Constructing functions with prescribed pathwise quadratic variation. (English) Zbl 1343.60066
Summary: We construct rich vector spaces of continuous functions with prescribed curved or linear pathwise quadratic variations. We also construct a class of functions whose quadratic variation may depend in a local and nonlinear way on the function value. These functions can then be used as integrators in Föllmer’s pathwise Itō calculus. Our construction of the latter class of functions relies on an extension of the Doss-Sussman method to a class of nonlinear Itō differential equations for the Föllmer integral. As an application, we provide a deterministic variant of the support theorem for diffusions. We also establish that many of the constructed functions are nowhere differentiable.
MSC:
| 60H05 | Stochastic integrals |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60J60 | Diffusion processes |
Keywords:
pathwise quadratic variation; Föllmer integral; pathwise Itō differential equation; Doss-Sussman method; diffusions; support theorem; nowhere differentiabilityReferences:
| [1] | Bender, C.; Sottinen, T.; Valkeila, E., Pricing by hedging and no-arbitrage beyond semimartingales, Finance Stoch., 12, 4, 441-468 (2008) · Zbl 1199.91170 |
| [2] | Bick, A.; Willinger, W., Dynamic spanning without probabilities, Stochastic Process. Appl., 50, 2, 349-374 (1994) · Zbl 0801.90010 |
| [3] | Ciesielski, Z., On the isomorphisms of the spaces \(H_\alpha\) and \(m\), Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., 8, 217-222 (1960) · Zbl 0093.12301 |
| [4] | 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 |
| [5] | Cont, R.; Fournié, D.-A., Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab., 41, 1, 109-133 (2013) · Zbl 1272.60031 |
| [6] | Davis, M.; Obłój, J.; Raval, V., Arbitrage bounds for prices of weighted variance swaps, Math. Finance, 24, 4, 821-854 (2014) · Zbl 1314.91209 |
| [7] | de Rham, G., Sur un exemple de fonction continue sans dérivée, Enseign. Math., 3, 71-72 (1957) · Zbl 0077.06104 |
| [8] | Doss, H., Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré Sect. B (N.S.), 13, 2, 99-125 (1977) · Zbl 0359.60087 |
| [9] | Dupire, B., Functional Itô calculus (2009), Bloomberg Portfolio Research Paper |
| [10] | 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 |
| [11] | Ekren, I.; Touzi, N.; Zhang, J., Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) |
| [12] | Engelbert, H. J.; Schmidt, W., On solutions of one-dimensional stochastic differential equations without drift, Z. Wahrsch. Verw. Gebiete, 68, 3, 287-314 (1985) · Zbl 0535.60049 |
| [13] | Ethier, S. N.; Kurtz, T. G., Markov Processes. Characterization and Convergence, Wiley Ser. Probab. Math. Stat.: Probab. Math. Stat. (1986), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York · Zbl 0592.60049 |
| [14] | Föllmer, H., Calcul d’Itô sans probabilités, (Seminar on Probability, XV. Seminar on Probability, XV, Univ. Strasbourg, Strasbourg, 1979/1980. Seminar on Probability, XV. Seminar on Probability, XV, Univ. Strasbourg, Strasbourg, 1979/1980, Lecture Notes in Math., vol. 850 (1981), Springer: Springer Berlin), 143-150 · Zbl 0461.60074 |
| [15] | 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 |
| [16] | Freedman, D., Brownian Motion and Diffusion (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0501.60070 |
| [17] | Gantert, N., Einige grosse Abweichungen der Brownschen Bewegung, Bonner Math. Schriften, vol. 224 (1991), Rheinische Friedrich-Wilhelms-Universität: Rheinische Friedrich-Wilhelms-Universität Bonn, Dissertation · Zbl 0746.60026 |
| [18] | 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 |
| [19] | Groh, J., A nonlinear Volterra-Stieltjes integral equation and a Gronwall inequality in one dimension, Illinois J. Math., 24, 2, 244-263 (1980) · Zbl 0454.45002 |
| [20] | Klingenhöfer, F.; Zähle, M., Ordinary differential equations with fractal noise, Proc. Amer. Math. Soc., 127, 4, 1021-1028 (1999) · Zbl 0915.34054 |
| [21] | Kuipers, L.; Niederreiter, H., Uniform Distribution of Sequences, Pure Appl. Math. (1974), Wiley-Interscience [John Wiley & Sons]: Wiley-Interscience [John Wiley & Sons] New York-London-Sydney · Zbl 0281.10001 |
| [22] | Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces. I, Ergeb. Math. Ihrer Grenzgeb., vol. 92 (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0362.46013 |
| [23] | Lyons, T. J., Uncertain volatility and the risk-free synthesis of derivatives, Appl. Math. Finance, 2, 2, 117-133 (1995) · Zbl 1466.91347 |
| [24] | Mureşan, M., A Concrete Approach to Classical Analysis, CMS Books Math./Ouvrages Math. SMC (2009), Springer: Springer New York · Zbl 1163.26001 |
| [25] | Rudin, W., Principles of Mathematical Analysis, Int. Ser. Pure Appl. Math. (1976), McGraw-Hill Book Co.: McGraw-Hill Book Co. New York-Auckland-Düsseldorf · Zbl 0148.02903 |
| [26] | Schied, A., Model-free CPPI, J. Econom. Dynam. Control, 40, 84-94 (2014) · Zbl 1402.91732 |
| [27] | Schied, A., On a class of generalized Takagi functions with linear pathwise quadratic variation, J. Math. Anal. Appl., 433, 974-990 (2016) · Zbl 1325.26020 |
| [28] | 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 |
| [29] | Schied, A.; Voloshchenko, I., Pathwise no-arbitrage in a class of delta hedging strategies (2015) · Zbl 1443.91299 |
| [30] | Sondermann, D., Introduction to Stochastic Calculus for Finance, Lecture Notes in Econom. and Math. Systems, vol. 579 (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1136.91014 |
| [31] | Stroock, D. W.; Varadhan, S. R.S., On the support of diffusion processes with applications to the strong maximum principle, (Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, Calif., 1970/1971. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, Calif., 1970/1971, Probab. Theory, vol. III (1972), Univ. California Press: Univ. California Press Berkeley, Calif.), 333-359 |
| [32] | Sussmann, H. J., On the gap between deterministic and stochastic ordinary differential equations, Ann. Probab., 6, 1, 19-41 (1978) · Zbl 0391.60056 |
| [33] | Teschl, G., Ordinary Differential Equations and Dynamical Systems, Grad. Stud. Math., vol. 140 (2012), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 1263.34002 |
| [34] | Vovk, V., Continuous-time trading and the emergence of probability, Finance Stoch., 16, 4, 561-609 (2012) · Zbl 1262.91163 |
| [35] | Widder, D. V., The Laplace Transform, Princeton Math. Ser., vol. 6 (1941), Princeton University Press: Princeton University Press Princeton, NJ · JFM 67.0384.01 |
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