A bunch of random ramblings on the Langevin equation, with some details on the Ito-Stratanovitch distinction

1. Luc_Faucheux_2020
Stochastic Calculus – ITO – IV
The example of the Langevin equation
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2. Luc_Faucheux_2020
Couple of notes on those slides
¨ Those are part IV of the the slides on stochastic calculus
¨ Since they are mostly devoted to the Langevin equation, they are somewhat “stand-alone”
and I have tried to keep them independent from the other as much as possible
¨ Some of the results are demonstrated in the other slide decks, but are used in this deck
sometimes without redoing the derivation
2

3. Luc_Faucheux_2020
A useful example
The Langevin equation (1908)
3

4. Luc_Faucheux_2020
A useful example – Langevin
¨ In 1908 Langevin introduced the Langevin equation in order to describe the velocity of a
particle in a viscous fluid, subject to random collisions from the surrounding fluid (thermal
noise)
¨ 𝑑𝑉 𝑡 = 𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + 𝑏 𝑉 𝑡 , 𝑡 . 𝑑𝑊
¨ 𝑎 𝑉 𝑡 , 𝑡 = −𝑘𝑉
¨ 𝑏 𝑉 𝑡 , 𝑡 = 𝜎
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ In 1930 Ornstein and Uhlenbeck put the equation on a firmer basis and explored a slightly
more general class of SDE known as OU processes
¨ OU processes have very nice properties (stationary, Gaussian, Markovian)
¨ In particular, unlike the Geometric Brownian motion, the Langevin process has stable
dynamics of moments
4

5. Luc_Faucheux_2020
Paul Langevin
5

6. Luc_Faucheux_2020
A useful example – Langevin - II
¨ 𝑑𝑉 𝑡 = 𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + 𝑏 𝑉 𝑡 , 𝑡 . 𝑑𝑊
¨ 𝑎 𝑉 𝑡 , 𝑡 = −𝑘𝑉
¨ 𝑏 𝑉 𝑡 , 𝑡 = 𝜎
¨ Note that here we looking at the velocity 𝑉 𝑡 of the particle as the stochastic variable 𝑋 𝑡
¨ Hopefully we are mentally flexible enough to do the jump
¨ The issue once again is how do you interpret: 𝑏 𝑉 𝑡 , 𝑡 . 𝑑𝑊
6

7. Luc_Faucheux_2020
A useful example – Langevin - III
¨ With the ITO convention:
¨ 𝑑𝑉 𝑡 = 𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + 𝑏 𝑉 𝑡 , 𝑡 . 𝑑𝑊 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ 𝑑𝑉 𝑡 = 𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + 𝑏 𝑉 𝑡 , 𝑡 . [ . 𝑑𝑊 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. [ . 𝑑𝑊
¨ So the PDF for the particle velocity follows the FORWARD ITO PDE:
¨
!"($,&|(!,&!)
!&
= −
!
!$
𝑎 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 𝑉*, 𝑡* −
!
!$
[
+
,
. [𝑏(𝑣, 𝑡), . 𝑝(𝑣, 𝑡|𝑉*, 𝑡*)]
¨
!"($,&|(!,&!)
!&
= −
!
!$
(−𝑘𝑣). 𝑝 𝑣, 𝑡 𝑉*, 𝑡* −
!
!$
[
+
,
. [𝜎, . 𝑝(𝑣, 𝑡|𝑉*, 𝑡*)]
7

8. Luc_Faucheux_2020
A useful example – Langevin - IV
¨ With the STRATO convention:
¨ 𝑑𝑉 𝑡 = 𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + 𝑏 𝑉 𝑡 , 𝑡 . 𝑑𝑊 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ 𝑑𝑉 𝑡 = 6𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + 7𝑏 𝑉 𝑡 , 𝑡 . ∘ . 𝑑𝑊 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. ∘ . 𝑑𝑊
¨ So the PDF for the particle velocity follows the FORWARD STRATO PDE:
¨
!"($,&|(!,&!)
!&
= −
!
!$
9
:
{6𝑎 𝑣, 𝑡 +
+
,
. 7𝑏 𝑣, 𝑡 .
!
!$
7𝑏 𝑣, 𝑡 }. 𝑝 𝑣, 𝑡 𝑉*, 𝑡* −
!
!$
[
+
,
. [7𝑏(𝑣, 𝑡), . 𝑝(𝑣, 𝑡|𝑉*, 𝑡*)]
¨ And since:
!
!(
7𝑏 𝑣, 𝑡 = 0
¨
!"($,&|(!,&!)
!&
= −
!
!$
(−𝑘𝑣). 𝑝 𝑣, 𝑡 𝑉*, 𝑡* −
!
!$
[
+
,
. [𝜎, . 𝑝(𝑣, 𝑡|𝑉*, 𝑡*)]
8

