January  2021, 26(1): 121-154. doi: 10.3934/dcdsb.2020230

Bilinear equations in Hilbert space driven by paths of low regularity

1. 

Charles University, Faculty of Mathematics and Physics, Sokolovská 83, Prague 8,186 75, Czech Republic

2. 

Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y Análisis numérico, Avda. Reina Mercedes s/n, 41012-Sevilla, Spain

* Corresponding author: María J. Garrido-Atienza

Received  January 2020 Revised  June 2020 Published  January 2021 Early access  July 2020

Fund Project: The first author is supported by the Czech Science Foundation, project GAČR 19-07140S. The second author is supported by Ministerio de Ciencia, Innovación y Universidades, Grant No. PGC2018-096540-I00

In the article, some bilinear evolution equations in Hilbert space driven by paths of low regularity are considered and solved explicitly. The driving paths are scalar-valued and continuous, and they are assumed to have a finite $ p $-th variation along a sequence of partitions in the sense given by Cont and Perkowski [Trans. Amer. Math. Soc. Ser. B, 6 (2019) 161–186] ($ p $ being an even positive integer). Typical functions that satisfy this condition are trajectories of the fractional Brownian motion with Hurst parameter $H=1 / p$. A strong solution to the bilinear problem is shown to exist if there is a solution to a certain non–autonomous initial value problem. Subsequently, sufficient conditions for the existence of the solution to this initial value problem are given. The abstract results are applied to several stochastic partial differential equations with multiplicative fractional noise, both of the parabolic and hyperbolic type, that are solved explicitly in a pathwise sense.

Citation: Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230
References:
[1] J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York/London, 1966. 
[2]

M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains, Springer International Publishing, 2015. doi: 10.1007/978-3-319-14648-5.

[3]

E. Alòs and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stoch. Stoch. Rep., 75 (2003), 129-152.  doi: 10.1080/1045112031000078917.

[4]

H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sci., 11 (1984), 593-676. 

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A. Ananova and R. Cont, Pathwise integration with respect to paths of finite quadratic variation, J. Math. Pures Appl., 107 (2017), 737-757.  doi: 10.1016/j.matpur.2016.10.004.

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M. Caruana and P. Friz, Partial differential equations driven by rough paths, J. Differ. Equ., 247 (2009), 140-173.  doi: 10.1016/j.jde.2009.01.026.

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M. CaruanaP. Friz and H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 28 (2011), 27-46.  doi: 10.1016/j.anihpc.2010.11.002.

[9]

P. Cheridito and D. Nualart, Stochastic integral of divergence type with respecto to fractional Brownian motion with Hurst parameter ${H}\in\left(0, \frac{1}{2}\right)$, Ann. I. H. Poincaré Probab. Stat., 41 (2005), 1049-1081.  doi: 10.1016/j.anihpb.2004.09.004.

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R. Cont and P. Das, Quadratic variation and quadratic roughness, preprint, arXiv: 1907.03115.

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R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259 (2010), 1043-1072.  doi: 10.1016/j.jfa.2010.04.017.

[12]

R. Cont and N. Perkowski, Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Trans. Amer. Math. Soc. Ser. B, 6 (2019), 161-186.  doi: 10.1090/btran/34.

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A. DeyaM. GubinelliM. Hofmanová and S. Tindel, One-dimensional reflected rough differential equations, Stoch. Proc. Appl., 129 (2019), 3261-3281.  doi: 10.1016/j.spa.2018.09.007.

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K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag New York, 2000.

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E. H. Essaky and D. Nualart, On the $\frac{1}{H}$-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter ${H}<\frac{1}{2}$, Stoch. Proc. Appl., 125 (2015), 4117-4141.  doi: 10.1016/j.spa.2015.06.001.

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M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3, 1/2]$, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2553-2581.  doi: 10.3934/dcdsb.2015.20.2553.

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M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.

[31]

M. J. Garrido-AtienzaB. Maslowski and J. Šnupárková, Semilinear stochastic equations with bilinear fractional noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3075-3094.  doi: 10.3934/dcdsb.2016088.

