Deriving first passage time from Gambler's ruin

Question

Quantity of interest: the first time a stochastic process hits a certain threshold $a\in \mathbb{R}$, if it starts at $0$: $$T_a = \min\{ t>0: X_t = a | X_0 = 0 \}$$

We are interested in the distribution of $T_a$ for either case,

1. a discrete-time stochastic process $X_t$ (called Gambler's ruin problem), or
2. a continuous-time, specifically a diffusion process $X_t$ given by $dX_t = adt + \sigma^2 dB_t$ (called first passage time).

How can we derive 2. (continuous) from 1. (discrete)?

I think it should be related to the derivation of continuous process from discrete processes with infinitesimal time step (i.e. the way Brownian motion is derived from random walk).