Given the process $log(A) = \sqrt{\rho}T\sigma Y + \sqrt{(1-\rho)T} \sigma Z^i - \frac{1}{2}T\sigma^2$ find $E[e^{\sqrt{(1-\rho)T}\sigma Z^i}\mathcal{1}_{A<c}]$ where $Z^i \sim N(0,1)$, c and Y are some positive constants, and $\mathcal{1}$ is the indicator function.
The indicator function conditions on process $A$ and the exponential function inside the expectation is a function of $Z^i$ so I am a bit puzzled how to proceed. Kindly help!
HINT Let's write $\ln A = \alpha + \beta Z^i$ for $\alpha, \beta$ constant and $\beta > 0$. Note that $$ A < c \iff \ln c > \ln A \iff Z^i < \frac{c - \alpha}{\beta}. $$
Can you now finish?
UPDATE
Let $\phi(z)$ and $\Phi(z)$ denote the pdf and cdf of the standard normal distribution. You have $$ \mathbb{E}\left[e^{\beta Z} \mathbb{I}_{A<c}\right] = \int_{-\infty}^{(c-\alpha)/\beta} e^{\beta z} \phi(z) dz $$ which after completing the square you should be able to express as $\Phi(\cdot)$ of some transformation of $\alpha, \beta, c$.