Abstract
This paper analyzes the stability and fluctuations of the exchange rate with a speculative bubble using the methods of evolutionary finance and stochastic differential equations. It constructs a hybrid stochastic system for the financial market involving a discrete time process and a continuous time process. The discrete process models the bubble and is meant to capture the behavior of less sophisticated investors who trade infrequently. The continuous time process is a stochastic differential equation for monetary policy together with a backward stochastic equation for the exchange rate. Monetary policy is affected by the bubble and in turn affects the exchange rate as well as speculation. The bubble and exchange rate exhibit a form of bifurcation. This means the bubble and exchange rate experience fluctuations as the propensity to chase trends or switch predictors changes.
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