Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets
Tóm tắt
Consider a discrete-time infinite horizon financial market model in which the logarithm of the stock price is a time discretization of a stochastic differential equation. Under conditions different from those given in (Mbele Bidima and Rásonyi in Ann. Oper. Res. 200:131–146, 2012), we prove the existence of investment opportunities producing an exponentially growing profit with probability tending to 1 geometrically fast. This is achieved using ergodic results on Markov chains and tools of large deviations theory. Furthermore, we discuss asymptotic arbitrage in the expected utility sense and its relationship to the first part of the paper.
Tài liệu tham khảo
Bhattacharya, R.N., Waymire, E.C.: Stochastic Processes with Applications. Wiley, New York (1990)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin (1998)
Dokuchaev, N.: Mean-reverting market models: speculative opportunities and non-arbitrage. Appl. Math. Finance 14, 319–337 (2007)
Du, K., Neufeld, A.D.: A note on asymptotic exponential arbitrage with exponentially decaying failure probability. J. Appl. Probab. 50, 801–809 (2013)
Föllmer, H., Schachermayer, W.: Asymptotic arbitrage and large deviations. Math. Financ. Econ. 1, 213–249 (2007)
Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 2nd edn. de Gruyter, Berlin (2004)
Kabanov, Y.M., Kramkov, D.O.: Asymptotic arbitrage in large financial markets. Finance Stoch. 2, 143–172 (1998)
Kontoyiannis, I., Meyn, S.: Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13, 304–362 (2003)
Kontoyiannis, I., Meyn, S.: Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10, 61–123 (2005)
Mbele Bidima, M.L.D., Rásonyi, M.: On long-term arbitrage opportunities in Markovian models of financial markets. Ann. Oper. Res. 200, 131–146 (2012)
Meyn, S., Tweedie, R.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge (2009)