The approach is the same in discrete and continuous time.
You have
$$\text{CVA} = E\left[e^{-\int_0^{\tau} r_u du} \text{EAD}_{\tau} (1-R)\mathbf{1}_{\tau \leq T}\right]$$
where
- $\tau$ = stochastic time of default ($+\infty$ if the counterparty never defaults)
- $T$ = deal maturity
- $\text{EAD}_t$ = exposure at default = exposure (stochastic) to the counterparty if the counterparty defaults on time $t$
- $R$ = recovery rate
This formula simply states that CVA is the present value of a flow that represents the Loss Given Default upon default.
If you now assume that default time and discounted EAD are independent (no "wrong way risk") then
$$\text{CVA} = \int_0^T E\left[e^{-\int_0^{t} r_u du} \text{EAD}_{t} (1-R)\right] \phi(t) dt$$
where $\phi()$ is the density of the distribution of $\tau$.
If you then discretize time with a discrete time line $t_i$, $t_0=0$, $t_N=T$, the integral is approximated as
$$\text{CVA} = \sum_{i=1}^{N} E\left[e^{-\int_0^{t_i} r_u du} \text{EAD}_{t_i} (1-R)\right] P(t_{i-1} < \tau \leq t_i)$$
which is the discrete model CVA formula you are familiar with.
