It\^o versus H\"anggi-Klimontovich

Carlos Escudero; Helder Rojas

arxiv: 2309.03654 · v2 · submitted 2023-09-07 · 🧮 math-ph · cond-mat.stat-mech· math.MP· math.PR

It\^o versus H\"anggi-Klimontovich

Carlos Escudero , Helder Rojas This is my paper

Pith reviewed 2026-05-24 06:49 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechmath.MPmath.PR

keywords stochastic differential equationsItô integralStratonovich integralHänggi-Klimontovich integralLangevin equationrelativistic Brownian motionnoise interpretationmultiplicative noise

0 comments

The pith

The Hänggi-Klimontovich integral is less well adapted than the Itô integral for modeling Langevin particle dispersal and relativistic Brownian motion.

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

The paper defines the Hänggi-Klimontovich stochastic integral precisely and compares its behavior to the Itô and Stratonovich integrals in two standard physical models. It shows that the Hänggi-Klimontovich choice produces results that diverge from expected physical behavior in random dispersal of Langevin particles and in relativistic Brownian motion. The authors conclude that, for these systems, the Itô interpretation (and sometimes Stratonovich) aligns better with the intended dynamics than the newer Hänggi-Klimontovich rule. This challenges the recent preference in parts of statistical mechanics for the Hänggi-Klimontovich convention as a default choice.

Core claim

When the Hänggi-Klimontovich integral is used to interpret the noise in the Langevin equation for particle dispersal and in the stochastic formulation of relativistic Brownian motion, the resulting equations and their solutions are less consistent with the underlying physical requirements than the corresponding Itô or Stratonovich formulations.

What carries the argument

The Hänggi-Klimontovich stochastic integral, which evaluates the integrand at the right endpoint of each partition interval and thereby produces a different chain rule and different drift terms than Itô or Stratonovich calculus.

If this is right

For the standard Langevin equation of Brownian motion with position-dependent friction, the Itô interpretation yields the physically expected equilibrium distribution while the Hänggi-Klimontovich version does not.
In the relativistic Brownian motion model, the Hänggi-Klimontovich integral produces spurious terms that violate expected relativistic invariance or energy bounds that the Itô form preserves.
The Stratonovich integral performs better than Hänggi-Klimontovich but is still inferior to Itô in at least one of the two examined cases.
Claims that the Hänggi-Klimontovich rule is universally preferable for systems in statistical mechanics are not supported by these classical examples.

Where Pith is reading between the lines

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

If the same ranking holds for other multiplicative-noise systems with state-dependent diffusion, many existing derivations that adopted the Hänggi-Klimontovich convention may need re-examination.
The result suggests that the choice of stochastic integral should be made case-by-case rather than fixed by a general physical principle.
A systematic comparison across a wider set of exactly solvable models could quantify how often each interpretation matches known limiting cases.

Load-bearing premise

The stochastic differential equations chosen for Langevin dispersal and relativistic Brownian motion are the correct models for those physical systems and therefore allow a fair, objective comparison among the three noise interpretations.

What would settle it

Direct numerical simulation or laboratory measurement showing that the stationary distribution or mean-square displacement predicted by the Hänggi-Klimontovich version of either model matches experiment better than the Itô version.

read the original abstract

Interpreting the noise in a stochastic differential equation, in particular the It\^o versus Stratonovich dilemma, is a problem that has generated a lot of debate in the physical literature. In the last decades, a third interpretation of noise, given by the so-called H\"anggi-Klimontovich integral, has been proposed as better adapted to describe certain physical systems, particularly in statistical mechanics. Herein, we introduce this integral in a precise mathematical manner and analyze its properties, signaling those that have made it appealing within the realm of physics. Subsequently, we employ this integral to model some statistical mechanical systems, such as the random dispersal of Langevin particles and the relativistic Brownian motion. We show that, for these classical examples, the H\"anggi-Klimontovich integral is worse adapted than the It\^o integral and even the Stratonovich one.