9. Luc_Faucheux_2020
A useful example – Langevin - V
¨ So BOTH ITO and STRATO interpretation of the Langevin equation will return the SAME PDE
(forward Fokker Planck) for the PDF:
¨
!"($,&|(!,&!)
!&
= −
!
!$
(−𝑘𝑣). 𝑝 𝑣, 𝑡 𝑉*, 𝑡* −
!
!(
[
+
,
. [𝜎, . 𝑝(𝑣, 𝑡|𝑉*, 𝑡*)]
¨ Again no real surprise there, this is to be expected since
!
!$
7𝑏 𝑡, 𝑉 𝑡 = 0
¨ So :
¨ 𝑎 𝑣, 𝑡 = 6𝑎 𝑣, 𝑡 +
+
,
. 7𝑏 𝑣, 𝑡 .
!
!$
7𝑏 𝑣, 𝑡 = 6𝑎 𝑣, 𝑡
¨ 𝑏 𝑣, 𝑡 = 7𝑏 𝑣, 𝑡
¨ So both ITO and STRATO are equivalent for homogeneous diffusion coefficients
9

10. Luc_Faucheux_2020
A useful example – Langevin - VI
¨ Let us know look at a function of the velocity, in both ITO and STRATO
¨ We will look at the kinetic energy 𝜉 =
-("
,
¨ Because this function is NOT a linear function of the velocity, we would expect to observe a
divergence between the ITO and the STRATO treatments
¨ Let us show that either one you choose, you will still get the same PDE and same PDF, as
long as you are consistent within your choice (if you choose ITO, you have to use ITO lemma)
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11. Luc_Faucheux_2020
A useful example – Langevin - VII
¨ Let’s use ITO interpretation and ITO calculus and ITO lemma
¨ 𝑑𝑉 𝑡 = 𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + 𝑏 𝑉 𝑡 , 𝑡 . [ . 𝑑𝑊(𝑡) = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. ([). 𝑑𝑊
¨ Note: from time to time we need to remember ourselves that this is really:
¨ We really are writing an SIE, because random processes are NOT differentiable
¨ 𝑉 𝑡. − 𝑉 𝑡* = ∫&/&*
&/&.
𝑑𝑉 𝑡 = ∫&/&*
&/&.
𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + ∫&/&*
&/&.
𝑏 𝑉 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡)
¨ 𝑉 𝑡. − 𝑉 𝑡* = ∫&/&*
&/&.
𝑑𝑉 𝑡 = ∫&/&*
&/&.
(−𝑘𝑉(𝑡)). 𝑑𝑡 + ∫&/&*
&/&.
𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝜉 =
-("
,
¨ ITO lemma is given by:
¨ 𝑓 𝑉 𝑡. − 𝑓 𝑉 𝑡* = ∫&/&*
&/&. !0
!(
. ([). 𝑑𝑉(𝑡) +
+
,
∫&/&*
&/&. !"1
!(" . 𝑑𝑉 𝑡
,
+ ∫&/&*
&/&. !0
!&
. 𝑑𝑡
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12. Luc_Faucheux_2020
A useful example – Langevin - VIII
¨ 𝑑𝑓 =
!0
!(
. [ . 𝑑𝑉 +
+
,
.
!"1
!(" . 𝑑𝑉,
¨ Let’s apply this to the energy:
¨ 𝜉 =
-("
,
¨
!2
!(
= 𝑚𝑉
¨
!"2
!(" = 𝑚
¨
!2
!&
= 0
¨ 𝑑𝜉 = 𝑚𝑉. [ . 𝑑𝑉 +
+
,
