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arxiv: 2509.08466 · v1 · pith:GRRAJ4YNnew · submitted 2025-09-10 · 🧮 math.PR

Limit theorems for stochastic Volterra processes

Luigi Amedeo Bianchi , Stefano Bonaccorsi , Ole Ca\~nadas , Martin Friesen This is my paper

Pith reviewed 2026-05-21 22:53 UTC · model grok-4.3

classification 🧮 math.PR
keywords stochastic Volterra equationsMarkovian liftslimit distributionslaw of large numberscentral limit theoremfractional processesHilbert space methods
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The pith

Stochastic Volterra processes admit Markovian lifts in Hilbert spaces that yield limit distributions, laws of large numbers with rates, and central limit theorems for time averages.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds an abstract framework that converts stochastic Volterra equations into Markov processes valued in Hilbert spaces through operator-valued kernels. This construction establishes the existence and characterization of limit distributions, including cases with multiple stationary processes. It further derives a law of large numbers that includes an explicit convergence rate and a central limit theorem for time averages when the driving noise lies in the Gaussian domain of attraction. The results are shown to cover fractional Volterra equations with additive or multiplicative Gaussian noise via concrete lifts such as Laplace transforms in weighted spaces or shift semigroups. A reader would care because these theorems describe the long-run statistical behavior of processes that carry memory of past states.

Core claim

By embedding stochastic Volterra equations into a Markovian framework using operator-valued kernels in Hilbert spaces, the authors show that limit distributions exist and can be characterized, possibly with multiple stationary processes. They derive a law of large numbers with convergence rate and a central limit theorem for time averages in the Gaussian domain of attraction. This is illustrated for fractional cases with additive or multiplicative noise using lifts based on Laplace transforms or shift semigroups.

What carries the argument

The abstract Hilbert space-valued Markovian lift of the stochastic Volterra equation via Laplace transforms or shift semigroups, which transfers limit theorems from the lifted space back to the original process.

If this is right

  • Limit distributions and stationary processes exist and can be characterized for the lifted Markovian version of the process.
  • Time averages obey a law of large numbers that includes an explicit rate of convergence.
  • A central limit theorem holds for time averages when the noise is in the Gaussian domain of attraction.
  • The results apply directly to fractional stochastic Volterra equations driven by additive or multiplicative Gaussian noise.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lifting technique could be checked on non-Gaussian driving noises to test whether the convergence statements survive.
  • Numerical schemes for long-time simulation might exploit the Markov property in the lifted space to generate stationary samples more efficiently.
  • The framework may connect to other memory models such as those with jumps or delay equations through analogous operator-valued kernels.

Load-bearing premise

The chosen Markovian lift must preserve the original Volterra dynamics closely enough that theorems proved in the lifted space carry over to the unlifted process.

What would settle it

A specific fractional Volterra equation with Gaussian noise where the time average fails to satisfy the predicted law of large numbers or central limit theorem after the lift is applied.

read the original abstract

We introduce an abstract Hilbert space-valued framework of Markovian lifts for stochastic Volterra equations with operator-valued Volterra kernels. Our main results address the existence and characterisation of possibly multiple limit distributions and stationary processes, a law of large numbers including a convergence rate, and the central limit theorem for time averages of the process within the Gaussian domain of attraction. As particular examples, we study Markovian lifts based on Laplace transforms in a weighted Hilbert space of densities and Markovian lifts based on the shift semigroup on the Filipovi\'c space. We illustrate our results for the case of fractional stochastic Volterra equations with additive or multiplicative Gaussian noise.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces an abstract Hilbert space-valued framework of Markovian lifts for stochastic Volterra equations with operator-valued Volterra kernels. Main results include existence and characterisation of possibly multiple limit distributions and stationary processes, a law of large numbers with convergence rate, and a central limit theorem for time averages within the Gaussian domain of attraction. Applications are given to lifts via Laplace transforms in weighted Hilbert spaces of densities and via the shift semigroup on the Filipović space, with illustrations for fractional stochastic Volterra equations driven by additive or multiplicative Gaussian noise.

