Assume I consider $N$ realizations of a stochastic process $\left\{X_t \right\}_{t \in (0,T)}$ with following dynamics: $$ dX_t = \mu dt + \sigma dW_t.$$
I further assume that:
- The process has no drift ($\mu=0$) and volatility $\sigma > 0$.
- There is an absorbing boundary at $X=0$.
- All $N$ realizations start from the same initial value $X_0 > 0$.
- All $N$ realizations run for the same finite time $T < \infty$ or until they hit the boundary.
We can calculate analytically what is the hitting probability $p_\text{hit} < 1$ that such a realized trajectory will hit the boundary. Denote by $K$ the number of realizations that have actually hit the boundary. (For large $N$, this is roughly $K \approx p_\text{hit} N$.)
Assume I show you only the subset of $K \leqslant N$ walks that have actually hit the boundary. I then ask you to infer what is the best estimate $\hat{\mu}$ of the underlying drift $\mu$ that generated these $K$ trajectories. Intuitively, I would assume that, at least for $N$ large enough, your statistical estimate will almost surely be $\hat{\mu} < 0$. In other words, I think that this conditioning on a sure hit can be mapped to the presence of an effective, negative drift (we pick only the most "unlucky" trajectories).
Here, I am not interested in the statistical inference of the drift, but more generally in the question: If I condition the above stochastic process on a sure hit within time $T$, can I represent this conditioned process as an (unconditioned) stochastic process with another effective drift function $\mu_\text{eff}(x,t)$ (and potentially also effective volatility function)?
Alternatively, this problem is relatively easy to solve from the perspective of probability distributions. Denote by $p(x,t)$ the probability density at $X=x$ at time $t$ (conditional on having been at $X_0$ at time $t=0$). Applying essentially just Baye's theorem to probability densities, we find that the probability density of the conditioned problem is equal to $$ p(x,t | \text{hit until }T) = \frac{ p_\text{hit}(x, T-t) ~ p(x,t) }{ p_\text{hit}(X_0, T) }. $$ where $p_\text{hit}(x,\tau)$ is the probability to hit the boundary within a time $\tau > 0$ if now at position $x$. Since both $p_\text{hit}$ and $p(x,t)$ are known analytically, we can calculate $p(x,t | \text{hit until }T)$ analytically. On the other hand, if $p(x,t | \text{hit until }T)$ can be mapped to some stochastic process $$ dX_t = \mu_\text{eff}(x,t) dt + \sigma_\text{eff}(x,t) dW_t.$$ with effective drift and volatility, then $p(x,t | \text{hit until }T)$ should also be the solution of a Fokker-Planck equation with $\mu_\text{eff}, \sigma_\text{eff}$. Therefore, my above question can be reformulated as: If given a solution to a Fokker-Planck equation, how can one reverse engineer its generating drift and volatility?
This is too long for a comment, but is not quite an answer to your question since you are asking about time and space dependent coefficients:
A random walk with drift $\mu$ will typically hit the boundary around $T_\mu=X_0/\mu$, with a strongly decaying probability to hit at times different than $T_\mu$ . An unbiased random walk (that is $\mu=0$ ) will have hitting times with a probability that decays asymptotically as a power law with $T$. I believe the probability to survive beyond $T$ is proportional to $T^{-1/2}$. Since these behaviors are qualitatively different, I don't think you can map one to the other, at least not if $\mu$ is constant.