Quadratic variation, p-variation and integration with applications to stock price modelling

The paper develops a calculus for a class of real-valued functions having a quadratic variation. The main result is a solution of the representation problem for a class of evolutions having a quadratic variation. The result is applied to build up an asset pricing model. Also in the paper there are some results concerning an extension of the class of all semimartingales.

[1]  H. Oodaira.  On Strassen's version of the law of the iterated logarithm for Gaussian processes , 1972 .

[2]  Rimas Norvaisa.  Modelling of stock price changes: A real analysis approach , 2000, Finance Stochastics.

[3]  George Soros,et al.  The Alchemy of Finance: Reading the Mind of the Market , 1987 .

[4]  N. Kôno.  On self-affine functions , 1986 .

[5]  L. C. Young,et al.  An inequality of the Hölder type, connected with Stieltjes integration , 1936 .

[6]  W. T. Martin,et al.  The behavior of measure and measurability under change of scale in Wiener space , 1947 .

[7]  F. Knight.  On the random walk and Brownian motion , 1962 .

[8]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[9]  J. Kahane.  Some Random Series of Functions , 1985 .

[10]  John D. Dollard,et al.  Product Integration with Applications to Differential Equations: Product Integration of Measures , 1984 .

[11]  Another Approach to Riemann-Stieltjes Integrals , 1980 .

[12]  Tze Leung Lai,et al.  On random Fourier series , 1980 .

[13]  M. Osborne.  Brownian Motion in the Stock Market , 1959 .

[14]  E. G. Gladyshev.  A New Limit Theorem for Stochastic Processes with Gaussian Increments , 1961 .

[15]  Properties of Sample Functions for Stochastic Processes and Embedding Theorems , 1974 .

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

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

[18]  Strong variation for the sample functions of a stable process , 1973 .

[19]  Adriaan C. Zaanen,et al.  An introduction to the theory of integration , 1959 .

[20]  B. Z. Vulikh,et al.  Introduction to the Theory of Partially Ordered Spaces , 1967 .

[21]  M. Marcus,et al.  $p$-Variation of the Local Times of Symmetric Stable Processes and of Gaussian Processes with Stationary Increments , 1992 .

[22]  H. Kunita,et al.  On Square Integrable Martingales , 1967, Nagoya Mathematical Journal.

[23]  David R. Cox,et al.  The Theory of Stochastic Processes , 1967, The Mathematical Gazette.

[24]  R. Dudley,et al.  Differentiability of Six Operators on Nonsmooth Functions and p-Variation , 1999 .

[25]  M. A. Freedman.  Operators of $P$-variation and the evolution representation problem , 1983 .

[26]  Iosif Ilitch Gikhman,et al.  Introduction to the theory of random processes , 1969 .

[27]  P. Samuelson.  Mathematics of Speculative Price , 1973 .

[28]  harald Cramer,et al.  Stationary And Related Stochastic Processes , 1967 .

[29]  Terry Lyons,et al.  DIFFERENTIAL EQUATIONS DRIVEN BY ROUGH SIGNALS (I): AN EXTENSION OF AN INEQUALITY OF L. C. YOUNG , 1994 .

[30]  J. Kurzweil.  Generalized ordinary differential equations and continuous dependence on a parameter , 1957 .

[31]  Gilles Pisier,et al.  The strongp-variation of martingales and orthogonal series , 1988 .

[32]  C. Doléans-Dade,et al.  Quelques applications de la formule de changement de variables pour les semimartingales , 1970 .

[33]  F. Kittaneh,et al.  for the V , 1986 .

[34]  Walter Willinger,et al.  Dynamic spanning without probabilities , 1994 .

[35]  John D. Dollard,et al.  Product Integration with Application to Differential Equations , 1984 .

[36]  Qi-Man Shao,et al.  p-Variation of Gaussian processes with stationary increments , 1996 .

[37]  L. Young.  General inequalities for Stieltjes integrals and the convergence of Fourier series , 1938 .

[38]  Hans Föllmer,et al.  Calcul d'ito sans probabilites , 1981 .

[39]  W. Vega,et al.  On Almost Sure Convergence of Quadratic Brownian Variation , 1974 .

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

[41]  Bert Fristedt,et al.  Sample Functions of Stochastic Processes with Stationary, Independent Increments. , 1972 .

[42]  P. Levy,et al.  Random functions : general theory with special reference to Laplacian random functions , 1953 .

[43]  E. Love.  Integration by Parts and Other Theorems for R3S-Integrals , 1999 .

