  We study the parabolic Anderson model in $(1+1)$ dimensions with nearest
neighbor jumps and space-time white noise (discrete space/continuous time). We
prove a contour integral formula for the second moment and compute the second
moment Lyapunov exponent. For the model with only jumps to the right, we prove
a contour integral formula for all moments and compute moment Lyapunov
exponents of all orders.
