Semimartingale dynamics for a backward exchange rate process

Abstract

Via a forward SDE solution $\left(k_{t}, t \geq 0\right)$ that captures money supply dynamics, a macroeconomic model known as the monetary model generates a backward exchange rate process $\left(y_{t}, t \geq 0\right)$. For any $t \geq 0$, $y_{t}=k_{t}+\alpha^{-1} \mu_{t}$ where $\left(\mu_{t}, t \geq 0\right)$ is a backward process and $\alpha>0$ is a constant. Thus, $\left(y_{t}, t \geq 0\right)$ does not satisfy a conventional BSDE. Our paper proves $\left(y_{t}, t \geq 0\right)$ is a continuous semimartingale when restrictions on the SDE for $\left(k_{t}, t \geq 0\right)$ capture anti-inflationary initiatives. This new result in economic dynamics does not require the filtration to be the Brownian filtration.

Keywords

Backward process / Semimartingale / Anti-inflationary SDE policy

Acknowledgements

Many thanks are due to my parents, Linda Gagnon and Philip Gagnon, for their constant encouragement and to the referee whose comments greatly improved the paper. The paper is dedicated to the memory of Napoleon Bonaparte, a great innovator both on the battlefield and in politics.

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