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.

Desk Editor's Note private letter to a colleague

The paper gives a clean mathematical definition of the Hänggi-Klimontovich integral and applies it to two models, concluding Ito and Stratonovich are better adapted, but the ranking depends on the SDEs being chosen independently of interpretation.

read the letter

The main takeaway is that the authors formalize the Hänggi-Klimontovich integral with a precise definition and then test it on Langevin dispersal and relativistic Brownian motion, finding it less suitable than Ito or Stratonovich for those cases. The definition itself is the useful step; it turns something that had stayed somewhat informal in the physics literature into something that can be manipulated directly. They also lay out the properties that made the integral appealing in the first place, which helps readers see the motivation without hunting through older papers. That part is straightforward and reproducible. The comparative claim is narrower. It rests on running the three integrals through two specific SDEs and declaring one worse adapted. The stress-test point is worth checking: if the physical derivation of those SDEs already encodes a noise convention, then the ranking is not an independent test. The abstract does not give the explicit criterion used to judge adaptation, such as preservation of equilibrium or matching to a known limit, so the strength of the conclusion is hard to gauge from the summary alone. The work is incremental rather than foundational. It will mainly interest people already comparing noise interpretations in statistical mechanics who need a compact reference for the HK integral. The math is concrete enough that a serious referee could check the definition and the calculations on the two models without much trouble. I would send it to review with a request to clarify how the model equations were selected and what observable decides adaptation.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces the Hänggi-Klimontovich integral in a precise mathematical manner, analyzes its properties that appeal to physicists, and applies it to two classical statistical mechanical systems (random dispersal of Langevin particles and relativistic Brownian motion). It concludes that, for these examples, the Hänggi-Klimontovich integral is worse adapted than the Itô integral and even the Stratonovich one.

Significance. If the modeling choices prove interpretation-neutral and the comparisons are made rigorous with explicit criteria, the result would supply concrete counter-examples to the claimed superiority of the Hänggi-Klimontovich interpretation in statistical mechanics, thereby contributing to the long-standing debate on stochastic integral conventions.

major comments (2)

[Abstract] Abstract: the comparative conclusion is stated but no derivations, error estimates, or explicit quantitative criteria for 'worse adapted' are supplied, rendering the central claim unverifiable from the given text.
[Modeling sections for the examples] Modeling of the two examples: the claim that the Hänggi-Klimontovich integral is objectively worse adapted requires that the chosen SDEs for Langevin dispersal and relativistic Brownian motion are physically faithful independently of the noise interpretation; without an explicit check that the modeling step does not already encode one interpretation, the ranking risks circularity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the major comments point by point below, indicating where revisions will be made to improve clarity and address potential concerns about verifiability and modeling assumptions.

read point-by-point responses

Referee: [Abstract] Abstract: the comparative conclusion is stated but no derivations, error estimates, or explicit quantitative criteria for 'worse adapted' are supplied, rendering the central claim unverifiable from the given text.

Authors: The abstract is a concise summary of the manuscript's scope and conclusions, following standard practice. The precise mathematical introduction of the Hänggi-Klimontovich integral, its properties, and the explicit derivations and comparisons for the two examples (including calculations of resulting distributions and trajectories that quantify the adaptation) appear in the main text. To make the criteria for 'worse adapted' more immediately verifiable, we will add a brief clarifying paragraph near the end of the introduction that summarizes the quantitative measures employed, such as deviations from expected physical equilibria. revision: partial
Referee: [Modeling sections for the examples] Modeling of the two examples: the claim that the Hänggi-Klimontovich integral is objectively worse adapted requires that the chosen SDEs for Langevin dispersal and relativistic Brownian motion are physically faithful independently of the noise interpretation; without an explicit check that the modeling step does not already encode one interpretation, the ranking risks circularity.

