- #1

- 586

- 1

Hi folks,

I have a question concerning the infinitesimal generator of a stochastic ;process, more specificaly of Brownian motion.

Let [itex]X_t[/itex] be a stochastic process, then the infinitesimal generator A acting on nice (e.g. bounded, twice differentiable) functions f is defined by

[tex]

(Af)(x)=\lim_{t\to 0}{\frac{1}{t}\left[E_x\left[X_t\right]-1\right]}

[/tex]

For (one-dimensional) Brownian motion this turns out to be just the second derivative operator.

What happens however, if I were to consider reflected Brownian motion (reflected at zero). In distribution this process is equal to [itex]|B_t|[/itex] where [itex]B_t[/itex] is a (non-reflected) Brownian motion. My feeling is that for [itex]x \neq 0[/itex] the infintesimal generator should still be the second derivative, but what happens at x=0?

Unfortunately I couldn't find this in any textbook.

Any help appreciated

-Pere

I have a question concerning the infinitesimal generator of a stochastic ;process, more specificaly of Brownian motion.

Let [itex]X_t[/itex] be a stochastic process, then the infinitesimal generator A acting on nice (e.g. bounded, twice differentiable) functions f is defined by

[tex]

(Af)(x)=\lim_{t\to 0}{\frac{1}{t}\left[E_x\left[X_t\right]-1\right]}

[/tex]

For (one-dimensional) Brownian motion this turns out to be just the second derivative operator.

What happens however, if I were to consider reflected Brownian motion (reflected at zero). In distribution this process is equal to [itex]|B_t|[/itex] where [itex]B_t[/itex] is a (non-reflected) Brownian motion. My feeling is that for [itex]x \neq 0[/itex] the infintesimal generator should still be the second derivative, but what happens at x=0?

Unfortunately I couldn't find this in any textbook.

Any help appreciated

-Pere

Last edited: