2. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking Salvatore Miccichè with Fabrizio Lillo, Rosario N. Mantegna http://lagash.dft.unipa.it Observatory of Complex Systems Università degli Studi di Palermo, Dipartimento di Fisica e Tecnologie Relative Fokker-Planck equations, algebraic correlations, long range correlations, and related questions INFORMAL WORKSHOP on Fokker-Planck equations, algebraic correlations, long range correlations, and related questions ENS - Lyon, 29-30 March 2005

3. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking Volatility in Financial Markets Persistencies Time Series : Persistencies Volatility Clustering Volatility Clustering Lognormal Empirical pdf : Lognormal - for intermediate values of volatility Power law Power law - for large values   4.8 of volatility (   4.8) Long-Range Empirical Autocorrelation : Long-Range correlated process Power-law   0.3 ? non-exponential asymptotic decay. Power-law (   0.3) ? single exponential Empirical Leverage : single exponential fitting curve Motivations

4. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking Motivations Stochastic Volatility Models discrete time discrete time ARCH-GARCH, (Engle, Granger, …) ARCH-GARCH, (Engle, Granger, …) continuous time continuous time based on Langevin stochastic differential equations (with linear mean- reverting drift coefficient (Hull-White, Heston, Stein-Stein., …). based on Langevin stochastic differential equations (with linear mean- reverting drift coefficient (Hull-White, Heston, Stein-Stein., …). based on multifractality (Muzy et al., … ) based on multifractality (Muzy et al., … ) based on fractional Brownian motion (Sircar et al., … ) based on fractional Brownian motion (Sircar et al., … ) … …

5. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking 1) Characterize the stationary M arkovian stochastic processes wherethere is not one single characteristic time-scales 1) Characterize the stationary M arkovian stochastic processes where there is not one single characteristic time-scales. 2) Rather, we are interested in stationary Markovian stochastic processes withmany characteristic time- scales 2) Rather, we are interested in stationary Markovian stochastic processes with many characteristic time- scales. 3) Moreover, we are interested in stationary Markovian stochastic processes with an infinite ( how infinite? ) number of time-scales. 4) Explicit form of the autocorrelation function. Aim of our research Description in terms of Langevin stochastic differential equations. NO memory terms, NO fractional derivatives, …

6. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking PART I - stochastic processes PART III - stochastic volatility Outline Characterize stationary Markovian stochastic processes with multiple/infinite time-scales. Use these results to build up a Markovian stochastic volatility model which incorporates the long-range memory of volatility. PART II - ergodicity breaking Ergodicity breaking & autocorrelation function Ergodicity breaking & moments/FPTD Links with hamiltonian models

7. Stationary Markovian Stochastic Processes with Multiple Time-Scales Part I Stationary Markovian Stochastic Processes with Multiple Time-Scales References References [1] A. Schenzle, H. Brand, Phys. Rev. A, 20(4),, (1979) [1] A. Schenzle, H. Brand, Phys. Rev. A, 20(4), 1628, (1979) [2] M. Suzuki, K. Kaneko, F. Sasagawa, Prog. Theor. Phys., 65(3),, (1981) [2] M. Suzuki, K. Kaneko, F. Sasagawa, Prog. Theor. Phys., 65(3), 828, (1981) [3] J. Farago, Europhys. Lett., 52(4),, (2000) [3] J. Farago, Europhys. Lett., 52(4), 379, (2000) [4] F. Lillo, S. Miccichè, R. N. Mantegna, cond-mat, 0203442, (2002)

8. Stationary Markovian Stochastic Processes with Multiple Time-Scales A) TOOL: We study the Autocorrelation function of a stochastic process described by a non-linear Langevin equation. METHODOLOGY : relationship between the Fokker-Planck equation and the Schrödinger equation with a potential V S. B) A simple example of the methodology used: the Ornstein-Uhlenbeck process. C) How the spectral properties of the (quantum) potential V S affect the structure of the autocorrelation function: processes with multiple time scales processes with infinite time scales Outline PATH : from exponential to non exponential autocorrelation. END-POINT : power-law? i.e. Long-Range Correlated processes ?