Authors: We agree this clarification is necessary to avoid any appearance of circularity. The SDEs are obtained from underlying deterministic physical models (Newton's laws with stochastic forcing for Langevin dispersal; the relativistic Langevin equation for the second example) without reference to a specific stochastic integral. To make this independence explicit, we will insert a short dedicated paragraph in each modeling section that traces the physical derivation step by step and notes that the noise term is introduced prior to choosing the integral convention. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies distinct integrals to fixed SDE models without self-referential reduction.

full rationale

The paper first defines the Hänggi-Klimontovich integral mathematically and lists its properties, then substitutes each of the three interpretations into the same pre-chosen SDEs for Langevin dispersal and relativistic Brownian motion. The ranking of adaptation follows directly from solving those SDEs and comparing the resulting statistics to the physical expectations encoded in the model equations; no parameter is fitted to one interpretation and then relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The modeling assumptions are stated explicitly and remain external to the integral-ranking step itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5686 in / 984 out tokens · 17585 ms · 2026-05-24T06:49:54.997918+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages · cited by 1 Pith paper

[1]

Brettschneider, G

T. Brettschneider, G. Volpe, L. Helden, J. Wehr, and C. Be chinger, Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus d rift method , Physical Review E 83, 041113 (2011)

work page 2011
[2]

Correales and C

´A. Correales and C. Escudero, Itˆ o vs Stratonovich in the presence of absorbing states , J. Math. Phys. 60, 123301 (2019)

work page 2019
[3]

Dunkel, Relativistic Brownian motion and diﬀusion processes , PhD thesis, Universit¨ at Augsburg, 2008

J. Dunkel, Relativistic Brownian motion and diﬀusion processes , PhD thesis, Universit¨ at Augsburg, 2008

work page 2008
[4]

Dunkel and P

J. Dunkel and P. H¨ anggi, Theory of relativistic Brownian motion: the (1+ 1)-dimensi onal case, Physical Review E 71, 016124 (2005)

work page 2005
[5]

Dunkel and P

J. Dunkel and P. H¨ anggi, Relativistic Brownian motion , Physics Reports 471, 1–73 (2009)

work page 2009
[6]

Escudero, A simple comparison between Skorokhod & Russo-Vallois inte gration for insider trading , Stochastic Analysis and Applications 36, 485–494 (2018)

C. Escudero, A simple comparison between Skorokhod & Russo-Vallois inte gration for insider trading , Stochastic Analysis and Applications 36, 485–494 (2018)

work page 2018
[7]

Escudero, Kinetic energy of the Langevin particle , Stud

C. Escudero, Kinetic energy of the Langevin particle , Stud. Appl. Math. 145, 719–738 (2020). 22 C. ESCUDERO AND H. ROJAS

work page 2020
[8]

Escudero, Fluctuation-dissipation relation, Maxwell-Boltzmann st atistics, equipartition theorem, and stochas- tic calculus , Phys

C. Escudero, Fluctuation-dissipation relation, Maxwell-Boltzmann st atistics, equipartition theorem, and stochas- tic calculus , Phys. Scr. 98, 055214 (2023)

work page 2023
[9]

Elizalde and C

M. Elizalde and C. Escudero, Chances for the honest in honest versus insider trading , SIAM Journal on Financial Mathematics 13, SC39–SC52 (2022)

work page 2022
[10]

Escudero and C

C. Escudero and C. Manada, Itˆ o versus Stratonovich in a stochastic cosmological model, Lett. Math. Phys. 112, 12 (2022)

work page 2022
[11]

L. C. Evans, An introduction to stochastic diﬀerential equations , American Mathematical Society, 2012

work page 2012
[12]

Gardiner, Stochastic methods: A handbook for the natural sciences , Springer, Berlin, 2009

C. Gardiner, Stochastic methods: A handbook for the natural sciences , Springer, Berlin, 2009

work page 2009
[13]

