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On the large-time behaviour of affine Volterra processes (2204.05270v2)

Published 11 Apr 2022 in math.PR

Abstract: We show the existence of a stationary measure for a class of multidimensional stochastic Volterra systems of affine type. These processes are in general not Markovian, a shortcoming which hinders their large-time analysis. We circumvent this issue by lifting the system to a measure-valued stochastic PDE introduced by Cuchiero and Teichmann, whence we retrieve the Markov property. Leveraging on the associated generalised Feller property, we extend the Krylov-Bogoliubov theorem to this infinite-dimensional setting and thus establish an approach to the existence of invariant measures. We present concrete examples, including the rough Heston model from Mathematical Finance.

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References (56)
  1. Markovian structure of the Volterra Heston model. Stat. Prob. Lett., 149:63–72, 2018.
  2. Multi-factor approximation of rough volatility models. SIAM J. Financial Math., 10(2):309–349, 2019.
  3. Affine Volterra processes. Ann. Appl. Probab., 29(5):3155–3200, 2017.
  4. L. Arnold. Random Dynamical Systems. Springer-Verlag, New York, 1998.
  5. Pricing under rough volatility. Quant. Finance, 16(6):887–904, 2016.
  6. P. Billingsey. Probability and Measure. Wiley series in probability and statistics, 1995.
  7. Typical dynamics and fluctuation analysis of slow-fast systems driven by fractional Brownian motion. Stoch. Dyn., 21(07):2150030, 2021.
  8. A. Budhiraja and P. Dupuis. Analysis and Approximation of Rare Events Representations and Weak Convergence Methods. Springer, 2019.
  9. Generalized couplings and ergodic rates for SPDEs and other Markov models. The Annals of Applied Probability, 30(1):1–39, 2020.
  10. Spatial ergodicity for SPDEs via Poincaré-type inequalities. Electron. J. Probab., 26:1–37, 2021.
  11. Law of large numbers for branching symmetric hunt processes with measure-valued branching rates. J. Theoret. Probab., 30:898–931, 2017.
  12. Multiscale systems, homogenization, and rough paths. In International Conference in Honor of the 75th Birthday of SRS Varadhan, pages 17–48. Springer, 2016.
  13. C. Cuchiero and S. Svaluto-Ferro. Infinite-dimensional polynomial processes. Finance Stoch., 25:383–426, 2021.
  14. J. Cuchiero and J. Teichmann. Markovian lifts of positive semidefinite affine Volterra-type processes. Decis. Econ. Finance, 42:407–448, 2019.
  15. J. Cuchiero and J. Teichmann. Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case. J. Evol. Equ., 20:1301–1348, 2020.
  16. G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. in: Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992.
  17. G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Note Series, Series Number 229, 1996.
  18. P. Dörsek and J. Teichmann. A semigroup point of view on splitting schemes for stochastic (partial) differential equations. Preprint available at: arXiv:1011.2651, 2010.
  19. Affine processes and applications in finance. Ann. Appl. Probab., 13(3):984–1053, 2003.
  20. N. Dunford and J. T. Schwartz. Linear Operators, Part 1: General Theory. John Wiley and Sons, 1988.
  21. P. Dupuis and R. S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. Wiley, 1997.
  22. W. E. and J.-C. Mattingly. Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm. Pure Appl. Math, 54(11):1386–1402, 2001.
  23. Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys., 224:83–106, 2001.
  24. The microstructural foundations of leverage effect and rough volatility. Finance and Stochastics, 22(2):241–280, 2018.
  25. O. El Euch and M. Rosenbaum. Perfect hedging in rough Heston models. Ann. Appl. Probab., 28(6):3813–3856, 2018.
  26. O. El Euch and M. Rosenbaum. The characteristic function of rough Heston models. Math. Finance, 29(1):3–38, 2019.
  27. Stochastic volterra integral equations and a class of first-order stochastic partial differential equations. Stochastics, 2022.