9. Stationary Markovian Stochastic Processes with Multiple Time-Scales Nonlinear Langevin Equation Fokker-Planck Equation AutoCorrelation function  (  ) / AutoCovariance function R(  ) Nonlinear Langevin and Fokker-Planck Equations Linear drift h=-  x => Exponential Autocorrelation exp(-   ) Ito / Stratonovich prescription

10. Stationary Markovian Stochastic Processes with Multiple Time-Scales Chapman-Kolmogorov Equation: Markovian Property In the context of continuous-time stochastic processes, this is the definition of a Markovian stochastic process we will consider.

11. Stationary Markovian Stochastic Processes with Multiple Time-Scales Relationship Fokker-Planck / Schrödinger Hereafter we will consider the case of Additive Noise: g(x)=1 Stationary solution of Fokker-Planck eqn. Schrödinger equation quantum potential Stationarity is ensured if there exists a normalizable eigenfunction  0 (x) corresponding to the eigenvalue E 0 =0.

12. Stationary Markovian Stochastic Processes with Multiple Time-Scales Relationship Fokker-Planck / Schrödinger The validity of this methodology is based upon the assumption that i.e. the eigenfunctions {  0,  n,  E } are a COMPLETE set of eigenfunctions in the SPACE of INTEGRABLE functions L 1. Analogously, the eigenfunctions {  0,  n,  E } must be a COMPLETE set of eigenfunctions in the SPACE of SQUARE-INTEGRABLE functions L 2.

13. Stationary Markovian Stochastic Processes with Multiple Time-Scales Relationship Fokker-Planck / Schrödinger Completeness in L 2 is equivalent to Is (2) enough to ensure completeness in L 1 ?? Does completeness in L 2 imply completeness in L 1 ?? (1) (2) which implies:

14. Stationary Markovian Stochastic Processes with Multiple Time-Scales AutoCovariance Function Relationship Fokker-Planck / Schrödinger 2-point probability density odd eigenfunctions

15. Stationary Markovian Stochastic Processes with Multiple Time-Scales AutoCorrelation Function Discrete Spectrum only Therefore, in order to have not-exponential AutoCovariance function we need to introduce a continuum part in the spectrum. discrete continuum how infinite?

16. Stationary Markovian Stochastic Processes with Multiple Time-Scales Ornstein-Uhlenbeck process Linear drift Exponential autocorrelation expected In this case we have one single time-scale. one single time-scale In this case we have one single time-scale. harmonic oscillatorcompletely solvable With this choice of variables, the Schrödinger equation is the same as the one associated to the harmonic oscillator potential, which is completely solvable: ground state discrete eigenfunctions H n Hermite polynomials

17. Stationary Markovian Stochastic Processes with Multiple Time-Scales Ornstein-Uhlenbeck process AutoCorrelation We can compute the AutoCorrelation Function AutoCovariance due to properties of Hermite polynomials Variance AutoCorrelation Time-Scale T=1/ 

18. Stationary Markovian Stochastic Processes with Multiple Time-Scales The Square Well numerable set of time-scales In this case we have a numerable set of time-scales. ground state antisymmetric discrete eigenfunctions n odd Infinite Square Well completely solvable This is the Infinite Square Well problem. It is completely solvable: T n =1/E n

19. Stationary Markovian Stochastic Processes with Multiple Time-Scales The delta-like potential continuum spectrum In this case we have a continuum spectrum: ground state antisymmetric continuum eigenfunctions n odd  -like Well completely solvable This is the  -like Well problem. It is completely solvable: GAP GAP

20. Stationary Markovian Stochastic Processes with Multiple Time-Scales The delta-like potential c E analytically R(  ) analytically All functions c E can be obtained analytically. Also the further integration to get R(  ) can be performed analytically. theoretical resultsnumerical simulation The picture shows a comparison between the theoretical results and a numerical simulation of the stochastic process, i.e. a numerical integration of the Langevin equation. The numerical simulation is performed starting from the only knowledge of the drift coefficient h(x). Therefore, it is completely independent from the theoretical procedure used to obtain the autocovariance function. “Completeness”