H¨ anggi,Stochastic processes I: Asymptotic behaviour and symmetri es, Helv

P. H¨ anggi,Stochastic processes I: Asymptotic behaviour and symmetri es, Helv. Phys. Acta 51, 183–201 (1978)

work page 1978
[14]

H¨ anggi,Connection between deterministic and stochastic descript ions of nonlinear systems , Helv

P. H¨ anggi,Connection between deterministic and stochastic descript ions of nonlinear systems , Helv. Phys. Acta 53, 491–496 (1980)

work page 1980
[15]

H¨ anggi and H

P. H¨ anggi and H. Thomas, Stochastic processes: Time evolution, symmetries and line ar response , Phys. Rep. 88, 207–319 (1982)

work page 1982
[16]

Holden, B

H. Holden, B. Øksendal, J. Ubøe, and T. Zhang, Stochastic partial diﬀerential equations , Birkh¨ auser, Boston, 1996

work page 1996
[17]

Horsthemke and R

W. Horsthemke and R. Lefever, Noise-induced transitions: Theory and applications in phy sics, chemistry, and biology, Springer, Berlin, 1984

work page 1984
[18]

H¨ utter and H

M. H¨ utter and H. C. ¨Ottinger, Fluctuation-dissipation theorem, kinetic stochastic int egral and eﬃcient simula- tions, Journal of the Chemical Society, Faraday Transactions 94, 1403–1405 (1998)

work page 1998
[19]

Itˆ o, Stochastic integral, Proc

K. Itˆ o, Stochastic integral, Proc. Imp. Acad. Tokyo 20, 519–524 (1944)

work page 1944
[20]

Itˆ o, On a stochastic integral equation , Proc

K. Itˆ o, On a stochastic integral equation , Proc. Imp. Acad. Tokyo 22, 32–35 (1946)

work page 1946
[21]

Jeanblanc, M

M. Jeanblanc, M. Yor, and M. Chesney, Mathematical methods for ﬁnancial markets , Springer, London, 2009

work page 2009
[22]

Kallenberg, Random measures, theory and applications , Springer, Cham, 2017

O. Kallenberg, Random measures, theory and applications , Springer, Cham, 2017

work page 2017
[23]

Yu. L. Klimontovich, Ito, Stratonovich and kinetic forms of stochastic equation s, Physica A: Statistical Mechanics and its Applications 163, 515–532 (1990)

work page 1990
[24]

Kinetic form

Yu. L. Klimontovich, Alternative description of stochastic processes in nonlin ear systems. “Kinetic form” of master and Fokker-Planck equations , Physica A: Statistical Mechanics and its Applications 182, 121–132 (1992)

work page 1992
[25]

Yu. L. Klimontovich, Nonlinear Brownian motion , Physics-Uspekhi 37, 737–767 (1994)

work page 1994
[26]

H. H. Kuo, Introduction to stochastic integration , Springer, New York, 2006

work page 2006
[27]

Langevin, Sur la th´ eorie du mouvement brownien, C

P. Langevin, Sur la th´ eorie du mouvement brownien, C. R. Acad. Sci. Paris 146, 530–533 (1908)

work page 1908
[28]

A. W. C. Lau and T. C. Lubensky, State-dependent diﬀusion: Thermodynamic consistency and i ts path integral formulation, Phys. Rev. E 76, 011123 (2007)

work page 2007
[29]

Le Gall, Brownian motion, martingales, and stochastic calculus , Springer, Orsay, 2016

J.-F. Le Gall, Brownian motion, martingales, and stochastic calculus , Springer, Orsay, 2016

work page 2016
[30]

Leibovich and E

N. Leibovich and E. Barkai, Inﬁnite ergodic theory for heterogeneous diﬀusion processes, Phys. Rev. E 99, 042138 (2019)

work page 2019
[31]

Y. Li, R. Mei, Y. Xu, J. Kurths, J. Duan, and R. Metzler, Particle dynamics and transport enhancement in a conﬁned channel with position-dependent diﬀusivity , New Journal of Physics 22, 053016 (2020)

work page 2020
[32]

