November  2018, 17(6): 2351-2378. doi: 10.3934/cpaa.2018112

The spectral expansion approach to index transforms and connections with the theory of diffusion processes

CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

* Corresponding author

Received  June 2017 Revised  February 2018 Published  June 2018

Many important index transforms can be constructed via the spectral theory of Sturm-Liouville differential operators. Using the spectral expansion method, we investigate the general connection between the index transforms and the associated parabolic partial differential equations.

We show that the notion of Yor integral, recently introduced by the second author, can be extended to the class of Sturm-Liouville integral transforms. We furthermore show that, by means of the Feynman-Kac theorem, index transforms can be used for studying Markovian diffusion processes. This gives rise to new applications of index transforms to problems in mathematical finance.

Citation: Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112
References:
[1]

A. N. Borodin and P. Salminen, Handbook of Brownian Motion: Facts and Formulae, 2nd edition, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8163-0.

[2]

D. L. Cohn, Measure Theory, 2nd edition, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6956-8.

[3]

M. Craddock, On an integral arising in mathematical finance, in Nonlinear Economic Dynamics and Financial Modelling (eds. R. Dieci, X. He and C. Hommes), Springer, (2014), 355–370. doi: 10.1007/978-3-319-07470-2_20.

[4]

N. Dunford and J. T. Schwartz, Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space, Interscience Publishers, New York and London, 1963.

[5]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms. Vol. I, McGraw-Hill, New York, 1954.

[6]

G. GasaneoS. Ovchinnikov and J. H. Macek, A Kontorovich–Lebedev representation for zero-range potential eigensolutions, J. Phys. A, Math. Gen., 34 (2001), 8941-8954.  doi: 10.1088/0305-4470/34/42/315.

[7]

H. Geman and M. Yor, Quelques relations entre processus de Bessel, options asiatiques et fonctions confluentes hypergéométriques, (French) [Some relations between Bessel processes, Asian options, and confluent hypergeometric functions], C. R. Acad. Sci. Paris Sér. I, 314 (1992), 471-474. 

[8]

A. Gulisashvili, Analytically Tractable Stochastic Stock Price Models, Springer, Berlin, 2012. doi: 10.1007/978-3-642-31214-4.

[9]

D. Heath and M. Schweizer, Martingales versus PDEs in finance: An equivalence result with examples, J. Appl. Probab., 37 (2000), 947-957.  doi: 10.1239/jap/1014843075.

[10]

J. Jung and T. G. Pedersen, Polarizability of supported metal nanoparticles: Mehler-Fock approach, J. Appl. Phys., 112 (2012), 064312. doi: 10.1063/1.4752427.

[11]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[12]

M. I. Kontorovich and N. N. Lebedev, On a method of solution of some problems in diffraction theory and other related problems (Russian), Journal of Experimental and Theoretical Physics, 8 (1938), 1192-1206. 

[13]

T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in Special functions: group theoretical aspects and applications (eds. R. A. Askey, T. H. Koornwinder and W. Schempp), D. Reidel Publishing Co., (1984), 1–85. doi: 10.1007/978-94-010-9787-1_1.

[14]

H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, in Écoled'été de probabilités de Saint-Flour, XII–1982 (ed. P. L. Hennequin), Springer, (1984), 143–303. doi: 10.1007/BFb0099433.

[15]

N. N. Lebedev, Special Functions and Their Applications, Revised English edition, Translated and edited by R. A. Silverman, Prentice-Hall, Englewood Cliffs, N. J., 1965.

[16]

V. Linetsky, Spectral expansions for Asian (average price) options, Oper. Res., 52 (2004), 856-867.  doi: 10.1287/opre.1040.0113.

[17]

V. Linetsky, Spectral methods in derivative pricing, in Handbook of Financial Engineering (eds. J. R. Birge and V. Linetsky), Elsevier, (2006), 223–299. doi: 10.1016/S0927-0507(07)15006-4.

[18]

H. P. Jr. McKean, Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc., 82 (1956), 519-548.  doi: 10.1090/S0002-9947-1956-0087012-3.

[19]

M. A. Naimark, Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space, Frederick Ungar Publishing Co., New York, 1968.

[20]

C. Nasim, The Mehler–Fock transform of general order and arbitrary index and its inversion, Internat. J. Math. Math. Sci., 7 (1984), 171-180.  doi: 10.1155/S016117128400017X.

[21]

Y. A. Neretin, Index hypergeometric transform and imitation of analysis of Berezin kernels on hyperbolic spaces, Sb. Math., 192 (2001), 402-432.  doi: 10.1070/SM2001v192n03ABEH000552.

[22]

NIST Digital Library of Mathematical Functions, Release 1.0.17 of 2017-12-22 (eds. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders), 2016. Available at: http://dlmf.nist.gov/.

[23]

A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and series. Vol. 1. Elementary Functions, Gordon & Breach Science Publishers, New York, 1986.

