I have a Poisson Process with stationary and independent increments. Therefore I know:
$$P(N_T – N_t = r) = \dfrac{\exp(-\lambda(T-t))(\lambda(T-t))^r}{r!} \mbox{ where } T>t.$$
Now suppose I am considering a process where:
$Y_t = \exp(N_t – ct)$
How do I calculate the conditional expectation of $E(Y_T|Y_t)$? And under what conditions would I have a Martingale process?
Thanks in advance.
Edit:
Sorry, I am new here and am unfamiliar with the process. Based on the Q&A I thought I was just supposed to stick with the question and avoid discussing what I'd tried. I'll provide a bit more explanation.
What I was given was the assumption of independent and stationary increments, so I used the general definition of a Poisson process to arrive at the probability equation above. Then I wrote out what I know about conditional expectation. Namely:
$E[Y_T|Y_t] = \sum_{j=1}^{n}y_jP(Y_T=y_j|Y_t)$
Then I took the fact that $P(Y_T=k|Y_t=n)=\frac{P(Y_T=k,Y_t=n)}{P(Y_t = n)}$
And then I tried to plug in the original equation of a Poisson process:
$P(N_t = k) = \dfrac{(\lambda*t)^k}{r!}*\exp(-\lambda(t))$
I decided that because of the independence assumption my
$E[Y_T|Y_t]$ was simply equal to $E[Y_{_{T-t}}]$
But I am struggling to see how to use this in my new process since the $N_t$ is embedded in the process:
$Y_t = \exp(N_t – ct)$.
Best Answer
For starters, note that, if $X_a$ is Poisson with parameter $a$, $$ E[\mathrm e^{X_a}]=\sum_n\mathrm e^{-a}\frac{a^n}{n!}\mathrm e^n=\mathrm e^{a(\mathrm e-1)}. $$ Then, decompose $Y_T$ as $Y_T=\mathrm e^{N_t-cT}\mathrm e^{N_T-N_t}=Y_t\mathrm e^{-c(T-t)}\mathrm e^{N_T-N_t}$ and note that $N_T-N_t$ is Poisson with parameter $a=\lambda(T-t)$ and independent of $Y_t$. Hence $$ E[Y_T\mid Y_t]=Y_t\mathrm e^{-c(T-t)}E[\mathrm e^{N_T-N_t}]=Y_t\mathrm e^{-c(T-t)}\mathrm e^{\lambda(T-t)(\mathrm e-1)}, $$ and $E[Y_T\mid Y_t]=Y_t$ if $c=$ $____$.