## Asymmetric information in fads models.(English)Zbl 1101.60047

The authors study fads models as plausible alternatives to the efficient markets/constant expected returns assumptions. Under these models, logarithms of asset prices embody both a martingale component, with permanent shocks, and a stationary component, with temporary shocks. The basic questions are: do the fads exist and if they exist, do they matter? The paper addresses these two problems in a continuous-time version of the fads models. Two types of agents are considered: informed agents who observe both fundamental and market values and uninformed agents who only observe market values. For both agents, the problem of logarithmic utility maximization from terminal wealth is studied and a closed-form solution for expected logarithmic utilities is obtained. Then the decomposition of the asset price dynamics for uninformed agent is exploited to test statistically the presence of fads from market data.

### MSC:

 60H30 Applications of stochastic analysis (to PDEs, etc.) 60G15 Gaussian processes 60G44 Martingales with continuous parameter 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 62M07 Non-Markovian processes: hypothesis testing 91B84 Economic time series analysis
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### References:

 [1] Amendinger, J.; Imkeller, P.; Schweizer, M., Additional logarithmic utility of an insider, Stoch Proc Appl, 75, 263-286 (1998) · Zbl 0934.91020 [2] Baudoin, F., Conditioned stochastic differential equations: theory, examples and application to finance, Stoch Proc Appl, 100, 109-145 (2002) · Zbl 1058.60040 [3] Campbell, J. Y.; Cochrane, J. H., By force of habit: a consumption-based explanation of aggregate stock market behavior, J Polit Econ, 107, 205-251 (1999) [4] Cheridito, P., Mixed fractional Brownian motion, Bernoulli, 7, 913-934 (2001) · Zbl 1005.60053 [5] Cheridito, P. , Representations of Gaussian measures that are equivalent to Wiener measure. In: Séminaire de Probabilités XXXVII, Lecture Notes in Mathematics, vol 1382, pp. 81-89 Berlin Heidelberg New York: Springer · Zbl 1044.60029 [6] Corcuera, J. M.; Imkeller, P.; Kohatsu-Higa, A.; Nualart, D., Additional utility of insiders with imperfect dynamical information, Finance Stoch, 8, 437-450 (2004) · Zbl 1064.60087 [7] Dunford, N.; Schwartz, J. T., Linear operators. Part II, Wiley Classics Library (1988), New York: Wiley, New York [8] Fama, E. F., Efficient capital markets: a review of theory and empirical work, J Finance, 25, 383-417 (1970) [9] Goll, T.; Kallsen, J., A complete explicit solution to the log-optimal portfolio problem, Ann Appl Probab, 13, 774-799 (2003) · Zbl 1034.60047 [10] Hitsuda, M., Representation of Gaussian processes equivalent to Wiener process, Osaka J Math, 5, 299-312 (1968) · Zbl 0174.49302 [11] Karatzas, I., Shreve, S.E. Brownian motion and stochastic calculus. In: Graduate texts in mathematics, 2nd edn. vol. 113. Berlin Heidelberg New York: Springer 1991 · Zbl 0734.60060 [12] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E.; Xu, G.-L., Martingale and duality methods for utility maximization in an incomplete market, SIAM J Control Optim, 29, 702-730 (1991) · Zbl 0733.93085 [13] Kohatsu-Higa, A.; Akahori, J.; Ogawa, S.; Watanabe, S., Enlargement of filtrations and models for insider trading, Stochastic processes and applications to mathematical finance, 151-165 (2004), Singapore: World Scientifc, Singapore [14] Kramkov, D.; Schachermayer, W., The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann Appl Probab, 9, 904-950 (1999) · Zbl 0967.91017 [15] LeRoy, S. F.; Porter, R. D., The present-value relation: tests based on implied variance bounds, Econometrica, 49, 555-574 (1981) · Zbl 0469.62096 [16] Pikovsky, I.; Karatzas, I., Anticipative portfolio optimization, Adv App Probab, 28, 1095-1122 (1996) · Zbl 0867.90013 [17] Poterba, J. M.; Summers, L. H., Mean reversion in stock prices: evidence and implications, J Financ Econ, 22, 27-59 (1988) [18] Revuz, D., Yor, M. Continuous martingales and Brownian motion, In: Grundlehren der mathematischen Wissenschaften, 3rd edn. vol. 293. Berlin Heidelberg New York: Springer 1999 · Zbl 0917.60006 [19] Shiller, R. J., Do stock prices move too much to be justified by subsequent changes in dividends, Am Econ Rev, 71, 421-436 (1981) [20] Shiller, R. J.; Perron, P., Testing the random walk hypothesis: power versus frequency of observation, Econ Lett, 18, 381-386 (1985) · Zbl 1273.91383 [21] Smithies F. (1958) Integral equations. In: Cambridge tracts in mathematics and mathematical Physics, vol. 49. Cambridge University Press, New York · Zbl 0082.31901 [22] Summers, L., Does the stock market rationally reflect fundamental values, J Finance, 41, 591-601 (1986)
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