Register to reply 
Infinitesimal generators of bridged stochastic process 
Share this thread: 
#1
Jan1813, 02:43 AM

P: 13

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 


#2
Jan1813, 02:48 AM

P: 4,572

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? 


#3
Jan1813, 03:18 AM

P: 13

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


#4
Jan1813, 08:48 PM

P: 4,572

Infinitesimal generators of bridged stochastic process
I wish I could you out but this is beyond my current knowledge and skill set.



#5
Jan1913, 08:04 PM

P: 523

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. 


#6
Jan2313, 02:36 PM

P: 13

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? 


#7
Jan2413, 04:19 AM

P: 523

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



Register to reply 
Related Discussions  
Stochastic Process, Poisson Process  Calculus & Beyond Homework  0  
Heat transferred during an infinitesimal quasi static process of an ideal gas  Advanced Physics Homework  1  
Stochastic Process  Calculus & Beyond Homework  0  
Stochastic process (renewal process)  Calculus & Beyond Homework  0  
Proof that a stochastic process isn't a Markov Process  Set Theory, Logic, Probability, Statistics  4 