"Generalized" Geometric Brownian Motion as a SDE system

Question

It is very well known that the equation $$d X_t = \mu X_t dt+\sigma X_tdW_t$$ has a solution $$X_t = X_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t},$$ and we say that $X_t$ follows Geometric Brownian Motion. What if we consider a linear system $$dX_t = A X_t dt+BX_tdW_t$$ where $X_t = (X_t^1,\ldots,X_t^n)$, $W_t$ is a Wiener process, $A, B$ are a $n\times n$ matrices. For example, let $$A = \left[\begin{array}{cc}0&1\\0&0\end{array}\right]\quad \mathrm{and}\quad B= \left[\begin{array}{cc}0&0\\1&0\end{array}\right].$$ Then we get $$\left\{\begin{array}{l}d X_t^1 = X_t^2dt\\d X_t^2 = X_t^1dW_t\end{array}\right.$$ There is of course a trivial solution $X_t\equiv 0$, but what can we say about the other solutions in general? I couldn't find examples of such systems of equations. I found papers on equations of type $$dX_t = A X_t dt+B dW_t,$$ but the techniques used there don't seem to transfer here. I would greatly appreciate if someone could refer me to some books or papers which might help me with that type of systems.

Answer 1

The general form of a vector-valued homogenous SDE is $$ \tag{1} dX(t) = A(t)X(t)dt\phantom{dt}\!\!\!\! + \sum_{j=1}^m B_j(t)X(t) dW_j, $$ where $A(t)$ and $B_j(t)$ are $d \times d$ matrices and $W(t) = (W_1(t), \ldots W_m(t))$ is an $m$-dimensional Wiener process.

Provided that the matrices $A(t), B(t)$ are bounded on a time interval of interest, say $[t_0, T]$, and the initial state $X_{t_0}$ is independent of $W(t) - W(t_0)$ for $t \in [t_0, T]$ then (1) has a unique solution.

If $A(t)$ and $B(t)$ commute, then there is an explicit solution, which is analogous to the scalar case (but matrix-valued). If we don't have commutativity, then, in general, there is no nice explicit solution.

See "Stochastic Differential Equations. Theory and Applications" by L. Arnold.