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doi: 10.3934/dcdsb.2020230

Bilinear equations in Hilbert space driven by paths of low regularity

Petr Čoupek 1, and  María J. Garrido-Atienza 2,,

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  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.

Keywords: Itô-Föllmer calculus, p-th variation along a sequence of partitions, multiplicative noise, fractional Brownian motion, rough paths.
Mathematics Subject Classification: Primary: 60H15, 60G22; Secondary: 34F05, 47D06.
Citation: Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020230
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M. J. Garrido-Atienza, K. 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.  Google Scholar

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M. J. Garrido-Atienza, B. 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.  Google Scholar

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show all references

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

M. Caruana, P. 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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

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[15]

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

[16]

M. Davis, J. 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.  Google Scholar

[17]

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[18]

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[19]

A. Deya, M. Gubinelli, M. 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.  Google Scholar

[20]

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

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[22]

T. E. Duncan, B. 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.  Google Scholar

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

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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.  Google Scholar

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

[26]

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

[27]

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

[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.  Google Scholar
[29]

M. J. Garrido-Atienza, K. 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.  Google Scholar

[30]

M. J. Garrido-Atienza, K. 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.  Google Scholar

[31]

M. J. Garrido-Atienza, B. 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.  Google Scholar

[32]

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

[33]

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

[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.  Google Scholar

[35]

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

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[45]

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

[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. Google Scholar

[47]

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