[32]

M. Gitterman, Classical harmonic oscillator with multiplicative noise, Physica A, 352 (2005), 309-334.  doi: 10.1016/j.physa.2005.01.008.

[33]

M. Gubinelli, Ramification of rough paths, J. Differ. Equ., 248 (2010), 693-721.  doi: 10.1016/j.jde.2009.11.015.

[34]

M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum of Mathematics, Pi, 3 (2015), e6, 75pp. doi: 10.1017/fmp.2015.2.

[35]

M. GubinelliA. Lejay and S. Tindel, Young integrals and SPDEs, Potential Anal., 25 (2006), 307-326.  doi: 10.1007/s11118-006-9013-5.

[36]

M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75.  doi: 10.1214/08-AOP437.

[37]

J. M. E. Guerra and D. Nualart, The $1/{H}$-variation of the divergence integral with respect to the fractional Brownian motion for ${H}>1/2$ and fractional Bessel processes, Stoch. Proc. Appl., 115 (2005), 91-115.  doi: 10.1016/j.spa.2004.07.008.

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R. Hesse and A. Neamţu, Local mild solutions for rough stochastic partial differential equations, J. Differential Equations, 267 (2019), 6480-6538.  doi: 10.1016/j.jde.2019.06.026.

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Y. Hirai, Remarks on Föllmer's pathwise Itô calculus, Osaka J. Math., 56 (2019), 631-660. 

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A. Hocquet and M. Hofmanová, An energy method for rough partial differential equations, J. Differ. Equ., 265 (2018), 1407-1466.  doi: 10.1016/j.jde.2018.04.006.

[42]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.

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D. Kim, Local times for continuous paths of arbitrary regularity, preprint, arXiv: 1904.07327.

[44]

K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654.  doi: 10.2969/jmsj/03140647.

[45]

M. Lemieux, On the Quadratic Variation of Semi-martingales, Master's thesis, University of British Columbia, 1983.

[46]

B. M. Levitan and G. L. Litvinov, Generalized displacement operators, in Encyclopaedia of Mathematics (ed. M. Hazewinkel), vol. 4, Springer Netherlands, 1989,224–228.

[47]

R. M. ŁochowskiN. Perkowski and D. J. Prömel, A superhedging approach to stochastic integration, Stoch. Proc. Appl., 128 (2018), 4078-4103.  doi: 10.1016/j.spa.2018.01.009.

[48]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.

[49]

B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305.  doi: 10.1016/S0022-1236(02)00065-4.

[50]

B. Maslowski and J. Šnupárková, Stochastic affine evolution equations with multiplicative fractional noise, Appl. Math., 63 (2018), 7-35.  doi: 10.21136/AM.2018.0036-17.

[51]

Y. Mishura and A. Schied, On (signed) Takagi-Landsberg functions: $p$th variation, maximum, and modulus of continuity, J. Math. Anal. Appl., 473 (2019), 258-272.  doi: 10.1016/j.jmaa.2018.12.047.

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D. Nualart, The Malliavin Calculus and Related Topics, Springer - Verlag Berlin Heidelberg, 2006.

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A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied mathematical sciences, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

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N. Perkowski and D. Prömel, Pathwise stochastic integrals for model free finance, Bernoulli, 22 (2016), 2486-2520.  doi: 10.3150/15-BEJ735.

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V. Pipiras and M. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli, 7 (2001), 873-897.  doi: 10.2307/3318624.

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M. Pratelli, A remark on the $1/{H}$-variation of the fractional Brownian motion, in Séminaire de Probabilités XLIII (eds. C. Donati-Martin, A. Lejay and A. Rauault), vol. 2006 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 2011,215–219. doi: 10.1007/978-3-642-15217-7_8.

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show all references

References:
[1] J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York/London, 1966. 
[2]

M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains, Springer International Publishing, 2015. doi: 10.1007/978-3-319-14648-5.

[3]

E. Alòs and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stoch. Stoch. Rep., 75 (2003), 129-152.  doi: 10.1080/1045112031000078917.

[4]

H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sci., 11 (1984), 593-676. 

