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CK backward equation for time homogenous vs inhomogeneous cases

P

pranay.mistry@barnett-wad

Made first post
Hi,

Is there an intuitive explanation as to why the RHS of the CK backward equation varies by a factor of -1 in the time-hom vs inhom cases, and also why the same rule is not true for the forward equations?

Thanks in advance.
 
Hi

Good question!

In the time-homogeneous case, there is a single input to the probability functions, which is duration. The backward and forward equations are two different differential equations for these probability functions, both with respective to the same (and only) input to the function.

In the time-inhomogeneous case, there are two inputs to the probability functions, a start time and an end time. The backwards and forward equations in this case are differential equations for the start and end time respectively. These two inputs have opposite effects on the duration of time considered and therefore on how the probability changes when the input changes. What I mean by this is that as the start time increases, the total duration considered reduces but as the end time increases, the total duration considered increases. This is why the backward equation has a negative sign.

Hope this helps!

Andy
 
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