📂Stochastic Differential Equations

Shoji-Ozaki Local Linearization Method

Build-up¹

$$d X_{t} = f \left( t, X_{t} \right) dt + g \left( X_{t} \right) d W_{t}$$ Let’s assume that the diffusion $g$ is dependent only on $X_{t}$ and independent of time $t$ as given by the following stochastic differential equation. If $Y_{t}$ represents some constant $\sigma$ such that $\phi ’ \left( X_{t} \right) g \left( X_{t} \right) = \sigma$ is $\phi \in C^{2}$, which is then expressed as $Y_{t} := \phi \left( X_{t} \right)$, then, according to the Ito formula, consider changing it for some function $b$, $$\begin{align*} d Y_{t} =& \left( f \phi’ + {{ 1 } \over { 2 }} g^{2} \phi’’ \right) dt + g \phi’ d W_{t} \\ =& b \left( X_{t} \right) dt + \sigma d W_{t} \end{align*}$$ or think about the Lamperti transformation that makes the diffusion a constant $1$.

Methods

$$d X_{t} = f \left( X_{t} \right) dt + \sigma d W_{t}$$

Below is a method to numerically solve the given differential equation at evenly spaced time points with interval $h$.

Ozaki Method²

$$X_{t + h} = \exp \left( L_{t} h \right) X_{t} + \sigma \sqrt{ {{ \exp \left( 2 L_{t} h \right) - 1 } \over { 2 L_{t} }} }$$ Here, $L_{t}$ is obtained as follows. $$\begin{align*} L_{t} =& {{ 1 } \over { h }} \log \left[ 1 + J_{t}^{-1} \left( \exp \left( J_{t} h \right) - 1 \right) F_{t} \right] \\ J_{t} =& \left( {{ \partial f(x) } \over { \partial x }} \right)_{x = x_{t}} \\ F_{t} =& {{ f \left( x_{t} \right) } \over { x_{t} }} \end{align*}$$ $J_{t}$ is the Jacobian.

Shoji-Ozaki Method³

$$\begin{align*} X_{t + h} =& X_{t} + {{ f \left( t, X_{t} \right) } \over { L_{t} }} \left( \exp \left( L_{t} h \right) -1 \right) + {{ M_{t} } \over { L_{t}^{2} }} \left[ \exp \left( L_{t} h - 1 \right) - L_{t} h \right] \\ & + \sigma \int_{t}^{t+h} \exp \left[ L_{t} \left( t + h - u \right) \right] d W_{u} \end{align*}$$ Here, both $L_{t}$ and $M_{t}$ are obtained as follows. $$\begin{align*} L_{t} =& {{ \partial f } \over { \partial x }} \left( t, X_{t} \right) \\ M_{t} =& {{ \sigma^{2} } \over { 2 }} {{ \partial^{2} f } \over { \partial x^{2} }} \left( t, X_{t} \right) + {{ \partial f } \over { \partial t }} \left( t, X_{t} \right) \end{align*}$$

Explanation

Local Linearization is an approach that approximates the drift to a linear function for a very short time interval $h$, mainly researched by Toru Ozaki and Isao Shoji. According to Shoji’s theorem⁴, the $p$th convergence rate of the Shoji-Ozaki local linearization method $\tilde{X}_{t}$ is $2$, meaning the following holds true. $$E_{s} \left| X_{t} - \tilde{X}_{t} \right|^{p} = O \left( (t-s)^{2p} \right)$$