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Stochastic impulse control of exchange rates with Freidlin–Wentzell perturbations

Part of: Mathematical finance Mathematical economics Stochastic processes

Published online by Cambridge University Press:  04 April 2017

Gregory Gagnon
Gregory Gagnon*
Affiliation:
University of Toronto Mississauga
*
* Postal address: Department of Economics, University of Toronto Mississauga, 3359 Mississauga Road, Mississauga, Ontario, L5L 1C6, Canada. Email address: [email protected]
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Abstract

This paper pioneers a Freidlin–Wentzell approach to stochastic impulse control of exchange rates when the central bank desires to maintain a target zone. Pressure to stimulate the economy forces the bank to implement diffusion monetary policy involving Freidlin–Wentzell perturbations indexed by a parameter ε∈ [0,1]. If ε=0, the policy keeps exchange rates in the target zone for all times t≥0. When ε>0, exchange rates continually exit the target zone almost surely, triggering central bank interventions which force currencies back into the zone or abandonment of all targets. Interventions and target zone deviations are costly, motivating the bank to minimize these joint costs for any ε∈ [0,1]. We prove convergence of the value functions as ε→0 achieving a value function approximation for small ε. Via sample path analysis and cost function bounds, intervention followed by target zone abandonment emerges as the optimal policy.

Keywords

Stochastic impulse controlFreidlin–Wentzell theoryvalue functiontarget zone

MSC classification

Primary: 91B64: Macro-economic models (monetary models, models of taxation) 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Secondary: 60G17: Sample path properties 91B55: Economic dynamics
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 23 - 41
Copyright
Copyright © Applied Probability Trust 2017 

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References

Cadenillas, A. and Zapatero, F. (1999).Optimal central bank intervention in the foreign exchange market.J. Econom. Theory 87,218–242.Google Scholar
Cadenillas, A. and Zapatero, F. (2000).Classical and stochastic impulse control of the exchange rate using interest rates and reserves.Math. Finance 10,141–156.Google Scholar
El Karoui, N., Peng, S. and Quenez, M. (1997).Backward stochastic differential equations in finance.Math. Finance 7,1–71.Google Scholar
Flood, R. P. and Garber, P. M. (1983).A model of stochastic process switching.Econometrica 51,537–551.CrossRefGoogle Scholar
Freidlin, M. I. and Wentzell, A. D. (1998).Random Perturbations of Dynamical Systems, 2nd edn. Springer,New York.Google Scholar
Freidlin, M. I. and Wentzell,, A. D. (2006).Long-time behaviour of weakly coupled oscillators.J. Statist. Phys. 123,1311–1337.Google Scholar
Gagnon, G. (2012).Exchange rate bifurcation in a stochastic evolutionary finance model.Decis. Econom. Finance 35,29–58.Google Scholar
Galves, A., Olivieri, E. and Vares, M. E. (1987).Metastability for a class of dynamical systems subject to small random perturbations.Ann. Prob. 15,1288–1305.Google Scholar
Jeanblanc-Picqué, M. (1993).Impulse control method and exchange rate.Math. Finance 3,161–177.Google Scholar
Mundaca, G. and Øksendal, B. (1998).Optimal stochastic intervention control with application to the exchange rate.J. Math. Econom. 29,225–243.Google Scholar
Wang, P. (2009).The Economics of Foreign Exchange and Global Finance.Springer,New York.Google Scholar
Yong, J. and Zhou, X. Y. (1999). Stochastic Controls.Springer,New York.CrossRefGoogle Scholar