On a class of generalized Takagi functions with linear pathwise quadratic variation. (English) Zbl 1325.26020
Summary: We consider a class \(\mathcal{X}\) of continuous functions on \([0, 1]\) that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in \(\mathcal{X}\) admits a linear pathwise quadratic variation and can thus serve as an integrator in Föllmer’s pathwise Itō calculus. We derive several uniform properties of the class \(\mathcal{X}\). For instance, we compute the overall pointwise maximum, the uniform maximal oscillation, and the exact uniform modulus of continuity for all functions in \(\mathcal{X}\). Furthermore, we give an example of a pair \(x, y \in \mathcal{X}\) for which the quadratic variation of the sum \(x + y\) does not exist.
MSC:
| 26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |
Keywords:
generalized Takagi function; uniform modulus of continuity; pathwise quadratic variation; pathwise covariation; pathwise Itō calculus; Föllmer integralReferences:
| [1] | Allaart, P. C., On a flexible class of continuous functions with uniform local structure, J. Math. Soc. Japan, 61, 1, 237-262 (2009) · Zbl 1161.26003 |
| [2] | Allaart, P. C.; Kawamura, K., The Takagi function: a survey, Real Anal. Exchange, 37, 1, 1-54 (2011) · Zbl 1248.26007 |
| [3] | Bender, C.; Sottinen, T.; Valkeila, E., Pricing by hedging and no-arbitrage beyond semimartingales, Finance Stoch., 12, 4, 441-468 (2008) · Zbl 1199.91170 |
| [4] | Bick, A.; Willinger, W., Dynamic spanning without probabilities, Stochastic Process. Appl., 50, 2, 349-374 (1994) · Zbl 0801.90010 |
| [5] | Billingsley, P., Van der Waerden’s continuous nowhere differentiable function, Amer. Math. Monthly, 89, 691 (1982) · Zbl 0598.26014 |
| [6] | Cont, R.; Fournié, D.-A., Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259, 4, 1043-1072 (2010) · Zbl 1201.60051 |
| [7] | Cont, R.; Fournié, D.-A., Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab., 41, 1, 109-133 (2013) · Zbl 1272.60031 |
| [8] | Davis, M.; Obłój, J.; Raval, V., Arbitrage bounds for prices of weighted variance swaps, Math. Finance, 24, 4, 821-854 (2014) · Zbl 1314.91209 |
| [9] | de Rham, G., Sur un exemple de fonction continue sans dérivée, Enseign. Math., 3, 71-72 (1957) · Zbl 0077.06104 |
| [10] | Dupire, B., Functional Itô calculus (2009), available at http://dx.doi.org/10.2139/ssrn.1435551 |
| [11] | Ekren, I.; Keller, C.; Touzi, N.; Zhang, J., On viscosity solutions of path dependent PDEs, Ann. Probab., 42, 1, 204-236 (2014) · Zbl 1320.35154 |
| [12] | Faber, G., Über die Orthogonalfunktionen des Herrn Haar, Jahresber. Dtsch. Math.-Ver., 19, 104-112 (1910) · JFM 41.0470.01 |
| [13] | Föllmer, H., Calcul d’Itô sans probabilités, (Seminar on Probability, XV, Univ. Strasbourg, Strasbourg, 1979/1980. Seminar on Probability, XV, Univ. Strasbourg, Strasbourg, 1979/1980, Lecture Notes in Math., vol. 850 (1981), Springer: Springer Berlin), 143-150 · Zbl 0461.60074 |
| [14] | Föllmer, H., Probabilistic aspects of financial risk, (European Congress of Mathematics, vol. I. European Congress of Mathematics, vol. I, Barcelona, 2000. European Congress of Mathematics, vol. I. European Congress of Mathematics, vol. I, Barcelona, 2000, Progr. Math., vol. 201 (2001), Birkhäuser: Birkhäuser Basel), 21-36 · Zbl 1047.91041 |
| [15] | Föllmer, H.; Schied, A., Probabilistic aspects of finance, Bernoulli, 19, 4, 1306-1326 (2013) · Zbl 1279.91053 |
| [16] | Freedman, D., Brownian Motion and Diffusion (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0501.60070 |
| [17] | Friz, P. K.; Hairer, M., A Course on Rough Paths (2014), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 1327.60013 |
| [18] | Gantert, N., Einige grosse Abweichungen der Brownschen Bewegung, Bonner Mathematische Schriften, vol. 224 (1991), Rheinische Friedrich-Wilhelms-Universität Bonn, Dissertation · Zbl 0746.60026 |
| [19] | Gantert, N., Self-similarity of Brownian motion and a large deviation principle for random fields on a binary tree, Probab. Theory Related Fields, 98, 1, 7-20 (1994) · Zbl 0794.60014 |
| [20] | Hata, M.; Yamaguti, M., The Takagi function and its generalization, Jpn. J. Appl. Math., 1, 1, 183-199 (1984) · Zbl 0604.26004 |
| [21] | Kahane, J.-P., Sur l’exemple, donné par M. de Rham, d’une fonction continue sans dérivée, Enseign. Math., 5, 53-57 (1959) · Zbl 0090.27202 |
| [22] | Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, vol. 113 (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0734.60060 |
| [23] | Kôno, N., On generalized Takagi functions, Acta Math. Hungar., 49, 3-4, 315-324 (1987) · Zbl 0627.26004 |
| [24] | Lagarias, J. C., The Takagi function and its properties, (Functions in Number Theory and Their Probabilistic Aspects. Functions in Number Theory and Their Probabilistic Aspects, RIMS Kôkyûroku Bessatsu, vol. B34 (2012), Res. Inst. Math. Sci. (RIMS): Res. Inst. Math. Sci. (RIMS) Kyoto), 153-189 · Zbl 1275.26011 |
| [25] | Ledrappier, F., On the dimension of some graphs, (Symbolic Dynamics and Its Applications. Symbolic Dynamics and Its Applications, New Haven, CT, 1991. Symbolic Dynamics and Its Applications. Symbolic Dynamics and Its Applications, New Haven, CT, 1991, Contemp. Math., vol. 135 (1992), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 285-293 · Zbl 0767.28006 |
| [26] | Schied, A., Model-free CPPI, J. Econom. Dynam. Control, 40, 84-94 (2014) · Zbl 1402.91732 |
| [27] | Schied, A.; Stadje, M., Robustness of delta hedging for path-dependent options in local volatility models, J. Appl. Probab., 44, 4, 865-879 (2007) · Zbl 1210.91136 |
| [28] | Sondermann, D., Introduction to Stochastic Calculus for Finance, Lecture Notes in Economics and Mathematical Systems, vol. 579 (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1136.91014 |
| [29] | Takagi, T., A simple example of the continuous function without derivative, Proc. Phys.-Math. Soc. Jpn., 1, 176-177 (1903) · JFM 34.0410.05 |
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