×

Unified signature cumulants and generalized Magnus expansions. (English) Zbl 1527.60083

Summary: The signature of a path can be described as its full non-commutative exponential. Following T. Lyons, we regard its expectation, the expected signature, as a path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions with motivations ranging from financial mathematics to statistical physics. From an affine semimartingale perspective, the functional relation may be interpreted as a type of generalized Riccati equation.

MSC:

60L10 Signatures and data streams
60L90 Applications of rough analysis
60E10 Characteristic functions; other transforms
60G44 Martingales with continuous parameter
60G48 Generalizations of martingales
60G51 Processes with independent increments; Lévy processes
60J76 Jump processes on general state spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abi Jaber, E., Larsson, M., and Pulido, S., Affine volterra processes, Ann. Appl. Probab.29 (2019), no. 5, 3155-3200. · Zbl 1441.60052
[2] Aït-Sahalia, Y. and Jacod, J., High-frequency financial econometrics, Princeton University Press, 2014. · Zbl 1298.91018
[3] Alos, E., Gatheral, J., and Radoičić, R., Exponentiation of conditional expectations under stochastic volatility, Quantitative Finance; SSRN (2017) 20 (2020), no. 1, 13-27. · Zbl 1431.91387
[4] Applebaum, D., Lévy processes and stochastic calculus, Cambridge University Press, 2009. · Zbl 1200.60001
[5] Arribas, I. P., Salvi, C., and Szpruch, L., Sig-sdes model for quantitative finance, 2020, arXiv:2006.00218 [q-fin.CP], 1st ACM International Conference on AI in Finance (ICAIF 2020).
[6] Blanes, S., Casas, F., Oteo, J., and Ros, J., The magnus expansion and some of its applications, Phys. Rep.470 (2009), no. 5-6, 151-238.
[7] Bonnier, P. and Oberhauser, H., Signature cumulants, ordered partitions, and independence of stochastic processes, Bernoulli26 (2020), no. 4, 2727-2757. · Zbl 1469.60364
[8] Bruned, Y., Curry, C., and Ebrahimi-Fard, K., Quasi-shuffle algebras and renormalisation of rough differential equations, Bulletin of the London Mathematical Society52 (2020), no. 1, 43-63. · Zbl 1457.16029
[9] Casas, F. and Murua, A., An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications, J. Math. Phys.50 (2009), no. 3, 033513, 23. · Zbl 1202.17004
[10] Celestino, A., Ebrahimi-Fard, K., Patras, F., and Perales, D., Cumulant-cumulant relations in free probability theory from Magnus’ expansion, (2021). · Zbl 1501.46055
[11] Chen, K.-T., Iterated integrals and exponential homomorphisms†,Proc. London Math. Soc. s3-4 (1954), no. 1, 502-512. · Zbl 0058.25603
[12] Chevyrev, I. and Friz, P. K., Canonical rdes and general semimartingales as rough paths, Ann. Probab.47 (2019), no. 1, 420-463. · Zbl 1475.60203
[13] Chevyrev, I. and Lyons, T., Characteristic functions of measures on geometric rough paths, Ann. Probab.44 (2016), no. 6, 4049-4082. · Zbl 1393.60008
[14] Cohen, S. and Elliott, R. J., Stochastic calculus and applications, 2nd ed., Birkhäuser, Basel, 2015. · Zbl 1338.60001
[15] Cont, R. and Tankov, P., Financial modelling with jump processes, Financial Mathematics Series, Chapman & Hall/CRC, 2004. · Zbl 1052.91043
[16] Cuchiero, C., Filipović, D., Mayerhofer, E., and Teichmann, J., Affine processes on positive semidefinite matrices, The Annals of Applied Probability21 (2011), no. 2, 397-463. · Zbl 1219.60068
[17] Cuchiero, C., Keller-Ressel, M., and Teichmann, J., Polynomial processes and their applications to mathematical finance, Finance and Stochastics16 (2009). · Zbl 1270.60079
[18] Cuchiero, C., Svaluto-Ferro, S., and Teichmann, J., Signature sdes from an affine and polynomial perspective, 2021, In preparation. · Zbl 1461.91310
[19] Duffie, D., Filipović, D., Schachermayer, W., et al.Affine processes and applications in finance, Ann. Appl. Probab.13 (2003), no. 3, 984-1053. · Zbl 1048.60059
[20] Estrade, A., Exponentielle stochastique et intégrale multiplicative discontinues, Ann. Inst. Henri Poincaré Probab. Stat.28 (1992), no. 1, 107-129. · Zbl 0760.60048
[21] Fawcett, T., Problems in stochastic analysis: connections between rough paths and non-commutative harmonic analysis, Ph.D. thesis, University of Oxford, 2002.
