Infinitesimal generators of bridged stochastic process

[river_rat]

I hope someone can put me on the right track here. I need to derive the infinitesimal generator for a bridged gamma process and have come a bit stuck (its for a curve following stochastic control problem - don't ask). Any tips, papers, books that could guide me out of my hole would be greatly appreciated.

RR

 

[chiro]

Hey river_rat and welcome to the forums.

I don't know the ianswer to your question, but I also don't fully follow it either.

Is this generator some kind of infinitesimal delta or operator that generates a specific stochastic process?

 

[river_rat]

Hi chiro

The formal definition is the operator $\mathcal{L}$ where acts on $\mathcal{C}^{2, 1}$ test functions so that $\mathcal{L} f(x, t) = \lim_{h\rightarrow 0^+} \frac{\mathbb{E}(f(X_{t+h}) | X_t = x) - f(x)}{h}$.

For general ito processes or levy processes it is easy to find, but for a bridged gamma process there is some trick I seem to be missing as I know you can do this in closed form.

 

[chiro]

I wish I could you out but this is beyond my current knowledge and skill set.

 

[bpet]

Have you tried a more tractable example yet, such as the Brownian bridge?

I haven't checked the details but perhaps you could apply the forward and backward Kolmogorov equations to the conditional joint distribution. From there it wouldn't be too difficult to modify with jump terms.

 

[river_rat]

Hi bpet

The methodology I know for the brownian bridge goes as follows: first prove the Brownian bridge is a gaussian process, then find an equivalent process that is adapted to the original filtration generated by your brownian motion and that is a scaled ito integral. Then using ito's lemma on this new scaled ito integral you can arrive at the infinitesimal generator of the brownian bridge.

However, each of those steps are rather bespoke for the process at hand, especially the form of the scaled ito integral required.

I am interested on your forward and backward equation idea, care to elaborate?

 

[bpet]

The idea was to write the (conditional) transition density as $\frac{f(t,u,x,y)f(u,v,y,z)}{f(t,v,x,z)}$ and differentiate wrt u with the Kolmogorov equations. Does that help?