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

• shaiguy6
In summary, you do not know how to get a probability distribution for the random variable V at every time.
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?

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.

shaiguy6 said:
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.

## 1. What is Stochastic DiffyQ?

Stochastic DiffyQ is a mathematical tool used to model and analyze systems that involve randomness or uncertainty. It combines elements of differential equations and probability theory to describe the evolution of a system over time.

## 2. How is Stochastic DiffyQ used in science?

Stochastic DiffyQ is used in a variety of fields including physics, biology, economics, and engineering. It can be used to model and predict the behavior of complex systems that involve random factors, such as stock market fluctuations, chemical reactions, and population dynamics.

## 3. What is a probability distribution in Stochastic DiffyQ?

A probability distribution in Stochastic DiffyQ is a function that describes the likelihood of different outcomes occurring in a system. It can be used to calculate the probability of a particular event or state of the system at a given time.

## 4. How can a probability distribution be obtained for V(t) in Stochastic DiffyQ?

To obtain a probability distribution for V(t) in Stochastic DiffyQ, the system must first be modeled using stochastic differential equations. Then, numerical methods or simulation techniques can be used to solve the equations and generate a probability distribution for V(t).

## 5. What are some limitations of using Stochastic DiffyQ?

Some limitations of using Stochastic DiffyQ include the assumption of constant parameters and the difficulty in accurately modeling complex systems with many variables. Additionally, it may be challenging to interpret the results of a simulation or predict the behavior of a system over a long period of time.

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