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 -> ∞.