[22] Friedman, A., Partial differential equations of parabolic type, R.E. Krieger Publishing Company, 1983.
[23] Friz, P. K., Gatheral, J., and Radoičić, R., Forests, cumulants, martingales, to appear in The Annals of Probability (2022). · Zbl 07527829
[24] Friz, P. K. and Hairer, M., A course on rough paths, 2nd ed., Universitext, Springer International Publishing, 2020. · Zbl 1437.60002
[25] Friz, P. K. and Shekhar, A., General rough integration, Lévy rough paths and a Lévy-Kintchine-type formula, Ann. Probab.45 (2017), no. 4, 2707-2765. · Zbl 1412.60103
[26] Friz, P. K. and Victoir, N. B., The burkholder-davis-gundy inequality for enhanced martingales, Lecture Notes in Mathematics1934 (2006). · Zbl 1156.60027
[27] Friz, P. K. and Victoir, N. B., Multidimensional stochastic processes as rough paths: Theory and applications, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010. · Zbl 1193.60053
[28] Fukasawa, M. and Matsushita, K., Realized cumulants for martingales, Electronic Communications in Probability26 (2021), 1-10. · Zbl 1485.60047
[29] Gatheral, J. and Keller-Ressel, M., Affine forward variance models, Finance Stoch.23 (2019), no. 3, 501-533. · Zbl 1430.91110
[30] Hairer, M., Solving the kpz equation, Annals of Mathematics178 (2013), no. 2, 559-664. · Zbl 1281.60060
[31] Hakim-Dowek, M. and Lépingle, D., L’exponentielle stochastique des groupes de Lie, Séminaire de Probabilités XX 1984/85, Springer, 1986, pp. 352-374. · Zbl 0609.60009
[32] Hausdorff, F., Die symbolische Exponentialformel in der Gruppentheorie, Ber. Verh. Kgl. Sächs. Ges. Wiss. Leipzig., Math.-phys. Kl.58 (1906), 19-48. · JFM 37.0176.02
[33] Iserles, A. and Nørsett, S. P., On the solution of linear differential equations in lie groups, Philos. Trans. Roy. Soc. A357 (1999), no. 1754, 983-1019. · Zbl 0958.65080
[34] Iserles, A., Munthe-Kaas, H., Nørsett, S., and Zanna, A., Lie-group methods, Acta numerica (2005). · Zbl 1064.65147
[35] Jacod, J., Calcul stochastique et problèmes de martingales, Lecture Notes in Mathematics, vol. 714, Springer Berlin Heidelberg, Berlin, Heidelberg, 1979 (eng). · Zbl 0414.60053
[36] Jacod, J. and Shiryaev, A. N., Limit theorems for stochastic processes., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], no. 488, SpringerBerlin, 2003. · Zbl 1018.60002
[37] Kamm, K., Pagliarani, S., and Pascucci, A., The stochastic magnus expansion, Journal of Scientific Computing89 (2021), 56. https://doi.org/10.1007/s10915-021-01633-6. · Zbl 1489.60112
[38] Karatzas, I. and Shreve, S., Brownian motion and stochastic calculus, 2 ed., Graduate Texts in Mathematics, vol. 113, Springer, New York, NY, 1998. · Zbl 0638.60065
[39] Keller-Ressel, M., Larsson, M., and Pulido, S., Affine Rough Models, arXiv e-prints (2018), arXiv:1812.08486.
