Continuous-time trading and the emergence of volatility

This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the case of continuous martingales.

[1]  Akimichi Takemura,et al.  Lévy’s Zero–One Law in Game-Theoretic Probability , 2009, Journal of Theoretical Probability.

[2]  Kei Takeuchi,et al.  Game-theoretic versions of strong law of large numbers for unbounded variables , 2006, math/0603184.

[3]  A. Dvoretzky,et al.  Nonincrease Everywhere of the Brownian Motion Process , 1961 .

[4]  D. Lépingle,et al.  La variation d'ordre p des semi-martingales , 1976 .

[5]  Stephen Taylor,et al.  Exact asymptotic estimates of Brownian path variation , 1972 .

[6]  G. Schwarz,et al.  On time-free functions , 1972 .

[7]  Vladimir Vovk,et al.  Forecasting point and continuous processes: Prequential analysis , 1993 .

[8]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[9]  Krzysztof Burdzy,et al.  On Nonincrease of Brownian Motion , 1990 .

[10]  Kei Takeuchi,et al.  Capital Process and Optimality Properties of a Bayesian Skeptic in Coin-Tossing Games , 2005, math/0510662.

[11]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[12]  Akimichi Takemura,et al.  On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game , 2005, math/0508190.

[13]  A. M. Davie,et al.  Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation , 2007, 0710.0772.

[14]  Rajeeva L. Karandikar,et al.  On the quadratic variation process of a continuous Martingale , 1983 .

[15]  G. Shafer,et al.  Algorithmic Learning in a Random World , 2005 .

[16]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[17]  L. Dubins,et al.  ON CONTINUOUS MARTINGALES. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[18]  P. M. Neumann,et al.  Groups and Geometry , 1994 .

[19]  K. E. Dambis,et al.  On the Decomposition of Continuous Submartingales , 1965 .

[20]  Peter,et al.  Continuous-time trading and emergence of randomness , 2007 .

[21]  T. Björk.  Arbitrage Theory in Continuous Time , 2019 .

[22]  Terry Lyons.  Di erential equations driven by rough signals , 1998 .

[23]  W. Rudin.  Real and complex analysis , 1968 .

[24]  M. Lévy.  Le Mouvement Brownien Plan , 1940 .

[25]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[26]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[27]  Akimichi Takemura,et al.  Implications of contrarian and one-sided strategies for the fair-coin game , 2007 .

[28]  Kei Takeuchi,et al.  A new formulation of asset trading games in continuous time with essential forcing of variation exponent , 2007, 0708.0275.

[29]  G. Shafer,et al.  Probability and Finance: It's Only a Game! , 2001 .

[30]  Richard M. Dudley,et al.  Sample Functions of the Gaussian Process , 1973 .

[31]  Vladimir Vovk,et al.  Continuous-time trading and the emergence of randomness , 2007, 0712.1275.

[32]  A. Dawid,et al.  Prequential probability: principles and properties , 1999 .

[33]  C. Stricker,et al.  Sur la p-variation des surmartingales , 1979 .

[34]  Vladimir Vovk,et al.  A Game-Theoretic Explanation of the √(dt) Effect , 2003 .

[35]  J. Hoffmann-jorgensen,et al.  The general marginal problem , 1987 .

[36]  G Schwarz,et al.  Time-free continuous processes. , 1968, Proceedings of the National Academy of Sciences of the United States of America.