Significance. If the transfer of limit theorems from the lifted Markovian dynamics back to the original Volterra process is rigorously justified, the framework would provide a valuable unified method for obtaining ergodic and fluctuation results for a wide class of memory-dependent stochastic equations. The approach leverages standard tools from functional analysis and ergodic theory on infinite-dimensional spaces. Credit is due for the concrete treatment of both additive and multiplicative noise cases in the fractional examples, which demonstrates the framework's flexibility.

major comments (2)
  1. [§3.2] §3.2 (Transfer via projection): The central claim that LLN and CLT for time averages in the lifted space imply the corresponding results for the original scalar Volterra process rests on a continuous linear projection. However, under multiplicative noise the paper does not explicitly verify that the Gaussian domain of attraction and the convergence rates descend without additional regularity on the kernel or the projection operator; the lift may introduce smoothing absent from the original equation. This is load-bearing for the applicability statements in the fractional examples.
  2. [Theorem 3.4] Theorem 3.4 (Characterisation of stationary processes): The existence of multiple limit distributions is asserted via the Markovian lift, but the proof sketch does not address whether the projection preserves the invariance properties when the noise is multiplicative, potentially affecting the characterisation for the original process.
minor comments (2)
  1. [§2.1] The notation for the weighted Hilbert space in the Laplace-transform lift could include an explicit example of the weight function to improve clarity for readers unfamiliar with the construction.
  2. [§4] A brief comparison table of the two lift constructions (Laplace vs. shift semigroup) would help highlight their respective advantages and limitations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We appreciate the positive assessment of the framework's potential value and address the two major comments point by point below. Where the referee correctly identifies the need for additional explicit verification, we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Transfer via projection): The central claim that LLN and CLT for time averages in the lifted space imply the corresponding results for the original scalar Volterra process rests on a continuous linear projection. However, under multiplicative noise the paper does not explicitly verify that the Gaussian domain of attraction and the convergence rates descend without additional regularity on the kernel or the projection operator; the lift may introduce smoothing absent from the original equation. This is load-bearing for the applicability statements in the fractional examples.

    Authors: We agree that the transfer of the LLN and CLT via the continuous linear projection requires explicit verification in the multiplicative noise case to confirm that the Gaussian domain of attraction and convergence rates are preserved. The manuscript relies on the linearity and continuity of the projection to map the time averages back to the original process. To address the referee's concern about possible smoothing introduced by the lift, we will revise §3.2 by adding a dedicated lemma that verifies the descent of the domain of attraction and rates under the stated assumptions on the operator-valued kernel and the projection (as used in the Laplace-transform and Filipović-space lifts). This will directly support the applicability claims for the fractional examples with multiplicative noise. revision: yes

  2. Referee: [Theorem 3.4] Theorem 3.4 (Characterisation of stationary processes): The existence of multiple limit distributions is asserted via the Markovian lift, but the proof sketch does not address whether the projection preserves the invariance properties when the noise is multiplicative, potentially affecting the characterisation for the original process.

    Authors: Theorem 3.4 characterises stationary processes and multiple limit distributions for the lifted Markovian dynamics in the Hilbert space. The original Volterra process is recovered by applying the continuous projection to these objects. We acknowledge that the current proof sketch does not explicitly treat the preservation of invariance under projection for the multiplicative case. We will expand the proof of Theorem 3.4 to include a step showing that the push-forward of an invariant measure under the projection remains invariant for the original equation, using the structure of the multiplicative noise term and the properties of the projection operator. This clarification will be added in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in abstract functional-analytic framework

full rationale

The paper constructs an abstract Markovian lift for stochastic Volterra equations in a Hilbert space setting (weighted densities or Filipović space) and derives existence of limit distributions, LLN with rate, and CLT for time averages directly from the lifted dynamics using standard semigroup and stochastic analysis tools. The original scalar process is recovered via a continuous linear projection operator whose properties are established independently of the limit theorems. No equation reduces to a fitted parameter renamed as prediction, no self-citation supplies a load-bearing uniqueness theorem or ansatz, and the transfer of ergodic averages relies on stated regularity assumptions rather than circular redefinition. The framework is therefore self-contained against external functional-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Theoretical paper; relies on standard assumptions of stochastic analysis and functional analysis rather than new fitted constants or invented entities.

axioms (2)
  • domain assumption Volterra kernels are operator-valued and satisfy conditions allowing Markovian lifts in Hilbert space.
    Invoked to justify the existence of the lift framework and transfer of limit theorems.
  • domain assumption The noise is Gaussian or lies in the Gaussian domain of attraction.
    Required for the central limit theorem statement.

