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Lecture 7: Langevin equations II: geometric Brownian motion and perturbation theory Outline: Geometric Brownian motion, Stratonovich and Ito models Ito.

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Presentation on theme: "Lecture 7: Langevin equations II: geometric Brownian motion and perturbation theory Outline: Geometric Brownian motion, Stratonovich and Ito models Ito."— Presentation transcript:

1 Lecture 7: Langevin equations II: geometric Brownian motion and perturbation theory Outline: Geometric Brownian motion, Stratonovich and Ito models Ito calculus method small noise correlation time method (Stratonovich only) solution using Fokker-Planck equation Perturbation theory for nonlinear Langevin equations diagrammatic expansion self-consistent approximations

2 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention

3 model of share prices

4 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: model of share prices

5 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

6 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

7 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

8 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

9 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention Start with equation in differential form: Apply Ito’s lemma with model of share prices

10 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention  F = log x(t) is normally distributed with mean ( r – ½σ 2 )t and variance σ 2 t Start with equation in differential form: Apply Ito’s lemma with model of share prices

11 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention  F = log x(t) is normally distributed with mean ( r – ½σ 2 )t and variance σ 2 t => x(t) is log-normally distributed: Start with equation in differential form: Apply Ito’s lemma with model of share prices

12 “Geometric Brownian motion” (GBM) with Ito calculus and Ito convention  F = log x(t) is normally distributed with mean ( r – ½σ 2 )t and variance σ 2 t => x(t) is log-normally distributed: Start with equation in differential form: Apply Ito’s lemma with model of share prices

13 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

14 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

15 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

16 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

17 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma:

18 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma: in particular,

19 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma: in particular,

20 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma: in particular,

21 moments of x(t) can get moments of x(t) from this or directly from the SDE and Ito’s lemma: in particular,

22 geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift

23 geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation

24 geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation

25 geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2

26 geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2

27 geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2

28 geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2 moments:

29 geometric Brownian motion, Ito calculus with Stratonovich convention Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ 2 moments:

30 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model.

31 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time:

32 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time:

33 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) :

34 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables:

35 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables:

36 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables:

37 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables:

38 GBM with Stratonovich, finite- τ noise This was based on the “midpoint prescription”. I claimed that this prescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model. give y a small correlation time: solve for x(t) : use identity for Gaussian variables: as we got before

39 GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written

40 GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written

41 GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written change variables:

42 GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written change variables: y(t) is Gaussian with

43 GBM, Stratonovich, with Fokker-Planck recall the FP equation with Stratonovich convention can be written change variables: y(t) is Gaussian with as obtained from working with differentials and using Ito’s lemma

44 GBM, Ito convention, using Fokker-Planck (Finally), the FP equation for the Ito convention

45 GBM, Ito convention, using Fokker-Planck (Finally), the FP equation for the Ito convention can be written

46 GBM, Ito convention, using Fokker-Planck (Finally), the FP equation for the Ito convention can be written Here:

47 GBM, Ito convention, using Fokker-Planck (Finally), the FP equation for the Ito convention can be written Here: But this is just the same equation as in the Stratonovich case, excpt for a reduced drift r - ½σ 2, in agreement with what we found using differentials and the Ito lemma.

48 Summary: Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other.

49 Summary: Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other. All ways of treating the problem with the Stratonovich convention (differentials + midpoint correction + Ito’s lemma, finite-τ noise, FP) agree with each other.

50 Summary: Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other. All ways of treating the problem with the Stratonovich convention (differentials + midpoint correction + Ito’s lemma, finite-τ noise, FP) agree with each other. Ito and Stratonovich problems are different (Stratonovich has a drift rate larger by ½σ 2 ).

51 Summary: Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other. All ways of treating the problem with the Stratonovich convention (differentials + midpoint correction + Ito’s lemma, finite-τ noise, FP) agree with each other. Ito and Stratonovich problems are different (Stratonovich has a drift rate larger by ½σ 2 ). I have shown this here for GBM, but it is true in general (except for constant G(x), in which case Stratonovich and Ito are equivalent).

