# Outline: Geometric Brownian motion, Stratonovich and Ito models

## Presentation on theme: "Outline: Geometric Brownian motion, Stratonovich and Ito models"— Presentation transcript:

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

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

“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention
model of share prices

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

“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

“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

“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

“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

“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

“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

“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:

“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:

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

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

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

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

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

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

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

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

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

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

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

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

geometric Brownian motion, Ito calculus with Stratonovich convention
Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ2

geometric Brownian motion, Ito calculus with Stratonovich convention
Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ2

geometric Brownian motion, Ito calculus with Stratonovich convention
Recall extra drift in our current notation same as for Ito convention except r -> r + ½σ2

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:

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:

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.

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:

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:

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):

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:

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:

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:

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:

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

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

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

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

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

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

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

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

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

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.

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

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.

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).

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).

Perturbation theory for nonlinear Langevin equations

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

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

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 :

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 :

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:

some definitions and notation
Write this as

some definitions and notation
Write this as where

some definitions and notation
Write this as where

some definitions and notation
Write this as where

some definitions and notation
Write this as where multiply by G0: in time domain:

some definitions and notation
Write this as where multiply by G0: in time domain: in frequency domain:

some definitions and notation
Write this as where multiply by G0: in time domain: in frequency domain: notation:

iteration of equation of motion
Define

iteration of equation of motion
Define equation of motion:

iteration of equation of motion
Define equation of motion:

iteration of equation of motion
Define equation of motion:

iteration of equation of motion
Define equation of motion: diagrammatic representation: key: : x0 = + : x : G0 : -g

iterate diagrams: = + 3 h

iterate diagrams: = + 3 : ξ h

iterate diagrams: = + 3 : ξ h +9 h + …

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

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

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

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

averaging the diagrams:
o o: = + 3

averaging the diagrams:
o o: = + 3 o o +9

averaging the diagrams:
o o: = + 3 o o +9 o +18 + … o

correlation function o

correlation function o o

in algebra, o = + 3 o o +9 o +18 o + …

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

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

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

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

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

1st-order approximation

1st-order approximation
or, in time domain,

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.

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:

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:

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:

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.

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):

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

self-consistent solution

self-consistent solution

self-consistent solution
self-consistent equation

self-consistent solution
self-consistent equation solution:

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 -> ∞.