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages · 1 internal anchor

  1. [1]

    Weak existence and uniqueness for affine stochastic Volterra equations withL 1- kernels.Bernoulli, 27(3):1583 – 1615, 2021.doi:10.3150/20-BEJ1284

    Eduardo Abi Jaber. Weak existence and uniqueness for affine stochastic Volterra equations withL 1- kernels.Bernoulli, 27(3):1583 – 1615, 2021.doi:10.3150/20-BEJ1284

    work page doi:10.3150/20-bej1284 2021

  2. [2]

    Markovian structure of the Volterra Heston model.Statist

    Eduardo Abi Jaber and Omar El Euch. Markovian structure of the Volterra Heston model.Statist. Probab. Lett., 149:63–72, 2019.doi:10.1016/j.spl.2019.01.024

    work page doi:10.1016/j.spl.2019.01.024 2019

  3. [3]

    Multifactor approximation of rough volatility models.SIAM J

    Eduardo Abi Jaber and Omar El Euch. Multifactor approximation of rough volatility models.SIAM J. Financial Math., 10(2):309–349, 2019.doi:10.1137/18M1170236

    work page doi:10.1137/18m1170236 2019

  4. [4]

    Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander.Vector-valued Laplace transforms and Cauchy problems, volume 96 ofMonographs in Mathematics. Birkh¨ auser/Springer Basel AG, Basel, second edition, 2011.doi:10.1007/978-3-0348-0087-7

    work page doi:10.1007/978-3-0348-0087-7 2011

  5. [5]

    Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise.J

    Boris Baeumer, Matthias Geissert, and Mih´ aly Kov´ acs. Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise.J. Differential Equations, 258(2):535–554, 2015.doi:10.1016/j.jde.2014.09.020

    work page doi:10.1016/j.jde.2014.09.020 2015

  6. [6]

    , Friz, P

    Christian Bayer, Peter Friz, and Jim Gatheral. Pricing under rough volatility.Quantitative Finance, 16(6):887–904, 2016.doi:10.1080/14697688.2015.1099717

    work page doi:10.1080/14697688.2015.1099717 2016

  7. [7]

    Ergodicity and Law of Large Numbers for the Volterra Cox-Iingersoll-Ross process, 2024.arXiv:2409.04496

    Mohamed Ben Alaya, Martin Friesen, and Jonas Kremer. Ergodicity and Law of Large Numbers for the Volterra Cox-Iingersoll-Ross process, 2024.arXiv:2409.04496

    work page arXiv 2024

  8. [8]

    Parameter estimation in the ergodic Volterra Ornstein-Uhlenbeck process, 2024.arXiv:2404.05554

    Mohamed Ben Alaya, Martin Friesen, and Jonas Kremer. Parameter estimation in the ergodic Volterra Ornstein-Uhlenbeck process, 2024.arXiv:2404.05554

    work page arXiv 2024

  9. [9]

    A rough multi-factor model of electricity spot prices.Energy Economics, 63:301– 313, 2017.doi:10.1016/j.eneco.2017.02.007

    Mikkel Bennedsen. A rough multi-factor model of electricity spot prices.Energy Economics, 63:301– 313, 2017.doi:10.1016/j.eneco.2017.02.007

    work page doi:10.1016/j.eneco.2017.02.007 2017

  10. [10]

    Representation and approximation of ambit fields in Hilbert space.Stochastics, 89(1):311–347, 2017.doi:10.1080/17442508.2016.1177057

    Fred Espen Benth and Heidar Eyjolfsson. Representation and approximation of ambit fields in Hilbert space.Stochastics, 89(1):311–347, 2017.doi:10.1080/17442508.2016.1177057

    work page doi:10.1080/17442508.2016.1177057 2017

  11. [11]