52 Perturbation theory for nonlinear Langevin equations (now back to additive noise)

53 Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F :

54 Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F :

55 Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F : Here I will concentrate on the example F(x) = -γx – gx 3 overdamped motion in a quartic potential, double-well potential for γ < 0 :

56 Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F : Here I will concentrate on the example F(x) = -γx – gx 3 overdamped motion in a quartic potential, double-well potential for γ < 0 :

57 Perturbation theory for nonlinear Langevin equations (now back to additive noise) Consider equations with a steady state, nonlinear F : Here I will concentrate on the example F(x) = -γx – gx 3 overdamped motion in a quartic potential, double-well potential for γ < 0 : add an external driving force:

58 some definitions and notation Write this as

59 some definitions and notation Write this as where

60 some definitions and notation Write this as where

61 some definitions and notation Write this as where

62 some definitions and notation Write this as where multiply by G 0 : in time domain:

63 some definitions and notation Write this as where multiply by G 0 : in time domain: in frequency domain:

64 some definitions and notation Write this as where multiply by G 0 : in time domain: in frequency domain: notation:

65 iteration of equation of motion Define

66 iteration of equation of motion Define equation of motion:

67 iteration of equation of motion Define equation of motion:

68 iteration of equation of motion Define equation of motion:

69 iteration of equation of motion Define equation of motion: diagrammatic representation: key: =+ : x 0 : x : G 0 : -g

70 iterate diagrams: =+ 3 h

71 iterate diagrams: =+ 3 h : ξ

72 iterate diagrams: = h : ξ h + …

73 averaging over noise: Recall: To average products of arbitrary numbers of factors of noise, pair in all ways (Wick’s theorem)

74 averaging over noise: Recall: To average products of arbitrary numbers of factors of noise, pair in all ways (Wick’s theorem) etc.

75 averaging over noise: Recall: To average products of arbitrary numbers of factors of noise, pair in all ways (Wick’s theorem) etc. Define the Green’s function

76 averaging over noise: Recall: To average products of arbitrary numbers of factors of noise, pair in all ways (Wick’s theorem) etc. Define the Green’s function

77 averaging the diagrams: =+ 3 o o:o:

78 averaging the diagrams: = o o o o:o:

79 averaging the diagrams: = o o o o o + … o:o:

80 correlation function o

81 o o

82 in algebra, = o o o o o + …

83 “self-energy” (“mass operator”) =+ + + …

84 “self-energy” (“mass operator”) =+ + + … =+

85 “self-energy” (“mass operator”) =+ + + … =+ Dyson equation

86 “self-energy” (“mass operator”) =+ + + … =+ Dyson equation or

87 “self-energy” (“mass operator”) =+ + + … =+ Dyson equation or = 3 o + 6 o o + … Σ =

88 1 st -order approximation

89 or, in time domain,

90 1 st -order approximation or, in time domain, A simple, low-order approximation for Σ sums an infinite number of terms in the series for G.

91 1 st -order approximation or, in time domain, A simple, low-order approximation for Σ sums an infinite number of terms in the series for G. lowest-order approximation for Σ :

92 1 st -order approximation or, in time domain, A simple, low-order approximation for Σ sums an infinite number of terms in the series for G. lowest-order approximation for Σ :

93 1 st -order approximation or, in time domain, A simple, low-order approximation for Σ sums an infinite number of terms in the series for G. lowest-order approximation for Σ :  increase in damping constant:

94 Hartree approximation Replace C 0 and G 0 in the expression for Σ by C and G. This sums up all self-energy insertions of this form on the internal lines in the self-energy diagrams.

95 Hartree approximation Replace C 0 and G 0 in the expression for Σ by C and G. This sums up all self-energy insertions of this form on the internal lines in the self-energy diagrams. lowest-order approximation (Hartree):

96 Hartree approximation Replace C 0 and G 0 in the expression for Σ by C and G. This sums up all self-energy insertions of this form on the internal lines in the self-energy diagrams. lowest-order approximation (Hartree): Σ = … o o o o o o o o o o

97 self-consistent solution

98

99  self-consistent equation

100 self-consistent solution  self-consistent equation solution:

101 self-consistent solution  self-consistent equation solution: This solution is approximate. But it is exact if x is a vector with n components, with in the limit n -> ∞.


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