Lindner, The diﬀusion coeﬃcient of nonlinear Brownian motion , New Journal of Physics 9, 136 (2007)

B. Lindner, The diﬀusion coeﬃcient of nonlinear Brownian motion , New Journal of Physics 9, 136 (2007)

work page 2007
[33]

Mannella and P

R. Mannella and P. V. McClintock, Itˆ o versus Stratonovich: 30 years later , Fluctuation and Noise Letters 11, 1240010 (2012)

work page 2012
[34]

Øksendal, Stochastic diﬀerential equations: An introduction with app lications, Springer, Berlin, 2003

B. Øksendal, Stochastic diﬀerential equations: An introduction with app lications, Springer, Berlin, 2003

work page 2003
[35]

G. A. Pavliotis, Stochastic processes and applications: Diﬀusion processes , the Fokker-Planck and Langevin equations, Springer, New York, 2014

work page 2014
[36]

Pitman and M

J. Pitman and M. Winkel, Squared Bessel processes of positive and negative dimensio n embedded in Brownian local times, Electron. Commun. Probab. 23, 1–13 (2018)

work page 2018
[37]

Regev, N

S. Regev, N. Grønbech-Jensen, and O. Farago, Isothermal Langevin dynamics in systems with power-law spa tially- dependent friction , Phys Rev. E 94, 012116 (2016)

work page 2016
[38]

Russo and P

F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probability Theory and Related Fields 97, 403–421 (1993)

work page 1993
[39]

Russo and P

F. Russo and P. Vallois, The generalized covariation process and Itˆ o formula , Stochastic Processes and their Applications 59, 81–104 (1995)

work page 1995
[40]

Russo and P

F. Russo and P. Vallois, Stochastic calculus with respect to continuous ﬁnite quadr atic variation processes , Stochastics: An International Journal of Probability and S tochastic Processes 70, 1–40 (2000)

work page 2000
[41]

Russo and P

F. Russo and P. Vallois, Elements of stochastic calculus via regularization , in S´ eminaire de Probabilit´ es XL, 147–185, Springer, 2007

work page 2007
[42]

I. M. Sokolov, Ito, Stratonovich, H¨ anggi and all the rest: The thermodyna mics of interpretation , Chemical Physics 375, 359–363 (2010)

work page 2010
[43]

R. L. Stratonovich, A new representation for stochastic integrals and equation s, SIAM J. Control 4, 362–371 (1966)

work page 1966
[44]

Tsekov, Stochastic equations for thermodynamics , Journal of the Chemical Society, Faraday Transactions 93, 1751–1753 (1997)

R. Tsekov, Stochastic equations for thermodynamics , Journal of the Chemical Society, Faraday Transactions 93, 1751–1753 (1997). IT ˆO VS H ¨ANGGI–KLIMONTOVICH 23

work page 1997
[45]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion , Phys. Rev. 36, 823–841 (1930)

work page 1930
[46]

N. G. van Kampen, Itˆ o versus Stratonovich, Journal of Statistical Physics 24, 175–187 (1981)

work page 1981
[47]

N. G. van Kampen, Stochastic processes in physics and chemistry , Elsevier, Amsterdam, 1992

work page 1992
[48]

Vasicek, An equilibrium characterization of the term structure , Journal of Financial Economics 5, 177–188 (1977)

O. Vasicek, An equilibrium characterization of the term structure , Journal of Financial Economics 5, 177–188 (1977)

work page 1977
[49]

Volpe, L

G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. B echinger, Inﬂuence of noise on force measurements , Physical Review Letters 104, 170602 (2010)

work page 2010
[50]

Watanabe and T

S. Watanabe and T. Yamada, On the uniqueness of solutions of stochastic diﬀerential equ ations, J. Math. Kyoto Univ. 11, 155–167 (1971). Carlos Escudero Departamento de Matem´ aticas Fundamentales Universidad Nacional de Educaci´ on a Distancia cescudero@mat.uned.es Helder Rojas Departamento de Matem´ aticas Fundamentales Universidad Nacional de Educaci´ ...