  28. A. M. Etheridge. Asymptotic behaviour of measure-valued critical branching processes. Proc. Amer. Math. Soc., 118(4), 1993.
  29. F. Flandoli and B. Maslowski. Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Comm. Math. Phys., 171:119–141, 1995.
  30. Small-time, large-time, and H→0→𝐻0H\to 0italic_H → 0 asymptotics for the rough Heston model. Math. Finance, 31(1), 2020.
  31. M. Friesen. Long-time behavior for subcritical measure-valued branching processes with immigration. Potential Analysis, 2022.
  32. M. Friesen and P. Jin. Volterra square-root process: Stationarity and regularity of the law. Preprint available at: https://arxiv.org/abs/2203.08677v1, 2022.
  33. Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion. Appl. Math. Optim, 60:151–172, 2009.
  34. J. Gehringer and X.-M. Li. Functional limit theorems for the fractional Ornstein–Uhlenbeck process. Journal of Theoretical Probability, 35(1):426–456, 2022.
  35. Functional limit theorems for Volterra processes and applications to homogenization. Preprint available at: arXiv:2104.06364, 2020.
  36. B. Gerencsér and M. Rásonyi. Invariant measures for multidimensional fractional stochastic volatility models. Stochastics and Partial Differential Equations: Analysis and Computations, 10(3):1132–1164, 2022.
  37. M. Hairer. Exponential mixing properties of stochastic pdes through asymptotic coupling. Probab. Theory Related Fields, 124:345–380, 2002.
  38. M. Hairer. Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab., 33(2):703–758, 2005.
  39. M. Hairer and X.-M. Li. Averaging dynamics driven by fractional Brownian motion. Ann. Probab., 48(4), 2020.
  40. M. Hairer and J.-C. Mattingly. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. of Math., 164:993–1032, 2006.
  41. M. Hairer and A. Ohashi. Ergodic theory for SDEs with extrinsic memory. Ann. Probab., 35(5):1950–1977, 2007.
  42. M. Hairer and N. S. Pillai. Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat., 47(2):601–628, 2011.
  43. Mittag-leffler functions and their applications. Journal of Applied Mathematics, Article ID 298628:1–51, 2011.
  44. I. Iscoe. Ergodic theory and a local occupation time for measure-valued critical branching Brownian motion. Stochastics, 18:197–243, 1986.
  45. M. Keller-Ressel and A. Mijatović. On the limit distributions of continuous-state branching processes with immigration. Stochastic Process. Appl., 122(6):2329–2345, 2012.
  46. Phase analysis for a family of stochastic reaction-diffusion equations. Preprint available at: arXiv:2012.12512, 2020.
  47. Survival of some measure-valued Markov branching processes. Stoch. Models, 35(2):221–233, 2019.
  48. N. Konno and T. Shiga. Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields, 79:201–225, 1988.
  49. On the (non-) stationary density of fractional-driven stochastic differential equations. arXiv preprint arXiv:2204.06329, 2022.
  50. X.-M. Li and J. Sieber. Slow-fast systems with fractional environment and dynamics. The Annals of Applied Probability, 32(5):3964–4003, 2022.
  51. Z. Li. Measure-Valued Branching Markov Processes. Springer, Probability and Its Applications, 2011.
  52. A degenerate stochastic partial differential equation for superprocesses with singular interaction. Probab. Theory Related Fields, 130:1–17, 2004.
  53. L. Mytnik and T. S. Salisbury. Uniqueness for Volterra-type stochastic integral equations. Preprint available at: arXiv:1502.05513, 2015.
  54. Averaging principles for mixed fast-slow systems driven by fractional Brownian motion. Preprint available at: arXiv:2001.06945, 2020.
  55. Averaging principle for fast-slow system driven by mixed fractional Brownian rough path. J. Differential Equations, 301:202–235, 2021.
  56. M. Varvenne. Ergodicity of fractional stochastic differential equations and related problems. PhD thesis, Université Paul Sabatier - Toulouse III, 2019.

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