    Limits of stochastic volterra equations driven by gaussian noise.Stochastics and Partial Differential Equations: Analysis and Computations, 13:585–630, 2025

    Luigi Amedeo Bianchi, Stefano Bonaccorsi, and Martin Friesen. Limits of stochastic volterra equations driven by gaussian noise.Stochastics and Partial Differential Equations: Analysis and Computations, 13:585–630, 2025. URL:https://doi.org/10.1007/s40072-024-00340-1

    work page doi:10.1007/s40072-024-00340-1 2025

  12. [12]

    Wiley Series in Probability and Statistics: Probability and Statistics

    Patrick Billingsley.Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. A Wiley- Interscience Publication

    work page 1999

  13. [13]

    Bobylev and Carlo Cercignani

    Alexander V. Bobylev and Carlo Cercignani. The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation.Appl. Math. Lett., 15(7):807– 813, 2002.doi:10.1016/S0893-9659(02)00046-0

    work page doi:10.1016/s0893-9659(02)00046-0 2002

  14. [14]

    Volterra equations in Banach spaces with completely mono- tone kernels.NoDEA Nonlinear Differential Equations Appl., 20(3):557–594, 2013.doi:10.1007/ s00030-012-0167-0

    Stefano Bonaccorsi and Gertrud Desch. Volterra equations in Banach spaces with completely mono- tone kernels.NoDEA Nonlinear Differential Equations Appl., 20(3):557–594, 2013.doi:10.1007/ s00030-012-0167-0

    work page 2013

  15. [15]

    An analytic approach to stochastic Volterra equations with completely monotone kernels.J

    Stefano Bonaccorsi and Elisa Mastrogiacomo. An analytic approach to stochastic Volterra equations with completely monotone kernels.J. Evol. Equ., 9(2):315–339, 2009.doi:10.1007/ s00028-009-0010-1

    work page 2009

  16. [16]

    Subgeometric rates of convergence of Markov processes in the Wasserstein metric

    Oleg Butkovsky. Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Ann. Appl. Probab., 24(2):526–552, 2014.doi:10.1214/13-AAP922

    work page doi:10.1214/13-aap922 2014

  17. [17]

    Generalized couplings and ergodic rates for SPDEs and other Markov models.Ann

    Oleg Butkovsky, Alexei Kulik, and Michael Scheutzow. Generalized couplings and ergodic rates for SPDEs and other Markov models.Ann. Appl. Probab., 30(1):1–39, 2020.doi:10.1214/19-AAP1485

    work page doi:10.1214/19-aap1485 2020

  18. [18]

    Fractional Brownian motion and the Markov property.Electron

    Philippe Carmona and Laure Coutin. Fractional Brownian motion and the Markov property.Electron. Comm. Probab., 3:95–107, 1998.doi:10.1214/ECP.v3-998

    work page doi:10.1214/ecp.v3-998 1998

  19. [19]

    Some results on stochastic convolutions arising in Volterra equations perturbed by noise.Atti Accad

    Philippe Cl´ ement and Giuseppe Da Prato. Some results on stochastic convolutions arising in Volterra equations perturbed by noise.Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 7(3):147–153, 1996. URL:http://www.bdim.eu/item?id=RLIN_1996_9_7_3_147_0

    work page 1996

  20. [20]

    White noise perturbation of the equations of linear parabolic viscoelasticity.Rend

    Philippe Cl´ ement, Giuseppe Da Prato, and Jan Pr¨ uss. White noise perturbation of the equations of linear parabolic viscoelasticity.Rend. Istit. Mat. Univ. Trieste, 29(1-2):207–220, 1997. URL:https: //rendiconti.dmi.units.it/volumi/29/14.pdf

    work page 1997

  21. [21]

    Philippe Cl´ ement, Wolfgang Desch, and Krista W. Homan. An analytic semigroup setting for a class of Volterra equations.J. Integral Equations Appl., 14(3):239–281, 2002.doi:10.1216/jiea/1181074916

    work page doi:10.1216/jiea/1181074916 2002

  22. [22]

