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