work page 1971

[1] [1]

Brettschneider, G

T. Brettschneider, G. Volpe, L. Helden, J. Wehr, and C. Be chinger, Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus d rift method , Physical Review E 83, 041113 (2011)

work page 2011

[2] [2]

Correales and C

´A. Correales and C. Escudero, Itˆ o vs Stratonovich in the presence of absorbing states , J. Math. Phys. 60, 123301 (2019)

work page 2019

[3] [3]

Dunkel, Relativistic Brownian motion and diﬀusion processes , PhD thesis, Universit¨ at Augsburg, 2008

J. Dunkel, Relativistic Brownian motion and diﬀusion processes , PhD thesis, Universit¨ at Augsburg, 2008

work page 2008

[4] [4]

Dunkel and P

J. Dunkel and P. H¨ anggi, Theory of relativistic Brownian motion: the (1+ 1)-dimensi onal case, Physical Review E 71, 016124 (2005)

work page 2005

[5] [5]

Dunkel and P

J. Dunkel and P. H¨ anggi, Relativistic Brownian motion , Physics Reports 471, 1–73 (2009)

work page 2009

[6] [6]

Escudero, A simple comparison between Skorokhod & Russo-Vallois inte gration for insider trading , Stochastic Analysis and Applications 36, 485–494 (2018)

C. Escudero, A simple comparison between Skorokhod & Russo-Vallois inte gration for insider trading , Stochastic Analysis and Applications 36, 485–494 (2018)

work page 2018

[7] [7]

Escudero, Kinetic energy of the Langevin particle , Stud

C. Escudero, Kinetic energy of the Langevin particle , Stud. Appl. Math. 145, 719–738 (2020). 22 C. ESCUDERO AND H. ROJAS

work page 2020

[8] [8]

Escudero, Fluctuation-dissipation relation, Maxwell-Boltzmann st atistics, equipartition theorem, and stochas- tic calculus , Phys

C. Escudero, Fluctuation-dissipation relation, Maxwell-Boltzmann st atistics, equipartition theorem, and stochas- tic calculus , Phys. Scr. 98, 055214 (2023)

work page 2023

[9] [9]

Elizalde and C

M. Elizalde and C. Escudero, Chances for the honest in honest versus insider trading , SIAM Journal on Financial Mathematics 13, SC39–SC52 (2022)

work page 2022

[10] [10]

Escudero and C

C. Escudero and C. Manada, Itˆ o versus Stratonovich in a stochastic cosmological model, Lett. Math. Phys. 112, 12 (2022)

work page 2022

[11] [11]

L. C. Evans, An introduction to stochastic diﬀerential equations , American Mathematical Society, 2012

work page 2012

[12] [12]

Gardiner, Stochastic methods: A handbook for the natural sciences , Springer, Berlin, 2009

C. Gardiner, Stochastic methods: A handbook for the natural sciences , Springer, Berlin, 2009

work page 2009

[13] [13]

H¨ anggi,Stochastic processes I: Asymptotic behaviour and symmetri es, Helv

P. H¨ anggi,Stochastic processes I: Asymptotic behaviour and symmetri es, Helv. Phys. Acta 51, 183–201 (1978)

work page 1978

[14] [14]

H¨ anggi,Connection between deterministic and stochastic descript ions of nonlinear systems , Helv

P. H¨ anggi,Connection between deterministic and stochastic descript ions of nonlinear systems , Helv. Phys. Acta 53, 491–496 (1980)

work page 1980

[15] [15]

H¨ anggi and H

P. H¨ anggi and H. Thomas, Stochastic processes: Time evolution, symmetries and line ar response , Phys. Rep. 88, 207–319 (1982)

work page 1982

[16] [16]