[5]

A. Ananova and R. Cont, Pathwise integration with respect to paths of finite quadratic variation, J. Math. Pures Appl., 107 (2017), 737-757.  doi: 10.1016/j.matpur.2016.10.004.

[6]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer-Verlag London, 2006.

[7]

M. Caruana and P. Friz, Partial differential equations driven by rough paths, J. Differ. Equ., 247 (2009), 140-173.  doi: 10.1016/j.jde.2009.01.026.

[8]

M. CaruanaP. Friz and H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 28 (2011), 27-46.  doi: 10.1016/j.anihpc.2010.11.002.

[9]

P. Cheridito and D. Nualart, Stochastic integral of divergence type with respecto to fractional Brownian motion with Hurst parameter ${H}\in\left(0, \frac{1}{2}\right)$, Ann. I. H. Poincaré Probab. Stat., 41 (2005), 1049-1081.  doi: 10.1016/j.anihpb.2004.09.004.

[10]

R. Cont and P. Das, Quadratic variation and quadratic roughness, preprint, arXiv: 1907.03115.

[11]

R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259 (2010), 1043-1072.  doi: 10.1016/j.jfa.2010.04.017.

[12]

R. Cont and N. Perkowski, Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Trans. Amer. Math. Soc. Ser. B, 6 (2019), 161-186.  doi: 10.1090/btran/34.

[13]

G. Da PratoM. Iannelli and L. Tubaro, Some results on linear stochastic differential equations in Hilbert spaces, Stochastics, 6 (1982), 105-116.  doi: 10.1080/17442508208833196.

[14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.
[15]

M. DavisJ. Obłój and V. Raval, Arbitrage boundes for prices of weighted variance swaps, Math. Fin., 24 (2014), 821-854.  doi: 10.1111/mafi.12021.

[16]

M. DavisJ. Obłój and P. Siorpaes, Pathwise stochastic calculus with local times, Ann. Inst. H. Poincaré Probab. Statist., 54 (2018), 1-21.  doi: 10.1214/16-AIHP792.

[17]

L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214.  doi: 10.1023/A:1008634027843.

[18]

A. DeyaM. GubinelliM. Hofmanová and S. Tindel, One-dimensional reflected rough differential equations, Stoch. Proc. Appl., 129 (2019), 3261-3281.  doi: 10.1016/j.spa.2018.09.007.

[19]

A. DeyaM. GubinelliM. Hofmanová and S. Tindel, A priori estimates for rough PDEs with application to rough conservation laws, J. Funct. Anal., 276 (2019), 3577-3645.  doi: 10.1016/j.jfa.2019.03.008.

[20]

A. DeyaM. Gubinelli and S. Tindel, Non-linear rough heat equations, Probab. Theory Relat. Fields, 153 (2012), 97-147.  doi: 10.1007/s00440-011-0341-z.

[21] J. Dieudonné, Foundations of Modern Analysis, vol. 1, Academic Press, New York/London, 1969. 
[22]

T. E. DuncanB. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stoch. Proc. Appl., 115 (2005), 1357-1383.  doi: 10.1016/j.spa.2005.03.011.

[23]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag New York, 2000.

[24]

E. H. Essaky and D. Nualart, On the $\frac{1}{H}$-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter ${H}<\frac{1}{2}$, Stoch. Proc. Appl., 125 (2015), 4117-4141.  doi: 10.1016/j.spa.2015.06.001.

[25]

H. Föllmer, Calcul d'Itô sans probabilités, in Séminaire de Probabilités XV, vol. 850 of Lecture Notes in Mathematics, Springer, Berlin, 1981,143–150.

[26]

H. Föllmer and A. Schied, Probabilistic aspects of finance, Bernoulli, 19 (2013), 1306-1326.  doi: 10.3150/12-BEJSP05.

[27]

P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014. doi: 10.1007/978-3-319-08332-2.

[28] P. K. Friz and N. B. Victoir., Multidimensional Stochastic Processes as Rough Paths, vol. 120 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.
[29]

M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3, 1/2]$, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2553-2581.  doi: 10.3934/dcdsb.2015.20.2553.