    ˇCoupek and B

    P. ˇCoupek and B. Maslowski. Stochastic evolution equations with Volterra noise.Stochastic Process. Appl., 127(3):877–900, 2017.doi:10.1016/j.spa.2016.07.003

    work page doi:10.1016/j.spa.2016.07.003 2017

  23. [23]

    Generalized Feller processes and Markovian lifts of sto- chastic Volterra processes: the affine case.J

    Christa Cuchiero and Josef Teichmann. Generalized Feller processes and Markovian lifts of sto- chastic Volterra processes: the affine case.J. Evol. Equ., 20(4):1301–1348, 2020.doi:10.1007/ s00028-020-00557-2. LIMIT THEOREMS FOR STOCHASTIC VOLTERRA PROCESSES 55

    work page 2020

  24. [24]

    1996 , Series =

    Giuseppe Da Prato and Jerzy Zabczyk.Ergodicity for infinite-dimensional systems, volume 229 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1996. doi:10.1017/CBO9780511662829

    work page doi:10.1017/cbo9780511662829 1996

  25. [25]

    Cambridge University Press, Cambridge, second edition, 2014.doi:10.1017/CBO9781107295513

    Giuseppe Da Prato and Jerzy Zabczyk.Stochastic equations in infinite dimensions, volume 152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 2014.doi:10.1017/CBO9781107295513

    work page doi:10.1017/cbo9781107295513 2014

  26. [26]

    Lifting of Volterra processes: Optimal control in UMD Banach spaces.arXiv:2306.14175v1, 2023

    Giulia Di Nunno and Michele Giordano. Lifting of Volterra processes: Optimal control in UMD Banach spaces.arXiv:2306.14175v1, 2023

    work page arXiv 2023

  27. [27]

    The microstructural founda- tions of leverage effect and rough volatility.Finance Stoch., 22(2):241–280, 2018.doi:10.1007/ s00780-018-0360-z

    Omar El Euch, Masaaki Fukasawa, and Mathieu Rosenbaum. The microstructural founda- tions of leverage effect and rough volatility.Finance Stoch., 22(2):241–280, 2018.doi:10.1007/ s00780-018-0360-z

    work page 2018

  28. [28]

    Stochastic Volterra integral equations and a class of first-order stochastic partial differential equations.Stochastics, 94(7):1054–1076, 2022.doi: 10.1080/17442508.2021.2019738

    Fred Espen Benth, Nils Detering, and Paul Kr¨ uhner. Stochastic Volterra integral equations and a class of first-order stochastic partial differential equations.Stochastics, 94(7):1054–1076, 2022.doi: 10.1080/17442508.2021.2019738

    work page doi:10.1080/17442508.2021.2019738 2022

  29. [29]

    Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise.Stoch

    Kistosil Fahim, Erika Hausenblas, and Mihaly Kov´ acs. Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise.Stoch. Partial Differ. Equ. Anal. Comput., 11(3):1044–1088, 2023.doi:10.1007/s40072-022-00250-0

    work page doi:10.1007/s40072-022-00250-0 2023

  30. [30]

    On a class of stochastic par- tial differential equations with multiple invariant measures.NoDEA Nonlinear Differential Equations Appl., 28(3):Paper No

    B´ alint Farkas, Martin Friesen, Barbara R¨ udiger, and Dennis Schroers. On a class of stochastic par- tial differential equations with multiple invariant measures.NoDEA Nonlinear Differential Equations Appl., 28(3):Paper No. 28, 46, 2021.doi:10.1007/s00030-021-00691-x

    work page doi:10.1007/s00030-021-00691-x 2021

  31. [31]

    Volterra square-root process: Stationarity and regularity of the law

    Martin Friesen and Peng Jin. Volterra square-root process: Stationarity and regularity of the law. The Annals of Applied Probability, 34(1A):318 – 356, 2024.doi:10.1214/23-AAP1965

    work page doi:10.1214/23-aap1965 2024

  32. [32]

    , Jaisson, T

    Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum. Volatility is rough.Quantitative Finance, 18(6):933–949, 2018.doi:10.1080/14697688.2017.1393551

    work page doi:10.1080/14697688.2017.1393551 2018

  33. [33]