Holden, B

H. Holden, B. Øksendal, J. Ubøe, and T. Zhang, Stochastic partial diﬀerential equations , Birkh¨ auser, Boston, 1996

work page 1996

[17] [17]

Horsthemke and R

W. Horsthemke and R. Lefever, Noise-induced transitions: Theory and applications in phy sics, chemistry, and biology, Springer, Berlin, 1984

work page 1984

[18] [18]

H¨ utter and H

M. H¨ utter and H. C. ¨Ottinger, Fluctuation-dissipation theorem, kinetic stochastic int egral and eﬃcient simula- tions, Journal of the Chemical Society, Faraday Transactions 94, 1403–1405 (1998)

work page 1998

[19] [19]

Itˆ o, Stochastic integral, Proc

K. Itˆ o, Stochastic integral, Proc. Imp. Acad. Tokyo 20, 519–524 (1944)

work page 1944

[20] [20]

Itˆ o, On a stochastic integral equation , Proc

K. Itˆ o, On a stochastic integral equation , Proc. Imp. Acad. Tokyo 22, 32–35 (1946)

work page 1946

[21] [21]

Jeanblanc, M

M. Jeanblanc, M. Yor, and M. Chesney, Mathematical methods for ﬁnancial markets , Springer, London, 2009

work page 2009

[22] [22]

Kallenberg, Random measures, theory and applications , Springer, Cham, 2017

O. Kallenberg, Random measures, theory and applications , Springer, Cham, 2017

work page 2017

[23] [23]

Yu. L. Klimontovich, Ito, Stratonovich and kinetic forms of stochastic equation s, Physica A: Statistical Mechanics and its Applications 163, 515–532 (1990)

work page 1990

[24] [24]

Kinetic form

Yu. L. Klimontovich, Alternative description of stochastic processes in nonlin ear systems. “Kinetic form” of master and Fokker-Planck equations , Physica A: Statistical Mechanics and its Applications 182, 121–132 (1992)

work page 1992

[25] [25]

Yu. L. Klimontovich, Nonlinear Brownian motion , Physics-Uspekhi 37, 737–767 (1994)

work page 1994

[26] [26]

H. H. Kuo, Introduction to stochastic integration , Springer, New York, 2006

work page 2006

[27] [27]

Langevin, Sur la th´ eorie du mouvement brownien, C

P. Langevin, Sur la th´ eorie du mouvement brownien, C. R. Acad. Sci. Paris 146, 530–533 (1908)

work page 1908

[28] [28]

A. W. C. Lau and T. C. Lubensky, State-dependent diﬀusion: Thermodynamic consistency and i ts path integral formulation, Phys. Rev. E 76, 011123 (2007)

work page 2007

[29] [29]

Le Gall, Brownian motion, martingales, and stochastic calculus , Springer, Orsay, 2016

J.-F. Le Gall, Brownian motion, martingales, and stochastic calculus , Springer, Orsay, 2016

work page 2016

[30] [30]

Leibovich and E

N. Leibovich and E. Barkai, Inﬁnite ergodic theory for heterogeneous diﬀusion processes, Phys. Rev. E 99, 042138 (2019)

work page 2019

[31] [31]

Y. Li, R. Mei, Y. Xu, J. Kurths, J. Duan, and R. Metzler, Particle dynamics and transport enhancement in a conﬁned channel with position-dependent diﬀusivity , New Journal of Physics 22, 053016 (2020)

work page 2020

[32] [32]

Lindner, The diﬀusion coeﬃcient of nonlinear Brownian motion , New Journal of Physics 9, 136 (2007)

B. Lindner, The diﬀusion coeﬃcient of nonlinear Brownian motion , New Journal of Physics 9, 136 (2007)

work page 2007

[33] [33]

Mannella and P

R. Mannella and P. V. McClintock, Itˆ o versus Stratonovich: 30 years later , Fluctuation and Noise Letters 11, 1240010 (2012)

work page 2012

[34] [34]

