📂Stochastic Differential Equations

Ito-Taylor Expansion Derivation

Theorem ¹

$$d X(t) = f \left( X_{t} \right) dt + g \left( X_{t} \right) d W_{t} \qquad , t \in [0, T]$$ Let’s assume the Ito Process is given as the solution to the above Autonomous Stochastic Differential Equation. If $f,g : \mathbb{R} \to \mathbb{R}$ satisfies the Linear Growth Condition, i.e., for some constant $K$, $\begin{cases} \left| f \left( X_{t} \right) \right| \le K \left( 1 + \left| X_{t} \right|^{2} \right) \\ \left| g \left( X_{t} \right) \right| \le K \left( 1 + \left| X_{t} \right|^{2} \right) \end{cases}$ is valid and sufficiently differentiable, then the following holds: $$X_{t} = X_{0} + f \left( X_{0} \right) \int_{0}^{t} ds + g \left( X_{0} \right) \int_{0}^{t} d W_{s} + R$$ Here, the remainder $R$ is as follows. $L^{k}$ is an operator that appears during the derivation process. $$R = \int_{0}^{t} L^{0} f \left( X_{z} \right) dz ds + \int_{0}^{t} L^{1} f \left( X_{z} \right) dW_{z} ds + \int_{0}^{t} L^{0} g \left( X_{z} \right) dz dW_{s} + \int_{0}^{t} L^{1} g \left( X_{z} \right) dW_{z} dW_{s}$$

Explanation

The Ito-Taylor expansion is also known as the Stochastic Taylor Formula. Formulaically, it can be considered as bringing out constants $f \circ X_{t}$ and $g \circ X_{t}$, which were inside the integral, evaluated at $t=0$, and bundling the resulting errors into $R$.

Derivation

$$X (t) = X_{0} + \int_{0}^{t} f(s) ds + \int_{0}^{t} g(s) d W_{s}$$ Let’s think of the integral form of the Ito Process as follows.

Ito’s Formula: Let’s assume the Ito Process $\left\{ X_{t} \right\}_{t \ge 0}$ is given. $$d X_{t} = u dt + v d W_{t}$$ If for function $V \left( t, X_{t} \right) = V \in C^{2} \left( [0,\infty) \times \mathbb{R} \right)$, $Y_{t} := V \left( t, X_{t} \right)$ is posed, then $\left\{ Y_{t} \right\}$ is also an Ito Process, and the following is valid. $$\begin{align*} d Y_{t} =& V_{t} dt + V_{x} d X_{t} + {{ 1 } \over { 2 }} V_{xx} \left( d X_{t} \right)^{2} \\ =& \left( V_{t} + V_{x} u + {{ 1 } \over { 2 }} V_{xx} v^{2} \right) dt + V_{x} v d W_{t} \end{align*}$$

If a function $h \in C^{2} \left( \mathbb{R} \right)$, which is twice differentiable and continuous, is applied to $X_{t}$, according to Ito’s Formula $$\begin{align*} h \left( X_{t} \right) =& h \left( X_{0} \right) + \int_{0}^{t} \left[ f \left( X_{s} \right) {{ \partial } \over { \partial X }} h \left( X_{s} \right) + {{ 1 } \over { 2 }} \left[ g \left( X_{s} \right) \right]^{2} {{ \partial^{2} } \over { \partial X^{2} }} h \left( X_{s} \right) \right] ds \\ & + \int_{0}^{t} g \left( X_{s} \right) {{ \partial } \over { \partial X }} h \left( X_{s} \right) d W_{s} \\ =& h \left( X_{0} \right) + \int_{0}^{t} L^{1} h \left( X_{s} \right) ds + \int_{0}^{t} L^{1} h \left( X_{s} \right) d W_{S} \end{align*}$$ Here, $L^{0}$ and $L^{1}$ are defined as follows Operator. $$\begin{align*} L^{0} &:= f {{ \partial } \over { \partial X }} + {{ 1 } \over { 2 }} g^{2} {{ \partial^{2} } \over { \partial X ^{2} }} \\ L^{1} &:= g {{ \partial } \over { \partial X }} \end{align*}$$ Formally applying them to $h = f$ and $h = g$, and then substituting into the original integral form of the given Ito Process’ $f \left( X_{t} \right)$ and $g \left( X_{t} \right)$, yields the following. $$\begin{align*} X (t) =& X_{0} + \int_{0}^{t} f(s) ds + \int_{0}^{t} g(s) d W_{s} \\ =& X_{0} + \int_{0}^{t} \left( {\color{Red} f \left( X_{0} \right)} + \int_{0}^{s} L^{1} f \left( X_{z} \right) dz + \int_{0}^{s} L^{1} f \left( X_{z} \right) d W_{z} \right) ds \\ & + \int_{0}^{t} \left( {\color{Red} g \left( X_{0} \right)} + \int_{0}^{s} L^{1} g \left( X_{z} \right) ds + \int_{0}^{s} L^{1} g \left( X_{z} \right) d W_{z} \right) d W_{s} \\ =& X_{0} + \int_{0}^{t} f \left( X_{0} \right) ds + \int_{0}^{t} g \left( X_{0} \right) d W_{s} + R \\ =& X_{0} + f \left( X_{0} \right) \int_{0}^{t} ds + g \left( X_{0} \right) \int_{0}^{t} d W_{s} + R \end{align*}$$

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