An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented, and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result, the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance.
REFERENCES
1.
A.
Antonov
and M.
Spector
, “Advanced analytics for the SABR model
,” Available at SSRN 2026350 (2012
).2.
A.
Arnaudon
, A. L.
Castro
, and D. D.
Holm
, “Noise and dissipation on coadjoint orbits
,” preprint arXiv:1601.02249 (2016
).3.
M.
Arnaudon
, X.
Chen
, and A. B.
Cruzeiro
, “Stochastic Euler-Poincaré reduction
,” J. Math. Phys.
55
(8
), 081507
(2014
).4.
F.
Avram
, N. N.
Leonenko
, and N.
Šuvak
, “On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions
,” Markov Process. Related Fields
19
(2
), 249
–298
(2013
), http://math-prf.org/journal/articles/id1297/.5.
M.
Barczy
, L.
Döring
, Z.
Li
, and G.
Pap
, “On parameter estimation for critical affine processes
,” Electron. J. Stat.
7
, 647
–696
(2013
).6.
D.
Belomestny
and M.
Reiß
, “Spectral calibration of exponential Lévy models
,” Finance Stoch.
10
(4
), 449
–474
(2006
).7.
J.-M.
Bismut
, Mécanique Aléatoire
, Lecture Notes in Mathematics
Vol. 866
(Springer-Verlag
, Berlin-New York
, 1981
).8.
L.
Borland
, “A theory of non-Gaussian option pricing
,” Quant. Finance
2
(6
), 415
–431
(2002
).9.
G.
Bormetti
and S.
Cazzaniga
, “Multiplicative noise, fast convolution and pricing
,” Quant. Finance
14
(3
), 481
–494
(2014
).10.
F.
Cordoni
and L.
Di Persio
, “Lie symmetry approach to the CEV model
,” Int. J. Differ. Equations Appl.
13
(3
), 109
–127
(2014
).11.
M.
Craddock
, “Fourier type transforms on Lie symmetry groups
,” J. Math. Phys.
56
(9
), 091501
(2015
).12.
M.
Craddock
and K. A.
Lennox
, “Lie symmetry methods for multi-dimensional parabolic PDEs and diffusions
,” J. Differ. Equations
252
(1
), 56
–90
(2012
).13.
C.
Cuchiero
, M.
Keller-Ressel
, and J.
Teichmann
, “Polynomial processes and their applications to mathematical finance
,” Finance Stoch.
16
(4
), 711
–740
(2012
).14.
F. C.
De Vecchi
, P.
Morando
, and S.
Ugolini
, “Symmetries of stochastic differential equations: A geometric approach
,” J. Math. Phys.
57
(6
) 063504
(2016
).15.
D.
Duffie
, D.
Filipović
, and W.
Schachermayer
, “Affine processes and applications in finance
,” Ann. Appl. Probab.
13
(3
), 984
–1053
(2003
).16.
J. L.
Forman
and M.
Sørensen
, “The Pearson diffusions: A class of statistically tractable diffusion processes
,” Scand. J. Stat.
35
(3
), 438
–465
(2008
).17.
R.
Friedrich
and J.
Peinke
, “Description of a turbulent cascade by a Fokker-Planck equation
,” Phys. Rev. Lett.
78
(5
), 863
(1997
).18.
G.
Gaeta
and N. R.
Quintero
, “Lie-point symmetries and stochastic differential equations
,” J. Phys. A
32
(48
), 8485
–8505
(1999
).19.
T. C.
Gard
, Introduction to Stochastic Differential Equations
, Monographs and Textbooks in Pure and Applied Mathematics
Vol. 114
(Marcel Dekker, Inc.
, New York
, 1988
).20.
F.
Ghasemi
, M.
Sahimi
, J.
Peinke
, and M. Reza Rahimi
Tabar
, “Analysis of non-stationary data for heart-rate fluctuations in terms of drift and diffusion coefficients
,” J. Biol. Phys.
32
(2
), 117
–128
(2006
).21.
P.
Hagan
, A.
Lesniewski
, and D.
Woodward
, “Probability distribution in the SABR model of stochastic volatility
,” in Large Deviations and Asymptotic Methods in Finance
, Springer Proceedings in Mathematics and Statistics
Vol. 110
(Springer
, Cham
, 2015
), pp. 1
–35
.22.
P. S.
Hagan
, D.
Kumar
, A. S.
Lesniewski
, and D. E.
Woodward
, “Managing smile risk
,” in The Best of Wilmott
(John Wiley & Sons
, 2002
), p. 249
.23.
S. L.
Heston
, “A closed-form solution for options with stochastic volatility with applications to bond and currency options
,” Rev. Financ. Stud.
6
(2
), 327
–343
(1993
).24.