21. Stationary Markovian Stochastic Processes with Multiple Time-Scales A quantum potential with gap continuum spectrum In this case we have a continuum spectrum: completely solvable. Bessel functions J (.)Y (.) This potential is still completely solvable. Eigenfunctions are expressed in terms of Bessel functions J (.) and Y (.). GAP GAP

22. Stationary Markovian Stochastic Processes with Multiple Time-Scales A quantum potential with gap c E analytically R(  )numerically c E  analytically. R(  )  numerically. cut-off effect smaller V 2  0 The cut-off effect given by the the exponential term gets smaller along with V 2  0. R(  )  exp{-V 2 2  } “Completeness “

23. continuum spectrumnon exponential So far: continuum spectrum  non exponential autocorrelation Stationary Markovian Stochastic Processes with Multiple Time-Scales A quantum potential without gap T inverse of the energy gap. T is related to the inverse of the energy gap. Single/Multiple (OU) Time-Scales  Infinite (shift) Time-Scales. What about reducing the energy gap to zero ?  NO GAP V1V1 -V 0

24. Eigenfunctions x>L Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA This condition is fulfilled if: It is worth noting that this is the only way to fulfill the following condition: Normalization conditions

25. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA We can prove that, for INVERSE SQUARED POTENTIALS : c E analyticallyR(  ) numerically asymptotically All functions c E can be obtained analytically. Further integration to get R(  ) can only be performed numerically. One can only show that asymptotically : Anomalous Diffusion 3<  <5  <1

26. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA

27. continuum spectrum + zero gap +V S  x -2  long-range processes not integrable continuum spectrum + zero gap +V S  x -2  long-range processes i.e. not integrable autocorrelation function Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA  >1  <1 Short-range correlated Long-range correlated Anomalous diffusion ?? Not Integrable Integrable

28. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA Is everything OK ? Almost! 1) Yet we do not have a proof of completeness !! 2) Simulations of the Autocorrelation are “ DIFFICULT ” x f(x)

29. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA Is there agreement between simulations and numerical integrations of the eigenfunctions ??

30. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA Is there agreement between simulations and numerical integrations of the eigenfunctions ??

31. Stationary Markovian Stochastic Processes with Multiple Time-Scales Another quantum potential without gap - I Consider the process: Do we get power-law decaying correlations? NO

32. Stationary Markovian Stochastic Processes with Multiple Time-Scales Another quantum potential without gap - II Consider the process: Red curves are CHIMERA. Black curves are a=0.5

33. Stationary Markovian Stochastic Processes with Multiple Time-Scales Another quantum potential without gap - II Limit a=0 ?

34. Stationary Markovian Stochastic Processes with Multiple Time-Scales Hamiltonian models: mean field theory We (TD, FB, SR, …) are interested in deriving a Fokker-Planck equation which describes the stochastic process of 1 particle in interaction with a bath of N-1 particles in equilibrium [4]. Consider the Hamiltonian [4] that describes a set of N particles governed by. that describes a set of N particles governed by long-range interactions. References References [4] F. Bouchet, PRE, 70, (2004) [4] F. Bouchet, PRE, 70, 036113, (2004) [5] F. Bouchet, T. Dauxois cond-mat, 0407703 (2004) - submitted to PRL [5] F. Bouchet, T. Dauxois cond-mat, 0407703, (2004) - submitted to PRL

35. Stationary Markovian Stochastic Processes with Multiple Time-Scales Hamiltonian models: mean field theory In [4] In [5] Eq. 10 This FP leads to “ … unsual algebraic correlation laws and to anomalous diffusion … ”. where f 0 (p) is some “given” equilibrium distribution of the N particles. QUESTIONS are: i) what are the (“physics”) differences between the two FPs? ii) why you consider  =3, i.e. processes for which variance is not well defined? iii) how do you write the cross-correlations between particles ? Eq. 11 Eq. 12 Eq. 9