[24]

M. M. Rodrigues and S. Yakubovich, On a heat kernel for the index Whittaker transform, Carpathian J. Math., 29 (2013), 231-238. 

[25]

A. N. Shiryaev, Probability, 2nd edition, Springer, New York, 1996. doi: 10.1007/978-1-4757-2539-1.

[26]

I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972.

[27]

H. M. SrivastavaY. V. Vasil'ev and S. Yakubovich, A class of index transforms with Whittaker's function as the kernel, Quart. J. Math. Oxford, 49 (1998), 375-394.  doi: 10.1093/qmathj/49.3.375.

[28]

H. M. SrivastavaV. K. Tuan and S. Yakubovich, The Cherry transform and its relationship with a singular Sturm–Liouville problem, Quart. J. Math. Oxford, 51 (2000), 371-383.  doi: 10.1093/qjmath/51.3.371.

[29]

E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations. Part I, 2nd edition, Clarendon Press, Oxford, 1962.

[30]

V. K. Tuan and A. I. Zayed, Paley-Wiener-type theorems for a class of integral transforms, J. Math. Anal. Appl., 266 (2002), 200-226.  doi: 10.1006/jmaa.2001.7740.

[31]

J. Weidmann, Spectral Theory of Ordinary Differential Operators, Springer, Berlin, 1987. doi: 10.1007/BFb0077960.

[32]

J. Wimp, A class of integral transforms, Proc. Edinb. Math. Soc., 14 (1964), 33-40.  doi: 10.1017/S0013091500011202.

[33]

S. Yakubovich, Index Transforms, World Scientific, Singapore, 1996. doi: 10.1142/9789812831064.

[34]

S. Yakubovich and J. de Graaf, On Parseval equalities and boundedness properties for Kontorovich-Lebedev type operators, Novi Sad J. Math., 29 (1999), 185-205. 

[35]

S. Yakubovich, On the least values of Lp-norms for the Kontorovich-Lebedev transform and its convolution, J. Approx. Theory, 131 (2004), 231-242.  doi: 10.1016/j.jat.2004.10.007.

[36]

S. Yakubovich, The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedev transform, Commun. Pure Appl. Anal., 10 (2011), 745-760.  doi: 10.3934/cpaa.2011.10.745.

[37]

S. Yakubovich, On the Yor integral and a system of polynomials related to the Kontorovich–Lebedev transform, Integral Transforms Spec. Funct., 24 (2013), 672-683.  doi: 10.1080/10652469.2012.750312.

[38]

M. Yor, Loi de l'indice du lacet Brownien et distribution de Hartman-Watson (French), Z. Wahrscheinlichkeits., 53 (1980), 71-95.  doi: 10.1007/BF00531612.

[39]

M. Yor, On Some Exponential Functionals of Brownian Motion, Adv. in Appl. Probab., 24 (1992), 509-531.  doi: 10.1017/S0001867800024381.

show all references

References:
[1]

A. N. Borodin and P. Salminen, Handbook of Brownian Motion: Facts and Formulae, 2nd edition, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8163-0.

[2]

D. L. Cohn, Measure Theory, 2nd edition, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6956-8.

[3]

M. Craddock, On an integral arising in mathematical finance, in Nonlinear Economic Dynamics and Financial Modelling (eds. R. Dieci, X. He and C. Hommes), Springer, (2014), 355–370. doi: 10.1007/978-3-319-07470-2_20.

[4]

N. Dunford and J. T. Schwartz, Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space, Interscience Publishers, New York and London, 1963.

[5]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms. Vol. I, McGraw-Hill, New York, 1954.

[6]

G. GasaneoS. Ovchinnikov and J. H. Macek, A Kontorovich–Lebedev representation for zero-range potential eigensolutions, J. Phys. A, Math. Gen., 34 (2001), 8941-8954.  doi: 10.1088/0305-4470/34/42/315.

[7]

H. Geman and M. Yor, Quelques relations entre processus de Bessel, options asiatiques et fonctions confluentes hypergéométriques, (French) [Some relations between Bessel processes, Asian options, and confluent hypergeometric functions], C. R. Acad. Sci. Paris Sér. I, 314 (1992), 471-474. 

[8]

A. Gulisashvili, Analytically Tractable Stochastic Stock Price Models, Springer, Berlin, 2012. doi: 10.1007/978-3-642-31214-4.

[9]

D. Heath and M. Schweizer, Martingales versus PDEs in finance: An equivalence result with examples, J. Appl. Probab., 37 (2000), 947-957.  doi: 10.1239/jap/1014843075.

[10]

J. Jung and T. G. Pedersen, Polarizability of supported metal nanoparticles: Mehler-Fock approach, J. Appl. Phys., 112 (2012), 064312. doi: 10.1063/1.4752427.

[11]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[12]

M. I. Kontorovich and N. N. Lebedev, On a method of solution of some problems in diffraction theory and other related problems (Russian), Journal of Experimental and Theoretical Physics, 8 (1938), 1192-1206. 

