Exploring Stochastic DiffyQ: How to Get a Probability Distribution for V(t)?

[shaiguy6]

Hello all,

I have run into this problem, and being that I know nothing about stochastic DiffyQ I am trying to toy around with it. Basically, the following is a boiled down version of my problem:

I have a probability density function that is given: p(t)

and let's say we pick 1 value from that density function (so that we get some value of time). I'm not exactly sure the proper notation for how to write the question, so the random variable that goes by the pdf p(t) i will call P. So then I will have a diffyQ that looks like this

\frac{dV(t)}{dt}=-V(t)+\delta (t-P)

where P is the time which is pulled from that probability distribution. My inuition tells me that I should get some probability distribution for V at every time. Is there a way to get that?

 

[MathematicalPhysicist]

Well the particular equation you wrote can be simplified to:
\frac{d(V(t) e^t)}{dt} = e^t \delta(t-P)

Does this give something to work with, I am not sure.

 

[Stephen Tashi]

My inuition tells me that I should get some probability distribution for V at every time. Is there a way to get that?

You haven't defined the random variable V(t) precisely.

On the one hand, it may be that you intend to make only a single random draw for the random variable P. If the function V(t) is some initial function V_o(t) then, after one random draw of P, V(t) can take at most two possible values and at each time t, there is a probability density that would involve at most two nonzero values.

On the other hand, you may want V(t) to represent some sort of limiting process as the number of independent random draws for P approaches infinity.

Assume P has a continuous probability density. Then both of these problems are interesting from an academic point of view. If you make only one random draw for P, V has probability 1 of remaining at V_o(t). This is an intuitively unpleasant result, but to avoid it, there would have to be some time P_1 that had a nonzero probability of exactly being drawn.

There are various definitions for "the limit of a sequence of probability distributions". However, I don't know any that would give you a different result than the one in the above paragraph as the number of draws for P approaches infinity.

I suspect your equation doesn't describe your problem. You'd do better to state the real problem.