36. Conclusions Stationary Markovian Stochastic Processes with Multiple Time-Scales Discrete SpectrumNumerable set of Time-Scales Short-Range Discrete Spectrum Numerable set of Time-Scales. Integrable AutoCorrelation..Short-Range Correlated Processes. Continuum Spectrum with GapInfinite set of Time-Scales Short-Range Continuum Spectrum with Gap Infinite set of Time-Scales. Integrable AutoCorrelation..Short-Range Correlated Processes. Continuum Spectrum without GapInfinite set of Time-Scales V S = V 1 /x  log(x) a  =2, a=0 Long-Range Continuum Spectrum without Gap Infinite set of Time-Scales. Not-Integrable AutoCorrelation.. V S = V 1 /x  log(x) a  =2, a=0 Long-Range Correlated Processes.                         “Completeness ??” Quantum Potential V S AutoCorrelation “Peculiarity!! ”

37. Ergodicity breaking Part II Ergodicity Breaking References References [1] E. Lutz, PRL, 93,, (2004) [1] E. Lutz, PRL, 93, 190602, (2004) [2] S. Miccichè, F. Lillo, R. N. Mantegna, in preparation [3] J.-P. Bouchaud, J. Phys. I France, 2,, (1992) [3] J.-P. Bouchaud, J. Phys. I France, 2, 1705, (1992) [4] G. Bel, E. Barkai, cond-mat/0502154

38. Ergodicity breaking A process is said to be Ergodic in the “ Mean Square sense ” iff (?) Definition of Ergodicity We will always consider: A n (t)=x(t) n

39. Ergodicity breaking Ergodicity for CHIMERA Ergodicity for CHIMERA holds for any  > 2n+1 n=1  >3 n=2  >5 n=3  >7 n=4  >9 Moments for CHIMERA are well defined for any  > n+1 n=1  >2 n=2  >3 n=3  >4 n=4  >5  =4n=1,2  =5n=1,2,3  =6n=1,2,3,4  =7n=1,2,3,4,5  =4n=1  =5n=1  =6n=1,2  =7n=1,2

40. Ergodicity breaking Moments of FPTD g 2 (t) for CHIMERA are well defined for any  > 2m-1  =4m=1,2  =5m=1,2  =6m=1,2,3  =7m=1,2,3 Value of nERGOMOMENTSFPTD n=1yesyesyes n=2  >5 yes yes n=4  >9  >5  >7... Ergodicity for CHIMERA m=1  >1 m=2  >3 m=3  >5 m=4  >7

41. Ergodicity breaking n=2 Let us consider n=2 : Ergodicity & Autocorrelation function   ErgoMomentsFPTD  =4  =0.5 long-range noup to 3 rd up to 2 nd  =4.8  =0.9 long-range noup to 3 rd up to 2 nd  =5.1  =1.1 short-range yesup to 4 th up to 3 rd  =6  =1.5 short-range yesup to 4 th up to 3 rd TIME-AVERAGE ENSEMBLE-AVERAGE x i (0) are distributed according to the stationary pdf.

42. Ergodicity breaking Ergodicity & Autocorrelation function

43. Ergodicity breaking Ergodicity & Autocorrelation function

44. Ergodicity breaking Multiplicative CHIMERA pdf Auto- correlation drift Coordinate transformation (asymptotically)

45. Ergodicity breaking Multiplicative CHIMERA

46. Ergodicity breaking Multiplicative CHIMERA Ergodicity for Multiplicative CHIMERA holds for any   > 2n+1 n=1   >3 n=2   >5 n=3   >7 n=4   >9   =4n=1   =5n=1   =6n=1,2   =7n=1,2 Short-Range & non-ergodic Long-Range & Ergodic

47. Ergodicity breaking Multiplicative CHIMERA   =0.5 (long-range)   =5.5 (ergodic)  =-0.5

48. Ergodicity breaking Multiplicative CHIMERA   =1.5 (short-range)   =3.5 (NON-ergodic)  =+0.5

49. Tentative Conclusions Ergodicity breaking Ergodicity  Short-Range Correlation Non-Ergodicity  Long-Range Correlation Is true only true for Stationary Markovian processes with additive noise 1) For Stationary Markovian processes with multiplicative noise one might have: 2) Non-Ergodicity & Short-Range Correlation Ergodicity  Long-Range Correlation MOMENTS of pdf are diverging ? or MOMENTS of FPTD are diverging - Sojourn times [3]? 3) QUESTION intimate 3) QUESTION is: what is the intimate source of Ergodicity Breaking ? In CHIMERA -like processes these features are both present. What about other (non markovian ? ) processes [4]?

