Infinitesimal generators for a family of polynomial processes : an algebraic approach

• Autorzy: Wesołowski, Jacek , Zięba, Agnieszka
• Języki publikacji: EN

Abstrakty

EN

Quadratic harnesses are time-inhomogeneous Markov polynomial processes with linear conditional expectations and quadratic conditional variances with respect to the past-future filtrations. Typically they are determined by five numerical constants η, θ, τ, σ and q hidden in the form of conditional variances. In this paper we derive infinitesimal generators of such processes in the case σ=0 extending previously known results. The infinitesimal generators are identified through a solution of a q-commutation equation in the algebra Q of infinite sequences of polynomials in one variable. The solution is a special element in Q whose coordinates satisfy a three-term recurrence and thus define a system of orthogonal polynomials. It turns out that the corresponding orthogonality measure υx,t. uniquely determines the infinitesimal generator (acting on polynomials or bounded functions with bounded continuous second derivative) as an integro-differential operator with an explicit kernel, where integration is with respect to the measure υ_x,t.

Słowa kluczowe

EN

polynomial process; quadratic harness; infinitesimal generator; orthogonal polynomials; algebra of polynomial sequences; three-step recurrence

• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2024
• Tom: Vol. 44, Fasc. 1
• Strony: 51--85
• Opis fizyczny: Bibliogr. 44 poz.

Twórcy

autor

Wesołowski, Jacek

• Faculty of Mathematics and Information Science Warsaw University of Technology 00-661 Warszawa, Poland,

autor

Zięba, Agnieszka

• Faculty of Mathematics and Information Science Warsaw University of Technology 00-661 Warszawa, Poland,

