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Consider a one dimensional diffusion process on the diffusion interval $I$originated in $x_0\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of$t$, $t>t_0$ with bounded derivatives and with $a(t)<b(t)$ and $a(t),b(t)\inI$, $\forall t>t_0$. We study the joint distribution of the two randomvariables $T_a$ and $T_b$, first hitting times of the diffusion process throughthe two boundaries $a(t)$ and $b(t)$, respectively. We express the jointdistribution of $T_a, T_b$ in terms of $P(T_a<t,T_a<T_b)$ and$P(T_b<t,T_a>T_b)$ and we determine a system of integral equations verified bythese last probabilities. We propose a numerical algorithm to solve this systemand we prove its convergence properties. Examples and modeling motivation forthis study are also discussed.

Joint densities of first hitting times of a diffusion process through two time dependent boundaries