# zbMATH — the first resource for mathematics

Discretely sampled signals and the rough Hoff process. (English) Zbl 1348.60058
Summary: We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead-lag increments. In particular, by sampling a $$d$$-dimensional continuous semimartingale $$X : [0, 1] \to \mathbb{R}^d$$ at a set of times $$D = \{t_i \}$$, we construct a piecewise linear, axis-directed process $$X^D : [0, 1] \to \mathbb{R}^{2 d}$$ comprised of a past and a future component. We call such an object the Hoff process associated with the discrete data $$\{X_t \}_{t_i \in D}$$. The Hoff process can be lifted to its natural rough path enhancement and we consider the question of convergence as the sampling frequency increases. We prove that the Itô integral can be recovered from a sequence of random ODEs driven by the components of $$X^D$$. This is in contrast to the usual Stratonovich integral limit suggested by the classical Wong-Zakai Theorem [E. Wong and M. Zakai, Int. J. Eng. Sci. 3, 213–229 (1965; Zbl 0131.16401)]. Such random ODEs have a natural interpretation in the context of mathematical finance.

##### MSC:
 60G17 Sample path properties 60G35 Signal detection and filtering (aspects of stochastic processes) 60G44 Martingales with continuous parameter 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text:
##### References:
 [1] C. Bayer, P. Friz, J. Gatheral, Pricing under rough volatility. Preprint, 2015. [2] D.L. Burkholder, B.J. Davis, R.F. Gundy, Integral inequalities for convex functions of operators on martingales, in: Proc. Sixth Berkeley Symp. Math. Statist. Prob., Vol. 2, 1972, pp. 223-240. · Zbl 0253.60056 [3] Chung, K. L., A course in probability theory, (2001), Academic Press · Zbl 0159.45701 [4] Comte, F.; Renault, E., Long memory in continuous-time stochastic volatility models, J. Math. Finance, 8, 4, 291-323, (1998) · Zbl 1020.91021 [5] Coutin, L.; Lejay, A., Semimartingales and rough paths theory, Electron. J. Probab., 10, 23, 761-785, (2005) · Zbl 1109.60035 [6] Friz, P., Examples from physics and economics where rough paths matter, Oberwolfach Rep., 9, 3, (2012) [7] Friz, P.; Gassiat, P.; Lyons, T. J., Physical Brownian motion in magnetic field as a rough path, Trans. Amer. Math. Soc., (2015) · Zbl 1390.60257 [8] Friz, P.; Hairer, M., A course on rough paths: with an introduction to regularity structures, (2014), Springer · Zbl 1327.60013 [9] Friz, P.; Oberhauser, H., Rough path limits of the Wong-zakai type with a modified drift term, J. Funct. Anal., 256, 10, 3236-3256, (2009) · Zbl 1169.60011 [10] Friz, P.; Victoir, N., A note on the notion of geometric rough paths, Probab. Theory Related Fields, 136, 3, 395-416, (2006) · Zbl 1108.34052 [11] Friz, P.; Victoir, N., Multidimensional stochastic processes as rough paths: theory and applications, vol. 120, (2010), Cambridge University Press [12] J. Gatheral, T. Jaisson, M. Rosenbaum, Volatility is rough. Preprint, 2014. · Zbl 1400.91590 [13] Gyurkó, G.; Lyons, T. J.; Kontkowski, M.; Field, J., Extracting information from the signature of a financial data stream. technical report, (2013) [14] Hambly, B.; Lyons, T. J., Uniqueness for the signature of a path of bounded variation and the reduced path group, Ann. of Math., 171, 1, 109-167, (2010) · Zbl 1276.58012 [15] Hoff, B., The Brownian frame process as a rough path, (2005), University of Oxford, (Ph.D. thesis) [16] Lejay, A., An introduction to rough paths, (Séminaire de probabilités, Vol. XXXVII, (2003), Springer), 1-59 · Zbl 1041.60051 [17] D. Levin, T.J. Lyons, H. Ni, Learning from the past, predicting the statistics for the future, learning an evolving system. Preprint, 2013. [18] Lyons, T. J., Differential equations driven by rough signals. I. an extension of an inequality of Young, Math. Res. Lett., 1, 4, 451-464, (1994) · Zbl 0835.34004 [19] T.J. Lyons, Rough paths, Signatures and the modelling of functions on streams, 2014. ArXiv Preprint arXiv:1405.4537. · Zbl 1373.93158 [20] Lyons, T. J.; Caruana, M.; Lévy, T., Differential equations driven by rough paths, (École d’Eté de Probabilités de Saint-Flour, Vol. XXXIV, (2004)), 1-93 [21] Lyons, T. J.; Ni, H.; Oberhauser, H., A feature set for streams and an application to high-frequency financial tick data, (Proceedings of the 2014 International Conference on Big Data Science and Computing, (2014), ACM), 5 [22] Lyons, T. J.; Qian, Z., System control and rough paths, (2002), Oxford University Press · Zbl 1029.93001 [23] H. Ni, A multidimensional stream and its signature representation. Preprint, 2015. [24] N. Perkowski, D.J. Prömel, Pathwise stochastic integrals for model free finance, 2013. ArXiv Preprint arXiv:1311.6187. · Zbl 1346.60078 [25] Rogers, L. C.G., Arbitrage with fractional Brownian motion, Math. Finance, 7, 1, 95-105, (1997) · Zbl 0884.90045 [26] Rogers, L. C.G.; Williams, D., Diffusions, Markov processes and martingales: volume 2, Itô calculus, (2000), Cambridge University Press · Zbl 0977.60005 [27] Sipiläinen, E. M., A pathwise view on solutions of stochastic differential equations, (1993), University of Edinburgh, (Ph.D. thesis) [28] Stein, E. M., Singular integrals and differentiability properties of functions, vol. 2, (1970), Princeton University Press [29] Williams, D. R.E., Path-wise solutions of stochastic differential equations driven by Lévy processes, Rev. Mat. Iberoam., 17, 2, 295-330, (2001) · Zbl 1002.60060 [30] Wong, E.; Zakai, M., On the relation between ordinary and stochastic differential equations, Internat. J. Engrg. Sci., 3, 2, 213-229, (1965) · Zbl 0131.16401 [31] Young, L. C., An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67, 1, 251-282, (1936) · Zbl 0016.10404
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.