[44]  C. Borell.  Analytic and Empirical Evidences of Isoperimetric Processes , 1990 .

[45]  Pál Révész,et al.  Random walk in random and non-random environments , 1990 .

[46]  Rimas Norvai.  Modelling of stock price changes: A real analysis approach , 2000 .

[47]  J. Bertoin.  Sur la mesure d’Occupation d’une classe de fonctions self-affines , 1988 .

[48]  T. H. Hildebrandt.  Definitions of Stieltjes Integrals of the Riemann Type , 1938 .

[49]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[50]  Donna Salopek,et al.  Tolerance to Arbitrage , 1998 .

[51]  W. B..  The Theory of Integration , 1928, Nature.

[52]  D. H. Goldenberg.  Sample path properties of futures prices , 1986 .

[53]  J. S. Mac Nerney.  Integral equations and semigroups , 1963 .

[54]  L. C. Young,et al.  On Fractional Integration by Parts , 1938 .

[55]  E. Giné,et al.  On Quadratic Variation of Processes with Gaussian Increments , 1975 .

[56]  J. Huston McCulloch,et al.  13 Financial applications of stable distributions , 1996 .

[57]  M. Meerschaert.  Regular Variation in R k , 1988 .

[58]  S. Berman.  Stationary and Related Stochastic Processes , 1967 .

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

[60]  Gaussian Measures in $B_p$ , 1983 .

[61]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[62]  Robert M. McLeod,et al.  The generalized Riemann integral , 1980 .

[63]  R. Gomory,et al.  Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E , 1997 .

[64]  On the variation of Gaussian processes , 1973 .

[65]  Norbert Wiener,et al.  The Quadratic Variation of a Function and its Fourier Coefficients , 1924 .

[66]  A. Yaglom.  Strong Limit Theorems for Stochastic Processes and Orthogonality Conditions for Probability Measures , 1965 .

[67]  Steven M. Melimis.  Numerical methods for stochastic processes , 1978 .

[68]  Peter L. Bernstein,et al.  Capital Ideas: The Improbable Origins of Modern Wall Street , 1991 .

[69]  Benoit B. Mandelbrot,et al.  Fractals and Scaling in Finance , 1997 .

[70]  N. Wiener,et al.  Notes on random functions , 1933 .

[71]  P. Protter.  Stochastic integration and differential equations : a new approach , 1990 .

[72]  N. Krylov,et al.  Introduction to the Theory of Random Processes , 2002 .

[73]  S. Kwapień,et al.  Random Series and Stochastic Integrals: Single and Multiple , 1992 .

[74]  G. Baxter,et al.  A strong limit theorem for Gaussian processes , 1956 .

[75]  E. Hewitt,et al.  Theory of functions of a real variable , 1960 .

[76]  Richard M. Dudley,et al.  Frechet Differentiability, $p$-Variation and Uniform Donsker Classes , 1992 .

[77]  D. Dahl,et al.  Capital ideas. , 1999, Materials management in health care.

[78]  P. Meyer,et al.  Probabilities and potential C , 1978 .

[79]  C. Tricot.  Curves and Fractal Dimension , 1994 .

[80]  Helly.  Fourier transforms in the complex domain , 1936 .

[81]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[82]  R. Norvaisa,et al.  Chain rules and p-variation , 2002 .

[83]  A Closer Look at the Implications of the Stable Paretian Hypotheses , 1975 .

[84]  P. Meyer,et al.  A decomposition theorem for supermartingales , 1962 .

[85]  T. Kamae.  A characterization of self-affine functions , 1986 .

[86]  C. Rogers,et al.  Absolute and Unconditional Convergence in Normed Linear Spaces. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[87]  P. Masani.  Product Integration with Applications to Differential Equations: The Place of Multiplicative Integration in Modern Analysis , 1984 .

[88]  Walter Willinger,et al.  Pathwise stochastic integration and applications to the theory of continuous trading , 1989 .

[89]  P. Meyer.  Decomposition of supermartingales: The uniqueness theorem , 1963 .

[90]  O. Barndorff-Nielsen.  Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling , 1997 .

[91]  On self-affine functions II , 1988 .

[92]  T. Andersen.  THE ECONOMETRICS OF FINANCIAL MARKETS , 1998, Econometric Theory.

[93]  The character space of the algebra of regulated functions. , 1978 .

[94]  N. Wiener,et al.  Fourier Transforms in the Complex Domain , 1934 .

[95]  P. Protter.  Stochastic integration and differential equations , 1990 .