We talking about A varying through time of course. If you
think of A as merely being uncertain rather than really
fluctuating, the formulation could be quite different I
suppose.
Reference:
Tuljapurkar, S. 1990. Population Dynamics in Variable
Environments. Lecture Notes in Biomathematics 85.
Springer-Verlag, NY.
Scott Ferson <scott@ramas.com>
Applied Biomathematics, 516-751-4350, fax -3435
Robert Dodier wrote:
>
> Fellow enthusiasts of uncertainty,
>
> I am considering the following problem, concerning the distribution
> over states in a system governed by linear differential equations
> with random coefficients.
>
> Let dx/dt = Ax be a system of differential equations, where the
> state x is a vector and A is a matrix. As we all know, the
> solution is x(t) = exp(At) x(0).
>
> Suppose x(0) is not known, and x(0) has a Gaussian distribution
> with mean mu and variance Sigma. Then x(t) is again Gaussian,
> with mean exp(At) mu and variance exp(At) Sigma exp(At)'.
>
> Now suppose that A is also not known. What is the marginal
> distribution over x(t)? Probably we need to assume some easily
> handled form for the distribution over A -- what will make
> exp(At) x(0) have some recognizable form?
>
> It is simple enough to find an expression for the joint distribution
> of x and A, but the difficulty is integrating over A to find the
> marginal over x. I have composed a one-dimensional problem, but
> even in this case I cannot carry out the integration; I tried
> lognormal, gamma, and Rayleigh, and Maxwell distributions for A.
>
> As one might suppose, I came across this problem in the course
> of studying uncertainty in the evolution of systems governed
> by differential equations. (Eventually I would like to be able
> to apply some results to nonlinear equations -- I am assuming
> that this works best through a linear approximation.) References to
> relevant works on uncertainty in continuous-time systems would be
> very much appreciated.
>
> I have consulted some textbooks on stochastic differential equations,
> but unfortunately I cannot tell if the answer to this problem is
> in there somewhere. What I gather from one stochastic mechanics book
> is that exact solutions for this problem are not known, and
> typically approximate solutions are computed through Monte Carlo or
> perturbation methods. I hope I am mistaken about the difficulty of
> this problem. :)
>
> Thanks for your time,
> Robert Dodier