7
$\begingroup$

Suppose one has the an Ito process of the form:

$$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$

The infinitesimal generator $LV(x)$ is defined by:

$$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) \right]-V(x)}{t}$$

One can show that $$LV(x) = \sum_{i}b_i(x)\frac{\partial V}{\partial x_i}(x) + \frac{1}{2}\sum_{i,j}(\sigma(x)\sigma(x)^T)_{ij}\frac{\partial^2V}{\partial x_i \partial x_j}(x)$$

I'm wondering if there's an equivalent one infinitesimal generator for $$dX_t = b(X_t)dt + \sigma(X_t)d\eta(t)$$

$$d\eta(t) = \lambda\eta(t) dt + \alpha dW(t)$$

This is a stochastic process where the perturbations are from an Ornstein-Uhlenbeck process instead of a Brownian Motion. Wiki gives the infinitesimal generator of an Ornstein-Uhlenbeck process to be:

$$LV(x) = -\lambda x V'(x) + \frac{\alpha^2}{2}V''(x)$$

But I don't know if there's a way to use that fact to combine it with $LV(x)$ for the Ito process to get the infinitesimal generator for the stochastic process perturbed by an Ornstein-Uhlenbeck process

$\endgroup$
3
$\begingroup$

You can look at your process $X_{t}$ as a two dimensional stochastic process

$$Y_{t}=\left[\begin{array}{cc}X_{t}\\ \eta_{t}\end{array}\right]$$

Then

$$dY_{t}=\left[\begin{array}{cc}dX_{t}\\ d\eta_{t}\end{array}\right]=\left[\begin{array}{cc}b(X_t)+\lambda\eta_{t}\sigma(X_{t})\\ \lambda\eta_{t}\end{array}\right]dt+\left[\begin{array}{cc}\alpha\sigma(X_t)&0\\ 0&\alpha\end{array}\right]\left[\begin{array}{cc}dW_{t}\\ dW_{t}\end{array}\right]$$

and the infinitesimal generator is of the form

$$LV(y)=LV(x,\eta)=\left(b(x)+\lambda\eta \sigma(x)\right)V'_{x}(x,\eta)+\lambda\eta V'_{\eta}(\eta,x)$$ $$+\frac{1}{2}\alpha^{2}\sigma^{2}(x)V''_{xx}(x,\eta)+\alpha^{2}\sigma(x)V''_{x\eta}(x,\eta)+\frac{1}{2}\alpha^{2}V''_{\eta\eta}(x,\eta)$$

By the way, the infinitesimal generator of an Ornstein-Uhlenbeck process of the form

$$d\eta_{t} = \lambda\eta_{t} dt + \alpha dW_{t}$$

is

$$LV(\eta)=\lambda \eta V'(\eta) + \frac{\alpha^2}{2}V''(\eta)$$

$\endgroup$
1
  • $\begingroup$ Can please give proof of the expression of the generator $LV(y)=LV(x,\eta)=....$ $\endgroup$ – Zbigniew Jan 5 '18 at 19:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.