[40] Keller-Ressel, M., Schachermayer, W., and Teichmann, J., Affine processes are regular, Probab. Theory Related Fields151 (2011), no. 3-4, 591-611. · Zbl 1235.60093
[41] Kurtz, T. G., Pardoux, E., and Protter, P., Stratonovich stochastic differential equations driven by general semimartingales, Ann. Inst. Henri Poincaré Probab. Stat.31 (1995), no. 2, 351-377. · Zbl 0823.60046
[42] Lacoin, H., Rhodes, R., and Vargas, V., A probabilistic approach of ultraviolet renormalisation in the boundary sine-gordon model, to appear in Probability Theory and Related Fields (2022).
[43] Le Gall, J.-F., Brownian motion, martingales, and stochastic calculus, Springer, 2016. · Zbl 1378.60002
[44] Lejan, Y. and Qian, Z., Stratonovich’s signatures of brownian motion determine brownian sample paths, Probability Theory and Related Fields157 (2011).
[45] Lyons, T., Rough paths, signatures and the modelling of functions on streams, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. IV, Kyung Moon Sa, Seoul, 2014, pp. 163-184. · Zbl 1373.93158
[46] Lyons, T. and Ni, H., Expected signature of brownian motion up to the first exit time from a bounded domain, Ann. Probab.43 (2015), no. 5, 2729-2762. · Zbl 1350.60086
[47] Lyons, T. and Victoir, N., Cubature on wiener space, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences460 (2004), no. 2041, 169-198. · Zbl 1055.60049
[48] Magnus, W., On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math.7 (1954), no. 4, 649-673. · Zbl 0056.34102
[49] Marcus, S. I., Modeling and analysis of stochastic differential equations driven by point processes, IEEE Trans. Inform. Theory24 (1978), no. 2, 164-172. · Zbl 0372.60084
[50] Marcus, S. I., Modeling and approximation of stochastic differential equations driven by semimartingales, Stochastics4 (1981), no. 3, 223-245. · Zbl 0456.60064
[51] Mckean, H. P., Stochastic integrals, AMS Chelsea Publishing Series, no. 353, American Mathematical Society, 1969. · Zbl 0191.46603
[52] Miller, W. Jr., Symmetry groups and their applications, Pure and Applied Mathematics, vol. 50, Academic Press, New York-London, 1972. · Zbl 0306.22001
[53] Mykland, P. A., Bartlett type identities for martingales, Ann. Statist.22 (1994), no. 1, 21-38. · Zbl 0808.62030
[54] Ni, H., The expected signature of a stochastic process, Ph.D. thesis, University of Oxford, 2012.
[55] Øksendal, B., Stochastic differential equations: An introduction with applications, 6 ed., Springer Berlin / Heidelberg, Berlin, Heidelberg, 2014 (eng). · Zbl 0567.60055
[56] Pham, H., Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Syst. Est. Control8 (1998), 1-27.
[57] Protter, P. E., Stochastic integration and differential equations, 2 ed., Stochastic Modelling and Applied Probability, Springer-Verlag Berlin Heidelberg, 2005.
[58] Reutenauer, C., Free lie algebras, Handbook of Algebra, vol. 3, Elsevier, 2003, pp. 887-903. · Zbl 1071.17003
[59] Revuz, D. and Yor, M., Continuous martingales and brownian motion, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2004.
[60] Stroock, D. W., Diffusion processes associated with lévy generators, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete32 (1975), no. 3, 209-244. · Zbl 0292.60122
[61] Stroock, D. W. and Varadhan, S. R. S., Diffusion processes with continuous coefficients, i, Communications on Pure and Applied Mathematics22 (1969), no. 3, 345-400. · Zbl 0167.43903
[62] Young, L. C., An inequality of the hölder type, connected with stieltjes integration, Acta Math.67 (1936), 251-282. · JFM 62.0250.02
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.