A (driftless) Geometric Brownian Motion is characterized by the scalar SDE:
$dS_t = \sigma S_tdB_t$
with $B_t$ a standard Brownian Motion (BM).
I am interested in the $d \times d$-dimensional SDE for $d \geq 2$:
$dX_t = \sigma X_t dW_t$
where $X_t$ is a $\mathbb{R}^{d \times d}$-valued stochastic process, $W_t$ is a $d \times d$ matrix of independent standard BMs, and $X_t dW_t$ stands for matrix multiplication, i.e. for $i,j = 1,\dots,d$:
$[dX_t]_{i,j} = \sigma \sum_{k=1}^d [X_t]_{i,k} [dW_t]_{k,j}$
$[X_T]_{i,j} = [X_0]_{i,j} + \sigma \sum_{k=1}^d \int_{0}^T [X_t]_{i,k} [dW_t]_{k,j}$
Has this process been studied already, and if so what would be a reference?
I am mainly interested in whether an explicit solution can be found.
