Difference between the "Hazard Rate" and the "Killing Function" of a diffusion model? I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. 
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What is the difference between the "Hazard Rate" and the "Killing Function" of a diffusion model?
Some Definitions: 
The Killing Function
The function k(t,x) is interpreted as the killing rate. Informally, this means that if, at time t, the particle is alive and is situated at the point x, then the probability that it dies in the next h units of time is approximately k(t,x)h when h is small.
\begin{gather*}
Pr(\rho  \leq t+h \mid \rho  > t, X(t) = x)\approx k(t, x)h & (1)
\end{gather*}
And,
\begin{gather*}
dX(t) = \mu dt+\sigma dW\ & (2)
\end{gather*}
Hazard Rate
\begin{gather*}
Pr(t \leq T \leq t+h \mid T > t) \approx \lambda (t)h & (3)
\end{gather*}
That is, λ(t)h represents the instantaneous chance that an individual will die in the interval (t, t + h) given that this individual is alive at age t. 
Lastly, to put it in perspective here is a picture of a diffusion with arbitrary Killing Function k(x) = a + Sqrt(t/b), where a, and b are some constants. 

I added the lines for later reference.

So, these results raise a lot more question. 


*

*How do I interpret "rho" in Equation (1) for example - if I am modeling a type of bird population for with X(t)? 

*How do I relate the Killing Function with the Hazard Rate? 

*Is it OK to say that if the f(t) is the density distribution of the First-Passage-Times (Refer to Fig-2), then the Hazard rate for the diffusion (2) is:
\begin{gather*}
\lambda (t) = \frac{f(t)}{1-F(t)} & (4)
\end{gather*}


*If I do not know the killing function - but I observe the first passage time distribution as in Fig-2: Is it possible to solve for the Killing Function?

*Lastly, in the definition k(t,x) is a function of both variables {x,t}. In the literature, most of the time is referred as k(x), which is really k(X(t)) since X() is a function of t. But if one was to actually apply it - as I did in Fig-1, say:
\begin{gather*}
k(x) = b[(x(t)-a)^{2}]\ & (5)
\end{gather*}
I would have to express it in terms of X(t):
\begin{gather*}
X(t) = a+\sqrt{\frac{t}{b}} & (6)
\end{gather*}
But X(t) is reserved for the diffusion model (2) so it makes it extra confusing. 
Note: Assume OP (original poster) is very unintelligent; hence, be very specific, use simple words, do not leave any algebra out, and do not hesitate to curse me out if I wrote something stupid above. 
Thank you so much in advance!  
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 A: I am thinking about some of the same questions you've posted here. I will try to offer partial answers and perhaps revive this question, and invite someone smarter than me to give full answers to your question. 

Firstly, lets clear up some notation. In your equation $3)$ you state $Pr(t \leq T \leq t+h \mid T > t)$ but as you condition on $T>t$, the only meaningful statement is $Pr(T \leq t+h \mid T > t)$ because $Pr(t\leq T \vert T>t)$ is simply zero. Usually this would read $Pr(t<T\leq t+\triangle t|T\geq t)$ leaving the possibility of $P(T\leq t)$ open.
Changing your equation $(3)$ to $Pr(T \leq t+h \mid T > t) \approx \lambda (t)h$ there is a minor misnomer. While $\lambda (t)h$ is indeed the approximate instantaneous probability of death, it is not the hazard rate. Only $\lambda(t)$ is a meaningful rate. However, both $k(t,x)h$ and $\lambda(t)h$ are meaningful objects. The first is the probability of passing a barrier, given time and position, while the second is probability of passing a barrier, given time.
Now the real difference between equation $(1)$ and $(3)$ stand out: The major difference is the extra piece of information in the killing function; The position $x$. This is a very important piece of information, because given two (or more) positions we can calculate $velocity$, which says something about the speed with which the particle is approaching the barrier. This information is not available in equation $(3)$. 
I hope this answers your first 3 questions.
Question 4 would probably be a no. However, given two different first passage time distributions, you might be able to say something about the parameters of the models. E.g. a more "right-skewed" FPT density would probably have a lower drift parameter etc. But saying something about position seems futile (however there are HMM approaches that models this).
Question 5 isn't really a question. If it is, then I'm confused about what you're asking.