S.
Hochgerner
and T. S.
Ratiu
, “Geometry of non-holonomic diffusion
,” J. Eur. Math. Soc.
17
(2
), 273
–319
(2015
).25.
D. D.
Holm
, “Variational principles for stochastic fluid dynamics
,” in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences
(The Royal Society
, 2015
), Vol. 471
, p. 20140963
.26.
R.
Kozlov
, “Symmetries of systems of stochastic differential equations with diffusion matrices of full rank
,” J. Phys. A
43
(24
), 245201
(2010
).27.
J.-A.
Lázaro-Camí
and J.-P.
Ortega
, “Stochastic Hamiltonian dynamical systems
,” Rep. Math. Phys.
61
(1
), 65
–122
(2008
).28.
J.-A.
Lázaro-Camí
and J.-P.
Ortega
, “Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations
,” Stoch. Dyn.
9
(1
), 1
–46
(2009
).29.
J.-A.
Lázaro-Camí
and J.-P.
Ortega
, “Superposition rules and stochastic Lie-Scheffers systems
,” Ann. Inst. Henri Poincaré Probab. Stat.
45
(4
), 910
–931
(2009
).30.
P.
Lescot
and J.-C.
Zambrini
, “Probabilistic deformation of contact geometry, diffusion processes and their quadratures
,” in Seminar on Stochastic Analysis, Random Fields and Applications V
, Progress in Probability
Vol. 59
(Birkhäuser
, Basel
, 2008
), pp. 203
–226
.31.
X.-M.
Li
, “An averaging principle for a completely integrable stochastic Hamiltonian system
,” Nonlinearity
21
(4
), 803
–822
(2008
).32.
M.
Lorig
, S.
Pagliarani
, and A.
Pascucci
, “A family of density expansions for Lévy-type processes
,” Ann. Appl. Probab.
25
(1
), 235
–267
(2015
).33.
S. J. A.
Malham
and A.
Wiese
, “Stochastic Lie group integrators
,” SIAM J. Sci. Comput.
30
(2
), 597
–617
(2008
).34.
U.
Marini Bettolo Marconi
, A.
Puglisi
, L.
Rondoni
, and A.
Vulpiani
, “Fluctuation–dissipation: Response theory in statistical physics
,” Phys. Rep.
461
(4
), 111
–195
(2008
).35.
S. V.
Meleshko
, Y. N.
Grigoriev
, N. K.
Ibragimov
, and V. F.
Kovalev
, Symmetries of Integro-differential Equations: With Applications in Mechanics and Plasma Physics
(Springer Science & Business Media
, 2010
), Vol. 806
.36.
I.
Moerdijk
and J.
Mrčun
, Introduction to Foliations and Lie Groupoids
, Cambridge Studies in Advanced Mathematics
Vol. 91
(Cambridge University Press
, Cambridge
, 2003
).37.
P. J.
Olver
, Applications of Lie Groups to Differential Equations
, 2nd ed. Graduate Texts in Mathematics
Vol. 107
(Springer-Verlag
, New York
, 1993
).38.
E.
Platen
and N.
Bruti-Liberati
, Numerical Solution of Stochastic Differential Equations with Jumps in Finance
, Stochastic Modelling and Applied Probability
Vol. 64
(Springer-Verlag
, Berlin
, 2010
).39.
N.
Privault
and J.-C.
Zambrini
, “Stochastic deformation of integrable dynamical systems and random time symmetry
,” J. Math. Phys.
51
(8
), 082104
(2010
).40.
U.
Seifert
, “Stochastic thermodynamics, fluctuation theorems and molecular machines
,” Rep. Progr. Phys.
75
(12
), 126001
(2012
).41.
W. T.
Shaw
and M.
Schofield
, “A model of returns for the post-credit-crunch reality: Hybrid brownian motion with price feedback
,” Quant. Finance
15
(6
), 975
–998
(2015
).42.
H.
Stephani
, Differential Equations: Their Solution Using Symmetries
(Cambridge University Press
, Cambridge
, 1989
).43.
È. B.
Vinberg
, Lie groups and Lie algebras, III
, Encyclopaedia of Mathematical Sciences
Vol. 41
(Springer-Verlag
, Berlin
, 1994
).44.
G.
Wilk
and Z.
Włodarczyk
, “Interpretation of the nonextensivity parameter q in some applications of tsallis statistics and Lévy distributions
,” Phys. Rev. Lett.
84
(13
), 2770
(2000
).45.
N. T.
Zung
and N. T.
Thien
, “Reduction and integrability of stochastic dynamical systems
,” preprint arXiv:1410.5492 (2014)
.© 2016 Author(s).
2016
Author(s)
You do not currently have access to this content.
