Abstract
In this paper we study a two-country economy and model a derivatives market where domestic T-bonds, foreign T-bonds and exchange rate derivatives are traded. We use the Heath-Jarrow-Morton approach, modeling domestic and foreign forward rates directly under each country martingale measure, assuming that they are both driven by a multidimensional Wiener process. Thus, the interest rate setting for both countries is defined by two infinite dimensional stochastic differential equations (SDEs). In addition we model the exchange rate stochastic process under the domestic martingale measure. The purpose of the paper is to understand when the inherently infinite dimensional domestic and foreign forward rate processes can be realized by means of a Markovian finite dimensional state space model. We assume that both forward rate volatilities and the volatility of the exchange rate can be arbitrary smooth functionals of both forward rate curves. We find necessary and sufficient conditions in terms of the exchange rate and interest rate volatilities to ensure that we obtain a finite dimensional realization (FDR) for both domestic and forward interest rates and we show how the exchange rate depends on such realizations. Finally, for the case when FDRs exist we show how to derive the dynamics of the underlying factors spanning the state space.