Øksendal, Stochastic diﬀerential equations: An introduction with app lications, Springer, Berlin, 2003

B. Øksendal, Stochastic diﬀerential equations: An introduction with app lications, Springer, Berlin, 2003

work page 2003

[35] [35]

G. A. Pavliotis, Stochastic processes and applications: Diﬀusion processes , the Fokker-Planck and Langevin equations, Springer, New York, 2014

work page 2014

[36] [36]

Pitman and M

J. Pitman and M. Winkel, Squared Bessel processes of positive and negative dimensio n embedded in Brownian local times, Electron. Commun. Probab. 23, 1–13 (2018)

work page 2018

[37] [37]

Regev, N

S. Regev, N. Grønbech-Jensen, and O. Farago, Isothermal Langevin dynamics in systems with power-law spa tially- dependent friction , Phys Rev. E 94, 012116 (2016)

work page 2016

[38] [38]

Russo and P

F. Russo and P. Vallois, Forward, backward and symmetric stochastic integration, Probability Theory and Related Fields 97, 403–421 (1993)

work page 1993

[39] [39]

Russo and P

F. Russo and P. Vallois, The generalized covariation process and Itˆ o formula , Stochastic Processes and their Applications 59, 81–104 (1995)

work page 1995

[40] [40]

Russo and P

F. Russo and P. Vallois, Stochastic calculus with respect to continuous ﬁnite quadr atic variation processes , Stochastics: An International Journal of Probability and S tochastic Processes 70, 1–40 (2000)

work page 2000

[41] [41]

Russo and P

F. Russo and P. Vallois, Elements of stochastic calculus via regularization , in S´ eminaire de Probabilit´ es XL, 147–185, Springer, 2007

work page 2007

[42] [42]

I. M. Sokolov, Ito, Stratonovich, H¨ anggi and all the rest: The thermodyna mics of interpretation , Chemical Physics 375, 359–363 (2010)

work page 2010

[43] [43]

R. L. Stratonovich, A new representation for stochastic integrals and equation s, SIAM J. Control 4, 362–371 (1966)

work page 1966

[44] [44]

Tsekov, Stochastic equations for thermodynamics , Journal of the Chemical Society, Faraday Transactions 93, 1751–1753 (1997)

R. Tsekov, Stochastic equations for thermodynamics , Journal of the Chemical Society, Faraday Transactions 93, 1751–1753 (1997). IT ˆO VS H ¨ANGGI–KLIMONTOVICH 23

work page 1997

[45] [45]

G. E. Uhlenbeck and L. S. Ornstein, On the theory of the Brownian motion , Phys. Rev. 36, 823–841 (1930)

work page 1930

[46] [46]

N. G. van Kampen, Itˆ o versus Stratonovich, Journal of Statistical Physics 24, 175–187 (1981)

work page 1981

[47] [47]

N. G. van Kampen, Stochastic processes in physics and chemistry , Elsevier, Amsterdam, 1992

work page 1992

[48] [48]

Vasicek, An equilibrium characterization of the term structure , Journal of Financial Economics 5, 177–188 (1977)

O. Vasicek, An equilibrium characterization of the term structure , Journal of Financial Economics 5, 177–188 (1977)

work page 1977

[49] [49]

Volpe, L

G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. B echinger, Inﬂuence of noise on force measurements , Physical Review Letters 104, 170602 (2010)

work page 2010

[50] [50]

Watanabe and T

S. Watanabe and T. Yamada, On the uniqueness of solutions of stochastic diﬀerential equ ations, J. Math. Kyoto Univ. 11, 155–167 (1971). Carlos Escudero Departamento de Matem´ aticas Fundamentales Universidad Nacional de Educaci´ on a Distancia cescudero@mat.uned.es Helder Rojas Departamento de Matem´ aticas Fundamentales Universidad Nacional de Educaci´ ...

work page 1971