Introduction
This blog would introduce some mathemetical tricks on Stochastic Differential Equations (SDEs).
Lamperti Transform
Consider the following standard SDE:
$$ \mathrm{d} x_t = f(x_t, t) \mathrm{d} t + L(x_t, t) \mathrm{d} w_t $$The Lamperti transform is to make the change-of-variable $y_t = h(x_t, t)$ in the SDE s.t. the SDE transforms into
$$ \mathrm{d} y_t = g(y_t, t) \mathrm{d} t + \mathrm{d} w_t $$That is, the multiplicative noise is transformed into additive noise. Numerically solving an additive-noise SDE is usually easier than solving a general SDE of the form. To illustrate the idea of the Lamperti transform, we can define a transform (change-of-variable) as follows:
$$ y_t = h(x_t, t) = \int_{\xi}^{x_t} \frac{1}{L(u, t)} \mathrm{d} u $$where $\xi$ is an arbitrary point. Applying the Itô’s formula on $h(x_t, t)$ gives
$$ \begin{aligned} \mathrm{d} y_t &= \frac{\partial h(x_t, t)}{\partial t} \mathrm{d} t + \frac{\partial h(x_t, t)}{\partial x_t} \mathrm{d} x_t + \frac{1}{2} \frac{\partial^2 h(x_t, t)}{\partial x_t^2} (\mathrm{d} x_t)^2 \\ &= \left[ \frac{\partial}{\partial t} \int_{\xi}^{x_t} \frac{1}{L(u, t)} \mathrm{d} u \right] \mathrm{d} t + \frac{1}{L(x_t, t)} \left[ f(x_t, t) \mathrm{d} t + L(x_t, t) \mathrm{d} w_t \right] + \frac{1}{2} \left[ - \frac{1}{L^2(x_t, t)} \frac{\partial L(x_t, t)}{\partial x_t} \right] L^2(x_t, t) \mathrm{d} t \\ &= \left( \frac{\partial}{\partial t} \int_{\xi}^{x_t} \frac{1}{L(u, t)} \mathrm{d} u + \frac{f(x_t, t)}{L(x_t, t)} - \frac{1}{2} \frac{\partial L(x_t, t)}{\partial x_t} \right) \mathrm{d} t + \mathrm{d} w_t \end{aligned} $$Provided that $x_t = h^{-1} (y_t, t)$ exists, we can express the SDE in termsa of $y_t$ by
$$ \begin{gathered} \mathrm{d} y_t = g(y_t, t) \mathrm{d} t + \mathrm{d} w_t \\ g(y_t, t) = \left( \frac{\partial}{\partial t} \int_{\xi}^{x_t} \frac{1}{L(u, t)} \mathrm{d} u + \frac{f(x_t, t)}{L(x_t, t)} - \frac{1}{2} \frac{\partial L(x_t, t)}{\partial x_t} \right) \Bigg|_{x_t = h^{-1} (y_t, t)} \end{gathered} $$More generally the transformation does not have a simple formula, but instead the transformation can only be defined as a solution to a system of partial differential equations (PDEs). To be continued…