Asset Dynamics:Geometric Brownian motion (GBM)
$$dA_t=r A_t \, dt+σ A_t \, dZ_t$$
$r$ is the risk free interest rate, $σ$ is the diffusion coefficient. $Z$ is a standard Brownian motion. Maturity is $T$. The initial asset value is $A_0$. We set a constant threshold $D$ ($0
My question is, how to get the following joint probability?
$$P\left(A_T≤D+S \;, \;\min_{0≤t≤T}A_t ≤D \right)$$
I need to calculate this result in my thesis. So I really need to know how to do it. Any comment or answer is very very appreciated!