Sydney, the AFMATH Conference in Brussels, the Cambridge-Princeton Conference and the TCF Workshop on Lessons from the Credit Crisis, and in particular to Kenneth Lindsay, for their comments and suggestions. This research was funded in part by the NSF under grant SES-0850533 (Aït-Sahalia) and by the NWO under grants Veni-2006 and Vidi-2009 (Laeven). Matlab code to implement the estimation procedure developed in this paper is available from the authors upon request. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

JEL classification: C58 G01 G15 C32Keywords: Jumps Contagion Crisis Hawkes process Self-and mutually exciting processes a b s t r a c t We propose a model to capture the dynamics of asset returns, with periods of crises that are characterized by contagion. In the model, a jump in one region of the world increases the intensity of jumps both in the same region (self-excitation) as well as in other regions (cross-excitation), generating episodes of highly clustered jumps across world markets that mimic the observed features of the data. We develop and implement moment-based estimation and testing procedures for this model. The estimates provide evidence of selfexcitation both in the US and the other world markets, and of asymmetric crossexcitation, with the US market typically having more influence on the jump intensity of other markets than the reverse. We propose filtered values of the jump intensities as a measure of market stress and examine their out-of-sample forecasting abilities.

We analyze the consumption-portfolio selection problem of an investor facing both Brownian and jump risks. We bring new tools, in the form of orthogonal decompositions, to bear on the problem in order to determine the optimal portfolio in closed form. We show that the optimal policy is for the investor to focus on controlling his exposure to the jump risk, while exploiting differences in the Brownian risk of the asset returns that lies in the orthogonal space.Comment: Published in at http://dx.doi.org/10.1214/08-AAP552 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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