    Cambridge University Press, Cam- bridge, 1990.doi:10.1017/CBO9780511662805

    Gustaf Gripenberg, Stig-Olof Londen, and Olof Staffans.Volterra integral and functional equations, volume 34 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cam- bridge, 1990.doi:10.1017/CBO9780511662805

    work page doi:10.1017/cbo9780511662805 1990

  34. [35]

    Structure-to-structure damage correlation for scenario-based re- gional seismic risk assessment.Structural Safety, 95, March 2022

    Philipp Harms and David Stefanovits. Affine representations of fractional processes with applica- tions in mathematical finance.Stochastic Process. Appl., 129(4):1185–1228, 2019.doi:10.1016/j. spa.2018.04.010

    work page doi:10.1016/j 2019

  35. [36]

    Markovian lifts of stochastic volterra equations in sobolev spaces: Solution theory, an ito formula and invariant measures, 2024.arXiv:2406.10352

    Florian Huber. Markovian lifts of stochastic volterra equations in sobolev spaces: Solution theory, an ito formula and invariant measures, 2024.arXiv:2406.10352

    work page arXiv 2024

  36. [37]

    Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation.Math

    Mih´ aly Kov´ acs and Jacques Printems. Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation.Math. Comp., 83(289):2325–2346, 2014.doi: 10.1090/S0025-5718-2014-02803-2

    work page doi:10.1090/s0025-5718-2014-02803-2 2014

  37. [38]

    De Gruyter, Berlin, Boston, 2018

    Alexei Kulik.Ergodic Behavior of Markov Processes: With Applications to Limit Theorems. De Gruyter, Berlin, Boston, 2018

    work page 2018

  38. [39]

    Kutoyants.Statistical inference for ergodic diffusion processes

    Yury A. Kutoyants.Statistical inference for ergodic diffusion processes. Springer Series in Statistics. Springer-Verlag London, Ltd., London, 2004.doi:10.1007/978-1-4471-3866-2

    work page doi:10.1007/978-1-4471-3866-2 2004

  39. [40]

    Uniqueness for Volterra-type stochastic integral equations

    Leonid Mytnik and Thomas S. Salisbury. Uniqueness for volterra-type stochastic integral equations, 2015.arXiv:1502.05513

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  40. [41]

    Dissertationes Math

    Martin Ondrej´ at.Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Math. (Rozprawy Mat.), 2004. URL:http://eudml.org/doc/286015

    work page 2004

  41. [42]

    Birkh¨ auser Verlag, Basel, 1993.doi:10.1007/978-3-0348-8570-6

    Jan Pr¨ uss.Evolutionary integral equations and applications, volume 87 ofMonographs in Mathematics. Birkh¨ auser Verlag, Basel, 1993.doi:10.1007/978-3-0348-8570-6

    work page doi:10.1007/978-3-0348-8570-6 1993

  42. [43]

    Mean reversion for HJMM forward rate models.Adv

    Anna Rusinek. Mean reversion for HJMM forward rate models.Adv. in Appl. Probab., 42(2):371–391, 2010.doi:10.1239/aap/1275055234

    work page doi:10.1239/aap/1275055234 2010

  43. [44]

    A note on invariant measures for HJM models.Finance Stoch., 9(3):389–398, 2005

    Michael Tehranchi. A note on invariant measures for HJM models.Finance Stoch., 9(3):389–398, 2005. doi:10.1007/s00780-004-0143-6

    work page doi:10.1007/s00780-004-0143-6 2005

  44. [45]

    Grundlehren der mathematischen Wissenschaften

    C´ edric Villani.Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2016

    work page 2016

  45. [46]

    Stochastic Volterra equations in Banach spaces and stochastic partial differential equation.J

    Xicheng Zhang. Stochastic Volterra equations in Banach spaces and stochastic partial differential equation.J. Funct. Anal., 258(4):1361–1425, 2010.doi:10.1016/j.jfa.2009.11.006. 56 LUIGI AMEDEO BIANCHI, STEFANO BONACCORSI, OLE CA ˜NADAS, AND MARTIN FRIESEN (Luigi Amedeo Bianchi)Department of Mathematics, University of Trento, Trento, Italy Email address:l...

    work page doi:10.1016/j.jfa.2009.11.006 2010