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Suppose $\lambda$ is a Markov process in $\mathbb{R}^n_+$, solving the s.d.e.: $$d\lambda_t = A(\lambda_\infty -\lambda_t)dt+B dZ_t$$ where for $i\in\{1,\dots,n\}$, $[Z_i,N_i]$ is a pure jump process whose jumps have a fixed probability distribution $(\zeta,\eta)\mapsto\nu_i(\zeta) 1[\eta\ge 1]$ on $\mathbb{R}^2$ and arrive with intensity $\lambda_{i,t}$. ($1[\cdots]$ is an indicator function.) This specification implies that $N_i$ is an (integer-valued) counting process, but $Z_i$ may not be.
My question is the following: for $i\in\{1,\dots,n\}$, what is the value of:
$$ \mathbb{E}\left[\exp{[c(N_{i,T}-N_{i,t})]}\middle|\lambda_t,\lambda_T\right] ?$$
Without the conditioning on $\lambda_T$, this would be a straightforward application of the results of appendix B of Duffie, Pan and Singleton (ECTA, 2000).
Is it tractable with the conditioning though?
If it helps, I'm particularly interested in two special cases:
- where $\nu_i(\zeta)=1[\zeta\ge 1]$, so $N_i=Z_i$, and,
- where $\nu_i(\zeta)=1-\exp{(-\zeta)}$, so the jump sizes are exponentially distributed.
In either case, a reasonable approximation (for small $T-t$) can be derived by assuming that all jumps occur at the very start or the very end of the interval. Can one do any better though?