# Calculate $\mathbb{E}[\int_o^T N_{t-}dS_t]$ - what went wrong?

First note, I had asked a similar question here, but the thread seems to have died, so I'll revive it here with more details. As a simplification of my real problem, I want to compute $\mathbb{E}[\int_o^T N_{t-}dS_t]$, where $S_t$ is a $(\mu,\sigma)$-geom. Brownian motion, and $N_t$ an independent Poisson process with const. intensity $\lambda$. Hence $\mathbb{E}[\int_o^T N_{t-}dS_t] = \mathbb{E}[\int_o^T N_{t-}S_t(\mu dt + \sigma dW_t)]$. Ito formula shows that the latter part of the integral should be square integrable, with integrable quadratic variation, so it is zero in expectation. So we should get

$\mathbb{E}[\int_o^T N_{t-}dS_t] = \mu\int_0^T\underbrace{\mathbb{E}[N_{t-}]}_{=\mathbb{E}[N_t-\mathbb{1}_{\Delta N_t \neq 0}]=\lambda t}\mathbb{E}[S_t]dt = \int_0^T \mu\lambda t e^{\mu t}dt$

$N_{t-}$ is a simple, left-continuous process, so I tried to confirm that numerically by calculating (MC)

$\mathbb{E}\left[ \sum_{i=0}^{N_t} i(S_{\tau_{i+1}}-S_{\tau_i})\right]$,

with $\tau_0=0$, $\tau_{N_t+1}=T$ and $\tau_i$ the exp-$(\lambda)$-distributed jump times. But values don't seem to converge to the analytic solution above. Is anything wrong?

• Your final formula doesn't involve T. But is this really of interest to research level mathematicians? It just seems like youre asking someone to check your calculations. Apr 5, 2012 at 18:42
• It probably isn't. But thank you anyway. I thought this requires only a glimpse. And it did. Indeed, there was only a $=$ missing. So again, thank you. Apr 5, 2012 at 19:09

I don't see why your approximation of the stochastic integral works. Shouldn't you take more points in the interval $[0,T]$ and not only the jump times to get a better approximation?
We have have the following approximation theorem for the stochastic integral: Let $X$ a semimartingale and $H$ a càdlag or càglàd adapted process. Let $t_0^n = 0 \leq t_1^n \leq \ldots \leq t_n^{p_n} = T$ a sequence of subdivisions $[0,T]$ such that $\lim_{n \to \infty} \sup_{k=0}^{p_n} t_{k+1}^n - t_{k}^n = 0$ then:
$$\lim_{n \to \infty} \sum_{k=0}^{p_n} H_{t_{k}^n} (X_{t_{k+1}^n} - X_{t_{k}^n}) = \int_0^T H_{s^-} dX_s \quad \text{in probability}$$
• Well in this special case, as $H$ is constant between two jumps $\tau_i,\tau_{i+1})$, most of the terms would cancel out. However, can you give me a reference of the above theorem? Shouldn't this limit exist only for finite variation process $X$? - cf Protter p. 44 "Stochastic Integration and Differential Equations". Apr 6, 2012 at 10:10