Bibliografia

• [1] M. Agoitia-Hurtado, Time-inhomogeneous polynomial processes in electricity spot price models, PhD thesis, Univ. of Freiburg, 2017.
• [2] M. Agoitia-Hurtado and T. Schmidt, Time-inhomonogeous polynomial processes, Stochastic Anal. Appl. 38 (2020), 527-564.
• [3] M. Anshelevich, q-Lévy processes, J. Reine Angew. Math. 576 (2004), 181-207.
• [4] M. Anshelevich, Generators of some non-commutative stochastic processes, Probab. Theory Related Fields 157 (2013), 777-815.
• [5] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, 55 pp.
• [6] F. Benth, N. Detering and P. Kruhner, Abstract polynomial processes, arXiv:2010.02483 (2020).
• [7] F. Benth, N. Detering and P. Kruhner, Independent increment processes: a multilinearity preserving property, Stochastics 93 (2021), 803-832.
• [8] N. Berestycki, E. Powell and G. Ray, (1+ε) moments suffice to characterise the GFF, Electron. J. Probab. 26 (2021), art. 44, 25 pp.
• [9] P. Biane, Quelques propriétés du mouvement brownien non-commutatif, Astérisque 23 (1996), 73-101.
• [10] P. Biane, Processes with free increments, Math. Z. 227 (1998), 143-174.
• [11] P. Biane, Quantum Markov processes and group representations, Quantum Probab. Comm. QP-PQ 227 (1998), 53-72.
• [12] P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Probab. Theory Related Fields 112 (1998), 373-409.
• [13] P. Billingsley, Probability and Measure, Wiley, 1995.
• [14] M. Bo˙zejko, B. Kümmerer and R. Speicher, q-Gaussian processes: Non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), 129–154.
• [15] W. Bryc, Markov processes with free-Meixner laws, Stochastic Process. Appl. 120 (2010), 1393-1403.
• [16] W. Bryc and W. Matysiak, Wilson’s 6 - j laws and stitched Markov processes, Theory Probab. Appl. 60 (2011), 333-356.
• [17] W. Bryc, W. Matysiak and J. Wesołowski, Quadratic harnesses, q-commutations, and orthogonal martingale polynomials, Trans. Amer. Math. Soc. 359 (2007), 5449–5483.
• [18] W. Bryc, W. Matysiak and J. Wesołowski, The bi-Poisson process: A quadratic harness, Ann. Probab. 36 (2008), 623-646.
• [19] W. Bryc and Y. Wang, Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries, Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), 2169-2194.
• [20] W. Bryc and J. Wesołowski, Conditional moments of q-Meixner processes, Probab. Theory Related Fields 131 (2005), 415-441
• [21] W. Bryc and J. Wesołowski, Askey-Wilson polynomials, quadratic harnesses and martingales, Ann. Probab. 38 (2010), 1221-1262.
• [22] W. Bryc and J. Wesołowski, Bridges of quadratic harnesses, Electron. J. Probab. 17 (2012), art. 105, 25 pp.
• [23] W. Bryc and J. Wesołowski, Infinitesimal generators of q-Meixner processes, Stochastic Process. Appl. 124 (2014), 915-926.
• [24] W. Bryc and J. Wesołowski, Infinitesimal generators for a class of polynomial processes, Studia Math. 229 (2015), 73-93.
• [25] W. Bryc and J. Wesołowski, Asymmetric simple exclusion process with open boundaries and quadratic harnesses, J. Statist. Phys. 167 (2017), 383-415.
• [26] T. Chihara, On co-recursive orthogonal polynomials, Proc. Amer. Math. Soc. 8 (1957), 899-905.
• [27] T. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978.
• [28] I. Corwin, Some recent progress on the stationary measure for the open KPZ equation, in: Toeplitz Operators and Random Matrices, Birkhäuser, 2023, 321-360.
• [29] I. Corwin and A. Knizel, Stationary measure of the open KPZ equation, arXiv:2103.12253 (2021).
• [30] J.-M. Courrège, Sur la forme intégro-différentielle des opérateurs de C∞k dans C0 satisfaisant au principe de maximum, Sém. Brelot-Choquet-Deny, Théorie Potent. 10 (1965/66), exp. 2, 38 pp.
• [31] C. Cuchiero, Affine and polynomial processes, PhD thesis, ETH Zürich, 2011.
• [32] C. Cuchiero, M, Keller-Ressel and J. Teichmann, Polynomial processes and their applications to mathematical finance, Finance Stoch. 16 (2012), 711-740.
• [33] C. Cuchiero, M. Larsson, and S. Svaluto-Ferro, Probability measure-valued polynomial diffusions, Electron. J. Probab. 24 (2019), art. 30, 32 pp.
• [34] C. Cuchiero and S. Svaluto-Ferro, Infinite-dimensional polynomial processes, Finance Stoch. 25 (2021), 383-426.
• [35] M. Dozzi, Two-parameter harnesses and the Wiener process, Z. Wahrsch. Verw. Gebiete 56 (1981), 507-514.
• [36] J. M. Hammersley, Harnesses, in: Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, CA, 1965/66), Vol. III: Physical Sciences, Univ. California Press, Berkeley, CA, 1967, 89-117.
• [37] E. Hoyle and L. Menguturk, Generalised Liouville processes and their properties, J. Appl. Probab. 57 (2020), 1088-1110.
• [38] M. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia Math. Appl. 98, Cambridge Univ. Press, 2005.
• [39] R. Koekoek, P. Lesky and R. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monogr. Math., Springer, 2010.
• [40] R. Mansuy and M. Yor, Harnesses, Lévy bridges and Monsieur Jourdain, Stochastic Process. Appl. 115 (2005), 329-338.
• [41] P. Szabłowski, A few remarks on quadratic harnesses, J. Difference Equations Appl. 20 (2014), 586-609.
• [42] P. Szabłowski, On Markov processes with polynomial conditional moments, Trans. Amer. Math. Soc. 367 (2015), 8487-8519.
• [43] P. Szabłowski, Markov processes, polynomial martingales and orthogonal polynomials, Stochastics 90 (2018), 61-77.
• [44] W. von Waldenfels, Fast positive Operatoren, Z. Wahrsch. Verw. Gebiete 4 (1965), 159-174.

Identyfikatory

DOI

10.37190/0208-4147.0015910.37190/0208-4147.00159