## A note on asymptotic exponential arbitrage with exponentially decaying failure probability.(English)Zbl 1286.91124

The main result of the paper states that if a price process $$S$$ is a continuous semimartingale and satisfies a large deviations estimate (a particular growth condition on the mean-variance tradeoff of $$S$$), then $$S$$ allows for asymptotic exponential arbitrage with exponentially decaying failure probability. This statement is a proof of a result conjectured by H. Föllmer and W. Schachermayer [Math. Financ. Econ. 1, No. 3–4, 213–249 (2008; Zbl 1153.91015)], but, in contrast to the latest conjecture, the authors’ result does not assume that $$S$$ is a diffusion and does not need any ergodicity assumption.

### MSC:

 91G10 Portfolio theory 60F10 Large deviations 60G44 Martingales with continuous parameter 60H30 Applications of stochastic analysis (to PDEs, etc.)

Zbl 1153.91015
Full Text:

### References:

 [1] Föllmer, H. and Schachermayer, W. (2007). Asymptotic arbitrage and large deviations. Math. Finance Econom. 1 , 213-249. · Zbl 1153.91015 [2] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Fundamental Principles Math. Sci. 288 ), 2nd edn. Springer, Berlin. · Zbl 1018.60002 [3] Karatzas, I. and Shreve, S. E. (2000). Brownian Motion and Stochastic Calculus , 2nd edn. Springer, Berlin. · Zbl 0734.60060 [4] Mbele Bidima, M. L. D. and Rásonyi, M. (2012). On long-term arbitrage opportunities in Markovian models of financial markets. Ann. Operat. Res. 200 , 131-146. · Zbl 1255.90084 [5] Schweizer, M. (1995). On the minimal martingale measure and the Föllmer-Schweizer decomposition. Stoch. Anal. Appl. 13 , 573-599. · Zbl 0837.60042
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