[13]

T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in Special functions: group theoretical aspects and applications (eds. R. A. Askey, T. H. Koornwinder and W. Schempp), D. Reidel Publishing Co., (1984), 1–85. doi: 10.1007/978-94-010-9787-1_1.

[14]

H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, in Écoled'été de probabilités de Saint-Flour, XII–1982 (ed. P. L. Hennequin), Springer, (1984), 143–303. doi: 10.1007/BFb0099433.

[15]

N. N. Lebedev, Special Functions and Their Applications, Revised English edition, Translated and edited by R. A. Silverman, Prentice-Hall, Englewood Cliffs, N. J., 1965.

[16]

V. Linetsky, Spectral expansions for Asian (average price) options, Oper. Res., 52 (2004), 856-867.  doi: 10.1287/opre.1040.0113.

[17]

V. Linetsky, Spectral methods in derivative pricing, in Handbook of Financial Engineering (eds. J. R. Birge and V. Linetsky), Elsevier, (2006), 223–299. doi: 10.1016/S0927-0507(07)15006-4.

[18]

H. P. Jr. McKean, Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc., 82 (1956), 519-548.  doi: 10.1090/S0002-9947-1956-0087012-3.

[19]

M. A. Naimark, Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space, Frederick Ungar Publishing Co., New York, 1968.

[20]

C. Nasim, The Mehler–Fock transform of general order and arbitrary index and its inversion, Internat. J. Math. Math. Sci., 7 (1984), 171-180.  doi: 10.1155/S016117128400017X.

[21]

Y. A. Neretin, Index hypergeometric transform and imitation of analysis of Berezin kernels on hyperbolic spaces, Sb. Math., 192 (2001), 402-432.  doi: 10.1070/SM2001v192n03ABEH000552.

[22]

NIST Digital Library of Mathematical Functions, Release 1.0.17 of 2017-12-22 (eds. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders), 2016. Available at: http://dlmf.nist.gov/.

[23]

A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and series. Vol. 1. Elementary Functions, Gordon & Breach Science Publishers, New York, 1986.

[24]

M. M. Rodrigues and S. Yakubovich, On a heat kernel for the index Whittaker transform, Carpathian J. Math., 29 (2013), 231-238. 

[25]

A. N. Shiryaev, Probability, 2nd edition, Springer, New York, 1996. doi: 10.1007/978-1-4757-2539-1.

[26]

I. N. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York, 1972.

[27]

H. M. SrivastavaY. V. Vasil'ev and S. Yakubovich, A class of index transforms with Whittaker's function as the kernel, Quart. J. Math. Oxford, 49 (1998), 375-394.  doi: 10.1093/qmathj/49.3.375.

[28]

H. M. SrivastavaV. K. Tuan and S. Yakubovich, The Cherry transform and its relationship with a singular Sturm–Liouville problem, Quart. J. Math. Oxford, 51 (2000), 371-383.  doi: 10.1093/qjmath/51.3.371.

[29]

E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations. Part I, 2nd edition, Clarendon Press, Oxford, 1962.

[30]

V. K. Tuan and A. I. Zayed, Paley-Wiener-type theorems for a class of integral transforms, J. Math. Anal. Appl., 266 (2002), 200-226.  doi: 10.1006/jmaa.2001.7740.

[31]

J. Weidmann, Spectral Theory of Ordinary Differential Operators, Springer, Berlin, 1987. doi: 10.1007/BFb0077960.

[32]

J. Wimp, A class of integral transforms, Proc. Edinb. Math. Soc., 14 (1964), 33-40.  doi: 10.1017/S0013091500011202.

[33]

S. Yakubovich, Index Transforms, World Scientific, Singapore, 1996. doi: 10.1142/9789812831064.

[34]

S. Yakubovich and J. de Graaf, On Parseval equalities and boundedness properties for Kontorovich-Lebedev type operators, Novi Sad J. Math., 29 (1999), 185-205. 

[35]

S. Yakubovich, On the least values of Lp-norms for the Kontorovich-Lebedev transform and its convolution, J. Approx. Theory, 131 (2004), 231-242.  doi: 10.1016/j.jat.2004.10.007.

[36]

S. Yakubovich, The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedev transform, Commun. Pure Appl. Anal., 10 (2011), 745-760.  doi: 10.3934/cpaa.2011.10.745.

[37]

S. Yakubovich, On the Yor integral and a system of polynomials related to the Kontorovich–Lebedev transform, Integral Transforms Spec. Funct., 24 (2013), 672-683.  doi: 10.1080/10652469.2012.750312.

[38]

M. Yor, Loi de l'indice du lacet Brownien et distribution de Hartman-Watson (French), Z. Wahrscheinlichkeits., 53 (1980), 71-95.  doi: 10.1007/BF00531612.

[39]

M. Yor, On Some Exponential Functionals of Brownian Motion, Adv. in Appl. Probab., 24 (1992), 509-531.  doi: 10.1017/S0001867800024381.

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