Let's consider a process $u - S_n$, where $S_n = Y_1 + \ldots + Y_n$ is a sum of iid random variables with the density $p(y)$. Let:
- $T_0 = \min\{n: u - S_n < 0 \}$,
- $T_b = \min \{n: u - S_n >b \}$ for $0 < u < b$,
- $T = \min \{T_0, T_b \}$.
We want to calculate $\psi = \mathbb{P}(T = T_0)$ using importance sampling (exponential change of measure).
We denote: $$ R = \begin{cases} \min\{n: S_n > u \} &\text{if such n exists}, \\ \infty &\text{otherwise}. \end{cases} $$
Therefore $\psi = \mathbb{P}(T = T_0) = \mathbb{P}(R < \infty).$
In order to define instrumental density we denote $q(y) = e^{ry}p(y).$
Let $\mathcal{R_n} = \{ (y_1, \ldots , y_n): y_1 \le u, y_1 > u- b, \ldots, y_1+...+y_{n-1} \le u, u-b > y_1 + \ldots + y_{n-1}, y_1 + \ldots +y_{n}>u, u-b > y_1 + \ldots + y_n \}.$
Then the event $\{ R = n \} $ occurs when $(Y_1, \cdots, Y_n) \in \mathcal{R_n}$, so
$$ \mathbb{P}_p(R=n) = \int \int_{\mathcal{R}_n} \cdots\int p(y_1) \ldots p(y_n) dy_1 ... dy_n = \int \int_{\mathcal{R}_n} \cdots \int e^{-r y_1} q(y_1) \ldots e^{-r y_n}q(y_n) dy_1 \ldots dy_n= $$ $$ = \int \int_{\mathcal{R}_n} \cdots \int e^{r (y_1 + \ldots + y_n)} q(y_1) \ldots q(y_n) dy_1 \ldots dy_n $$
I wonder how to express the above integral interms of $\mathbb{E}_q$. Can anyone help me, please?