  In this paper, motivated by modelling currency exchange markets with
matrix-valued stochastic processes, matrix-valued stochastic differential
equations (SDEs) are formulated. This is done based on the matrix trace, as for
the purpose of modelling currency exchange markets. To be more precise, we set
up a Hilbert space structure for $n\times n$ square matrices via the trace of
the Hadamard product of two matrices. With the help of this framework, one can
then define stochastic integral of It\^o type and It\^o SDEs. Two types of
sufficient conditions are discussed for the existence and uniqueness of
solutions to the matrix-valued SDEs.
