Homogeneous linear stochastic DE with noncommuting coefficients

Question

The system I am studying can be reduced to a Stratonovich vector stochastic differential equation

$dX = A X \; dt + \sum B_k X \circ dW_k$

with $W_k$, $k=1..m$ the Brownian motion in $m$ dimensions, $X$ the unknown process in $n$ dimensions, and $A$ and $B_k$ matrices that in the simple case are constant. However, they do not commute, so that we cannot express the solution as a simple exponent $X = \exp(A t + \sum_k B_k W_k)$.

Are there any general methods to solve such systems, or to prove interesting properties of the solutions, or perhaps to express solutions in term of some other standard processes (rather than $W_k$)? Any pointers to books or papers would be appreciated.

In particular, if the explicit solution is not possible, are there any techniques to compute/write an ODE for the expectation $E X_1/X_2$ (i.e. ratio of two components of the process)?

I have briefly looked through the book ``Stochastic Flows and Stochastic Differential Equations'' by H. Kunita; from what I could understand it seems that an explicit solution in terms of exponentials of combinations of $W_k$ is possible when the Lie algebra corresponding to the $A$ and $B_k$ is solvable. Unfortunately in my case it is not, and the book does not seem to comment on the general case.

As a simple illustration of the problem consider the following equation on the unit sphere in 3d:

$dX = a \times X \; dt + b \times X \circ dW$

where $a$ and $b$ are some non-collinear constant vectors, and $\times$ is the vector cross product. Informally, at each time point we rotate $X$ around $a$ proportionally to $|a| dt$ and around $b$ proportionally to $|b| dW$. What can be said about the resulting solution $X$? Can the resulting random walk be expressed explicitly in some form?

Answer 1

It may be useful to look at the Magnus series for the solution, especially if you're in a Lie algebra setting such as your example on the sphere. This series writes the solution as $X = \exp(\Omega(t))$ where $\Omega(t)$ is an infinite series which starts with the terms you wrote down: $\Omega(t) = At + \sum_k B_k W_t + \cdots$. I don't know whether this approach is useful in your setting and I don't have time at the moment, but you can find more in the recent review paper Blanes et al., "The Magnus expansion and some of its application" and references therein.