[30]

M. J. Garrido-AtienzaK. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.

[31]

M. J. Garrido-AtienzaB. Maslowski and J. Šnupárková, Semilinear stochastic equations with bilinear fractional noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3075-3094.  doi: 10.3934/dcdsb.2016088.

[32]

M. Gitterman, Classical harmonic oscillator with multiplicative noise, Physica A, 352 (2005), 309-334.  doi: 10.1016/j.physa.2005.01.008.

[33]

M. Gubinelli, Ramification of rough paths, J. Differ. Equ., 248 (2010), 693-721.  doi: 10.1016/j.jde.2009.11.015.

[34]

M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum of Mathematics, Pi, 3 (2015), e6, 75pp. doi: 10.1017/fmp.2015.2.

[35]

M. GubinelliA. Lejay and S. Tindel, Young integrals and SPDEs, Potential Anal., 25 (2006), 307-326.  doi: 10.1007/s11118-006-9013-5.

[36]

M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75.  doi: 10.1214/08-AOP437.

[37]

J. M. E. Guerra and D. Nualart, The $1/{H}$-variation of the divergence integral with respect to the fractional Brownian motion for ${H}>1/2$ and fractional Bessel processes, Stoch. Proc. Appl., 115 (2005), 91-115.  doi: 10.1016/j.spa.2004.07.008.

[38]

M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley Series in Physics Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1962.

[39]

R. Hesse and A. Neamţu, Local mild solutions for rough stochastic partial differential equations, J. Differential Equations, 267 (2019), 6480-6538.  doi: 10.1016/j.jde.2019.06.026.

[40]

Y. Hirai, Remarks on Föllmer's pathwise Itô calculus, Osaka J. Math., 56 (2019), 631-660. 

[41]

A. Hocquet and M. Hofmanová, An energy method for rough partial differential equations, J. Differ. Equ., 265 (2018), 1407-1466.  doi: 10.1016/j.jde.2018.04.006.

[42]

Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.

[43]

D. Kim, Local times for continuous paths of arbitrary regularity, preprint, arXiv: 1904.07327.

[44]

K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654.  doi: 10.2969/jmsj/03140647.

[45]

M. Lemieux, On the Quadratic Variation of Semi-martingales, Master's thesis, University of British Columbia, 1983.

[46]

B. M. Levitan and G. L. Litvinov, Generalized displacement operators, in Encyclopaedia of Mathematics (ed. M. Hazewinkel), vol. 4, Springer Netherlands, 1989,224–228.

[47]

R. M. ŁochowskiN. Perkowski and D. J. Prömel, A superhedging approach to stochastic integration, Stoch. Proc. Appl., 128 (2018), 4078-4103.  doi: 10.1016/j.spa.2018.01.009.

[48]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.

[49]

B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305.  doi: 10.1016/S0022-1236(02)00065-4.

[50]

B. Maslowski and J. Šnupárková, Stochastic affine evolution equations with multiplicative fractional noise, Appl. Math., 63 (2018), 7-35.  doi: 10.21136/AM.2018.0036-17.

[51]

Y. Mishura and A. Schied, On (signed) Takagi-Landsberg functions: $p$th variation, maximum, and modulus of continuity, J. Math. Anal. Appl., 473 (2019), 258-272.  doi: 10.1016/j.jmaa.2018.12.047.

[52]

D. Nualart, The Malliavin Calculus and Related Topics, Springer - Verlag Berlin Heidelberg, 2006.

[53]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied mathematical sciences, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[54]

N. Perkowski and D. Prömel, Pathwise stochastic integrals for model free finance, Bernoulli, 22 (2016), 2486-2520.  doi: 10.3150/15-BEJ735.

[55]

V. Pipiras and M. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli, 7 (2001), 873-897.  doi: 10.2307/3318624.

[56]

M. Pratelli, A remark on the $1/{H}$-variation of the fractional Brownian motion, in Séminaire de Probabilités XLIII (eds. C. Donati-Martin, A. Lejay and A. Rauault), vol. 2006 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 2011,215–219. doi: 10.1007/978-3-642-15217-7_8.

[57]

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