Matrix-valued SDEs arising from currency exchange markets

Abstract

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 × 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 Itô type and Itô SDEs. Two types of sufficient conditions are discussed for the existence and uniqueness of solutions to the matrixvalued SDEs.

Path independence of additive functionals for stochastic differential equations under G-framework

“…Then (S d , •, • HS , • HS ) is a Hilbert space; see e.g. [20]. Let spec(•) be the spectrum of a matrix •, and let B t = ( B i , B j t ) ij .…”