Matrix-valued SDEs arising from currency exchange markets

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.