Probability equation

Question

I want to solve $P(e^{X_1}+Ke^{X_2}<c) = \alpha$ for $K$, where $c$ is a constant and $X_1,X_2\sim N(0,1)$. To find $P(e^{X_1}+Ke^{X_2}<c)$, I can write it as an expectation $E(I(e^{X_1}+Ke^{X_2}<c))$,where $I()$ is an indicator function then do a Monte Carlo simulation. How can I determine K?

Answer 1

As long as you can evaluate $P(...) = f(K)$ for any fixed value of $K$, which you suggest doing by Monte-Carlo, you now have a 1-dimensional root-finding problem, e.g. find the value of $K$ such that $f(K)=\alpha$, where $\alpha$ and $f$ are known.

Numerically, such problems are typically solved by Newton's method http://en.wikipedia.org/wiki/Newton's_method for well-behaving $f$ (need existence of and ability to compute $f'(K)$, which you have) or by the Secant or Bisection methods http://en.wikipedia.org/wiki/Bisection_method and http://en.wikipedia.org/wiki/Secant_method.