50. A Two-Region Stochastic Volatility Model Part III Econo Physics from -PHYSICS to ECONO- (?)

51. A Two-Region Stochastic Volatility Model Issue 1) mean reverting OU-like driving processes Issue 2) long-range fBmMultifractal... Macroscopic / Phenomenological Model

52. A Two-Region Stochastic Volatility Model The set of investigated stocks Standard&Poor’s 100 We consider the 100 most capitalized stocks traded at NYSE. 95 of them enter the Standard&Poor’s 100 (SP100) stock index. intraday INTC  11900MKG  121 synchronizehomogenize We consider high-frequency (intraday) data. Transactions do not occur at the same time for all stocks. INTC  11900 transactions per day MKG  121 transactions per day We have to synchronize/homogenize the data: 100 1011 For each stock i=1,..., 100 For 1011 trading days 121950 12 intervals of 1950 seconds each TAQTAQ1995-1998 Trades And Quotes (TAQ) database maintained by NYSE (1995-1998)

53. Empirical Facts A Two-Region Stochastic Volatility Model lognormal power-law

54. Empirical Facts A Two-Region Stochastic Volatility Model  (  )   -    0.3

55. Empirical Facts A Two-Region Stochastic Volatility Model  t    1.7  t    1.7 () -() -  <1

56. Empirical Facts A Two-Region Stochastic Volatility Model  (q)  0.17+0.74 q  (q)  0.85 q 0.93

57. Empirical Facts A Two-Region Stochastic Volatility Model Leverage Anti-Correlation between returns and future volatility Bouchaud et al. PRL 87, 228701, (2001)

58. Models of Stochastic Volatility A Two-Region Stochastic Volatility Model Drift coefficient h(  ) Diffusion coefficient g(  ) What are the appropriate i) Drift coefficient h(  ) ii) Diffusion coefficient g(  ) able to reproduce the previous empirical stylized facts ? We are looking for models of stochastic volatility: dS/S =  dt +  d z d  = h(  ) dt + g(  ) d z .

59. A Two-Region Stochastic Volatility Model The proposed model: a Two-Region Model g(  )=1 Additive noise , L control the power-law V1 controls the log-normality d  = h(  ) dt + g(  ) d z 

60. A Two-Region Stochastic Volatility Model The proposed model: a Two-Region Model Lognormal Power-law Fokker-Planck equation - stationary solution

61. A Two-Region Stochastic Volatility Model The proposed model: a Two-Region Model

62. A Two-Region Stochastic Volatility Model dinamical properties of volatility are not well reproduced Nevertheless, the dinamical properties of volatility are not well reproduced by this simple model. The proposed model: a Two-Region Model  emp  4.8  emp  1.7 Volatility shows an empirical pdf that has power-law tails with exponent  emp  4.8 and an empirical mean squared displacement that is asymptotically power-law with exponent  emp  1.7, i.e.  emp  0.3  emp  0.3 in the autocorrelation function. power- law decaying autocorrelation One can prove that this simple Two-Region model admits a power- law decaying autocorrelation function with exponent:  4.8 would imply  0.9 i.e.  = 2-  1.1

63. A Two-Region Stochastic Volatility Model The proposed model: a Two-Region Model gapT -1 Version of the model with a gap T -1 in the energy spectrum

64. A Two-Region Stochastic Volatility Model The proposed model: MultiplicativeTwo-Region Model Multiplicative-Noise version of the model

65. A Two-Region Stochastic Volatility Model The proposed model: a Two-Region Model Leverage ( very preliminary ) very preliminary very preliminary d  = h(  ) dt + g(  ) d z 

