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RR

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- Thread starter river_rat
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- #1

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RR

- #2

chiro

Science Advisor

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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

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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

chiro

Science Advisor

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

- #5

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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

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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

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