Dealing with Random Coefficients in Linear Differential Equations

[thegreenlaser]

How does one go about dealing with a linear differential equation with random but constant coefficients (e.g. X''(t) + A*X'(t) + B*X(t) = 0 where A and B are random variables, but are constant with time)? I've searched for things like random differential equations and stochastic differential equations, but I always seem to find cases where the coefficients are noise processes, which is well above my head. Is there some way to take a differential equation like the one I mentioned above and extract everything there is to know about the "randomness" of X(t)?

(Sorry for completely rewriting this post, but I think my original question was not a very good one)

 

[Stephen Tashi]

How does one go about dealing with a linear differential equation with random but constant coefficients (e.g. X''(t) + A*X'(t) + B*X(t) = 0 where A and B are random variables, but are constant with time)?

Be precise about what you mean by "dealing with". If a situation is described by such a random equation, what is it that you want to find out about the situation? Usually, the coefficients of a differential equation don't define a specific situation by themselves. You also have to specify initial conditions or boundary conditions.

 

[thegreenlaser]

Be precise about what you mean by "dealing with". If a situation is described by such a random equation, what is it that you want to find out about the situation? Usually, the coefficients of a differential equation don't define a specific situation by themselves. You also have to specify initial conditions or boundary conditions.

Boundary conditions or initial conditions would be known. Given those, by "deal with," I guess I mean to ask what sort of information can I get about the unknown function (e.g. mean/variance or PDF as functions of time) from the differential equation and boundary conditions/initial conditions themselves? What sort of approaches are there for getting that information?

Sorry that this is kind of vague and general, I'm trying to learn more about this type of thing, and I'm just having trouble finding some places to start.

 

[Stephen Tashi]

I've never seen a treatment of the general subject of differential equations with random coefficients. In thinking about it, I can only offer the following inutitions.

One way to look at a "stochastic process" that takes place in time is that it generates a "random trajectory".

Suppose we have process that begins by selecting the random coefficients for a differential equation (in functions of time). Some differential equations do not have solutions. Let's assume we select coefficients in way that always produces a solution. That solution defines the particular realization of the process.

Each random selection of coefficients produces a deterministic trajectory. Such a trajectory is a "random trajectory" in the sense it is selected at random. But compared to the type of trajectory studied in books on stochastic processes, it is unusual - at least by my intuition. In the usual sort of stochastic process, if we know the current trajectory up to time t = T then there are many possible trajectories that can occur that continue the process to later times. But with the situation you describe, that may not be the case. If two deterministic functions match everywhere in some open interval then they may have to match at all times if they are "nice" functions (such as analytic functions).

There is a type of stochastic process called "stationary random functions". My understanding of this type of process is that you can approximate the trajectories of such a process by sums of waves (summed over various frequencies). To approximate the trajectory of such a process, you pick the coefficients of the waves indepenently at random and then compute the sum. Trying to fit what you want to do into this scenario seems rather artificial. You'd need a differential equations which always had a sum of waves as a soluiton and then you'd have to try to make the random coeffients of the equation produce random independent coefficients for the waves.

 

[atyy]

I'm not sure, but here's a thought. Let's just have one random coefficient "A" for simplicity.

Let's say the initial condition is fixed. If I can solve the differential equation for any value of the coefficient, I will get X(t,A). For any particular t, this will be X_t(A).

If I'm given P(A), then P(X_t) should be given by the change of variables formula, which involves a Jacobian, like in http://www.math.uiuc.edu/~r-ash/Stat/StatLec1-5.pdf.

 

[chiro]

If A and B are constant and independent of t, then you just the DE and treat your solution as a function of random variables with said distributions.

If this not the case, then you are dealing with stochastic calculus which means it will be a lot more general and as a result, a lot harder.

 

[thegreenlaser]

Thanks, this helps.