66. A Two-Region Stochastic Volatility Model Conclusions References References S. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Physica A, 314, 756-761, (2002)S. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Physica A, 314, 756-761, (2002) S. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Proceedings of: "The Second Nikkey Econophysics Research Workshop and Symposium", 12-14 November 2002, Tokio, Japan Springer Verlag, Tokio, edited by H. TakayasuS. Miccichè, G. Bonanno, F. Lillo, R. N. Mantegna, Proceedings of: "The Second Nikkey Econophysics Research Workshop and Symposium", 12-14 November 2002, Tokio, Japan Springer Verlag, Tokio, edited by H. Takayasu empirical pdf This simple model reproduces the empirical pdf quite well.  g(  )    () The empirical Autocorrelation function is power-law. The  and  exponents can be made independent from each other by generalizing the model as to considering a diffusion coefficient like g(  )    (multiplicative noise). leverage effect The leverage effect... structure of smileoption pricing The structure of smile, option pricing,...

67. A Two-Region Stochastic Volatility Model Additional Slides

68. Stationary Markovian Stochastic Processes with Multiple Time-Scales Probability Current We have checked that: x=  L : is continuous in x=  L : Continuity Equation

69. Stationary Markovian Stochastic Processes with Multiple Time-Scales Schenzle-Brand Phys. Rev. A, 20(4), 1628, (1979) Completeness In particular Presentation of the methodology along the lines of Risken. Difference between stationary and transient W 2 Explicitly find autocorrelation functions that are not-exponential. In general Mentioned, as in Risken. Schenzle-Brand Phys. Lett. A, 69(5), 313, (1979) Presentation of the methodology along the lines of Risken. Mention that: “… [In the multiplicative case] the Fokker-Planck equation is not of the Sturm- Liouville type contrary to the case of Fokker-Planck equations describing additive noise[9]. … [9] H. Risken, in: Progress in optics, Vol. 8, ed. E. Wolf (North Holland, Amsterdam, 1970)” In general

70. Stationary Markovian Stochastic Processes with Multiple Time-Scales Suzuki-Kaneko-Sasagawa Prog. Theor. Phys., 65(3), 828, (1981) In general Against S-B Divergent Modes I In order to have correct solutions we need to fulfill the following 3 conditions: 1)1) |  E (x)| is squared-integrable 2)2) |  E (x)| is integrable 3)3) j(x,t)=0 at infinity (natural boundary conditions) only1)fulfilled In S-B only condition 1) is fulfilled. If we apply 2) or 3) then the spectrum is different: there exists a maximal eigenvalue!!!! 2) and 3) should be equivalent. L 1  L 2 2) e non 1)divergent modes Since L 1  L 2, allora posso avere soddisfatta 2) e non 1). These are so-called divergent modes. a n Se esistono modi divergenti, this methodology can not be applied. In particular, one can not compute the a n coefficients. The existence of divergent modes strictly depends upon the chosen initial condition.

71. Stationary Markovian Stochastic Processes with Multiple Time-Scales Suzuki-Kaneko-Sasagawa Prog. Theor. Phys., 65(3), 828, (1981) Relation with our work L 1 and not L 2. In general one could have initial conditions that are L 1 and not L 2. a n In such case I can not compute the a n coefficients, and therefore the whole procedure fails. Divergent Modes II It can also occur if one looks for particular variables, i.e. Let us suppose that the quantity in round brackets diverges. Then, even though aE vanishes, the mode E can give a finite contribution. Example of L 1 and not L 2

72. Stationary Markovian Stochastic Processes with Multiple Time-Scales Graham-Schenzle Phys. Rev. A, 25(3), 1731, (1982) “ … The weaker boundary conditions he proposes (L 1 integrability of all solutions) is not sufficient to impose a Hilbert structure on the eigenvalue problem associated with the Fokker-Planck equation, and to formulate a completeness relation for its eigenfunctions. …” Verify that: Suzuki et al. Refuse Suzuki et al. criticism on the grounds that : Alternative I :

73. Stationary Markovian Stochastic Processes with Multiple Time-Scales Bacward Kolmogorov As a general rule, given Stationary solution of backward Kolmogorov eqn. then K(x,t) is a solution of the backward Kolmogorov equation. The functions  E (x) are eigenfunctions of the backward Kolmogorov equation. They can also be written as: