Matrix-valued exponential martingale

Problem's Description

Let $X$ be a matrix-valued Ito process $$dX_{t}=u_{t}dt+v_{t}dW_{t}$$ where $u$ and $v$ are $d\times d$-matrices, with $d\ge 2$, and $W$ is a real Brownian motion. Consider the exponential $$Q_{t}=\text{Exp}(X_{t}).$$ Under what conditions on $u,v$ is $Q$ a local martingale?

1. Book Continuous martingales and Brownian motion

   • Daniel Revuz,
   • Marc Yor

   pp. xiv+602, year of publication: 1999fulltext