aligned}$$

(17)

Therefore, we obtained the Ito-Taylor expansion for process (3) as (16). Using Ito’s lemma again, we have

$$\begin{aligned} \int_{t_{0}}^{t} \int_{t_{0}}^{\tau_{1}} dW( \tau_{2}) \,dW( \tau_{1})=\frac {1}{2}\bigl[W(t)-W(t_{0}) \bigr]^{2} -\frac{1}{2}(t-t_{0} ), \end{aligned}$$

(18)

and writing (18) into (16), we obtain the stochastic Taylor expansion

$$\begin{aligned} \begin{aligned}[b] X(t)&=X(t_{0})+f \bigl[X(t_{0})\bigr] \int_{t_{0}}^{t}{d\tau_{1}}+g \bigl[X(t_{0})\bigr] \int _{t_{0}}^{t}{dW( \tau_{1} )} \\ &\quad{}+g\bigl[X(t_{0})\bigr]g'\bigl[X(t_{0}) \bigr]\biggl\{ \frac{1}{2}\bigl[W(t)-W(t_{0})\bigr]^{2} - \frac {1}{2}(t-t_{0} ) \biggr\} +\mathcal{\widetilde{R}}. \end{aligned} \end{aligned}$$

(19)

Therefore, we can produce the numerical integration scheme for the SDE from Ito-Taylor expansion (19) with a time discretization \(0=t_{0}< t_{1}<\cdots<t_{n}<\cdots<t_{N}=T\) of a time interval [0,T] as follows:

$$\begin{aligned} \begin{aligned}[b] X(t_{i+1})&=X(t_{i})+f \bigl(X(t_{i})\bigr)\Delta t +g\bigl(X(t_{i})\bigr)\Delta W_{i} \\ &\quad{} +\frac{1}{2}g\bigl(X(t_{i})\bigr)g' \bigl(X(t_{i})\bigr)\bigl[(\Delta{W_{i}})^{2}- \Delta t\bigr]+\mathcal{\widetilde{R}}, \end{aligned} \end{aligned}$$

(20)

where \(\Delta t=t_{i+1}-t_{i} \) and \(\Delta W_{i}=W(t_{i+1})-W(t_{i})\) for \(i=0,1,2, \ldots,N-1 \) with the initial condition \(X(t_{0})=X_{0}\). The random variables \(\Delta W_{i}\) are independent \(N(0,\Delta t)\) normally distributed random variables.

### Euler-Maruyama method

One of the simplest numerical approximations for the SDE is the Euler-Maruyama method. If we truncate Ito’s formula of the stochastic Taylor series after the first order terms, we obtain the Euler method or Euler-Maruyama method as follows:

$$\begin{aligned} X(t_{i+1})=X(t_{i})+f\bigl(X(t_{i}) \bigr)\Delta t +g\bigl(X(t_{i})\bigr)\Delta W_{i} \end{aligned}$$

(21)

for \(i=0,1,2,\ldots,N-1\) with the initial value \(X(t_{0})=X_{0}\). Euler-Maruyama approximation converges with strong order 0.5 under Lipschitz and bounded growth conditions on the coefficients *f* and *g*, which were shown in [15]. [16] and [17] showed that an Euler-Maruyama approximation of an Ito process converges with weak order 1.0 under conditions of sufficient smoothness. It is clear that weak order of convergence is greater than strong order of convergence in the Euler-Maruyama method.

### Milstein method

The other numerical approximation method for the SDE is Milstein method. If we truncate the stochastic Taylor series after second order terms, we obtain the Milstein method as follows:

$$\begin{aligned} \begin{aligned}[b] X(t_{i+1})&=X(t_{i})+f \bigl(X(t_{i})\bigr)\Delta t +g\bigl(X(t_{i})\bigr)\Delta W_{i} \\ &\quad{}+\frac{1}{2}g\bigl(X(t_{i})\bigr)g' \bigl(X(t_{i})\bigr)\bigl[(\Delta{W_{i}})^{2}- \Delta t\bigr] \end{aligned} \end{aligned}$$

(22)

for \(i=0,1,2,\ldots,N-1\) with the initial value \(X(t_{0})=X_{0}\). Milstein approximation converges with strong order 1.0 under the \(E[X(0)]^{2}<\infty\) assumption, where *f* and *g* are twice continuously differentiable, and *f*, \(f'\), *g*, \(g'\) satisfy a uniform Lipschitz condition.

Note that \(g'(X(t_{i}))\) is differentiation of \(g(X(t_{i}))\), and if the type of SDE is an additive noise SDE, then the Milstein method leads to the Euler-Maruyama method.