Download presentation

Published byJaven Peay Modified over 2 years ago

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

model of share prices

4
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form:

5
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form: Apply Ito’s lemma with

6
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form: Apply Ito’s lemma with

7
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form: Apply Ito’s lemma with

8
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form: Apply Ito’s lemma with

9
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form: Apply Ito’s lemma with

10
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form: Apply Ito’s lemma with F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t

11
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form: Apply Ito’s lemma with F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t => x(t) is log-normally distributed:

12
**“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention**

model of share prices Start with equation in differential form: Apply Ito’s lemma with F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t => x(t) is log-normally distributed:

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 + ½σ2

26
**geometric Brownian motion, Ito calculus with Stratonovich convention**

Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ2

27
**geometric Brownian motion, Ito calculus with Stratonovich convention**

Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ2

28
**geometric Brownian motion, Ito calculus with Stratonovich convention**

Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ2 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 + ½σ2 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 - ½σ2, 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 ½σ2).

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 ½σ2). 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 – gx3 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 – gx3 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 – gx3 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 G0: in time domain:

63
**some definitions and notation**

Write this as where multiply by G0: in time domain: in frequency domain:

64
**some definitions and notation**

Write this as where multiply by G0: 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: : x0 = + : x : G0 : -g

70
iterate diagrams: = + 3 h

71
iterate diagrams: = + 3 : ξ h

72
iterate diagrams: = + 3 : ξ h +9 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:**

o o: = + 3

78
**averaging the diagrams:**

o o: = + 3 o o +9

79
**averaging the diagrams:**

o o: = + 3 o o +9 o +18 + … o

80
correlation function o

81
correlation function o o

82
in algebra, o = + 3 o o +9 o +18 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 o o Σ = = 3 + 6 + … o

88
**1st-order approximation**

89
**1st-order approximation**

or, in time domain,

90
**1st-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
**1st-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
**1st-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
**1st-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 C0 and G0 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 C0 and G0 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 C0 and G0 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
**self-consistent solution**

99
**self-consistent solution**

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 -> ∞.

Similar presentations

OK

Autumn 2008 EEE8013 Revision lecture 1 Ordinary Differential Equations.

Autumn 2008 EEE8013 Revision lecture 1 Ordinary Differential Equations.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on hindu religion facts Ppt on limits and continuity examples Ppt on indian textile industries in india Download ppt on mind controlled robotic arms and legs Ppt on porter's five forces article Ppt on hiv aids basics Ppt on sources of water for kindergarten Ppt on public health nutrition Formal backgrounds for ppt on social media Ppt on potential energy for class 9