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Probability of first-passage by a particular time (and conditional distribution of first-passage-time) of a diffusion process with negative drift

(This is a variation on Expectation of first-passage-time of a diffusion process with negative drift where more than just the particular expectation or Laplace transform seems necessary).

Take the stochastic process $X_0 = 0$ and $X_t = \nu t + \sigma W_t$ where $W_t$ is standard Brownian motion and $\nu$ is a drift which may be negative (this is the key complexity to the question)

Let $\alpha > 0$ be a fixed level, then define the first passage time as the random variable: $T = \inf\{ 0 < t \mid X_t=\alpha \}$.

Finally, define $0 < \bar{T} < \infty$ as a terminal time.

I want to compute two things:

  1. $\mathbb{P}(T < \bar{T})$
  2. $\mathbb{E}(e^{-r T} | T < \bar{T})$

Solving the $\nu \geq 0$ case: I believe this turns out to be easy as this follows the example in https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution#Relationship_with_Brownian_motion . Hence, $T$ is distributed as an Inverse Gaussian: $T\sim IG(\tfrac\alpha\nu, \tfrac {\alpha^2} {\sigma^2})$ in that notation.

Let $f(T;\tfrac\alpha\nu, \tfrac {\alpha^2} {\sigma^2})$ be the PDF and $F(T;\tfrac\alpha\nu, \tfrac {\alpha^2} {\sigma^2})$ be the CDF of the Inverse Gaussian, then we have our simple answer:

$$\mathbb{P}(T < \bar{T}) = F(\bar{T})$$

and $$\mathbb{E}(e^{-r T} | T < \bar{T}) = \frac{1}{ F(\bar{T})} \int_0^{\bar{T}}e^{-r T}f(T) dT$$

Of course, to eliminate the use of the PDF we could also write the expectation in terms of the counter-cdf of the right truncated CDF instead of the PDF.

Solving the $\nu < 0$ case: As Karlin and Taylor puts it: "When $\nu < 0$, $T$ has a defective probability distribution, that is $T$ is infinite with positive probability".

For that reason, a PDF cannot be used directly and the CDF would need to be used with care. But can we use the PDF conditionally, as above, adjusting for the probability of $T$ being infinite? Any ideas on how to do the equivalent claculations to the $\nu > 0$ case?

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