. 𝑚. 𝜎, 𝑑𝑡 = 𝑚𝑉. [ . (−𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. ([). 𝑑𝑊) +
+
,
. 𝑚. 𝜎, 𝑑𝑡
¨ 𝑑𝜉 = −𝑘𝑚𝑉,. 𝑑𝑡 + 𝑚𝑉 𝜎. ([). 𝑑𝑊 +
+
,
. 𝑚. 𝜎, 𝑑𝑡
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13. Luc_Faucheux_2020
A useful example – Langevin - IX
¨ 𝑑𝜉 = −𝑘𝑚𝑉,. 𝑑𝑡 + 𝑚𝑉 𝜎. ([). 𝑑𝑊 +
+
,
. 𝑚. 𝜎, 𝑑𝑡
¨ Expressing this equation in terms of 𝜉 =
-("
,
¨ 𝑑𝜉 = −2𝑘𝜉,. 𝑑𝑡 +
+
,
. 𝑚. 𝜎, 𝑑𝑡 + 𝜎 2𝑚𝜉. [ . 𝑑𝑊
¨ Of the form:
¨ 𝑑𝜉 = 𝑎 𝜉, 𝑡 . 𝑑𝑡 + 𝑏 𝜉, 𝑡 . [ . 𝑑𝑊
¨ With
¨ 𝑎 𝜉, 𝑡 = −2𝑘𝜉, +
+
,
. 𝑚. 𝜎,
¨ 𝑏 𝜉, 𝑡 = 𝜎 2𝑚𝜉
¨ 𝑏 𝜉, 𝑡 , = 2𝑚𝜉𝜎,
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14. Luc_Faucheux_2020
A useful example – Langevin - X
¨ So the PDF for the energy should follow the ITO FORWARD PDE:
¨
!"(2,&|3!,&!)
!&
= −
!
!2
𝑎 𝜉 𝑡 , 𝑡 . 𝑝 𝜉, 𝑡 𝜉*, 𝑡* −
!
!2
[
+
,
. [𝑏(𝜉 𝑡 , 𝑡), . 𝑝(𝜉, 𝑡|𝜉*, 𝑡*)]
¨
!"
!&
= −
!
!2
(−2𝑘𝜉, +
+
,
𝑚𝜎,). 𝑝 −
!
!2
[𝑚𝜉𝜎,. 𝑝]
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15. Luc_Faucheux_2020
A useful example – Langevin - XI
¨ Let’s use STRATO interpretation and STRATO calculus and STRATO lemma
¨ 𝑑𝑉 𝑡 = 𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + 𝑏 𝑉 𝑡 , 𝑡 . ∘ . 𝑑𝑊(𝑡) = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. (∘). 𝑑𝑊(𝑡)
¨ Note: from time to time we need to remember ourselves that this is really:
¨ We really are writing an SIE, because random processes are NOT differentiable
¨ 𝑉 𝑡. − 𝑉 𝑡* = ∫&/&*
&/&.
𝑑𝑉 𝑡 = ∫&/&*
&/&.
𝑎 𝑉 𝑡 , 𝑡 . 𝑑𝑡 + ∫&/&*
&/&.
𝑏 𝑉 𝑡 , 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ 𝑉 𝑡. − 𝑉 𝑡* = ∫&/&*
&/&.
𝑑𝑉 𝑡 = ∫&/&*
&/&.
(−𝑘𝑉(𝑡)). 𝑑𝑡 + ∫&/&*
&/&.
𝜎. (∘). 𝑑𝑊(𝑡)
¨ 𝜉 =
-("
,
¨ STRATO lemma is given by:
¨ 𝑓 𝑉 𝑡. − 𝑓 𝑉 𝑡* = ∫&/&*
&/&. !0
!(
. (∘). 𝑑𝑉(𝑡) + ∫&/&*
&/&. !0
!&
. 𝑑𝑡
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16. Luc_Faucheux_2020
A useful example – Langevin - XII
¨ 𝑑𝑓 =
!0
!(
. ∘ . 𝑑𝑉
¨ Let’s apply this to the energy:
¨ 𝜉 =
-("
,
¨
!2
!(
= 𝑚𝑉
¨
!2
!&
= 0
¨ 𝑑𝜉 = 𝑚𝑉. ∘ . 𝑑𝑉 = 𝑚𝑉. ∘ . (−𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. (∘). 𝑑𝑊)
¨ 𝑑𝜉 = −𝑘𝑚𝑉,. 𝑑𝑡 + 𝑚𝑉𝜎. (∘). 𝑑𝑊
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17. Luc_Faucheux_2020
A useful example – Langevin - XIII
¨ 𝑑𝜉 = −𝑘𝑚𝑉,. 𝑑𝑡 + 𝑚𝑉𝜎. (∘). 𝑑𝑊
¨ Expressing this equation in terms of 𝜉 =
-("
,
¨ 𝑑𝜉 = −2𝑘𝜉,. 𝑑𝑡 + 𝜎 2𝑚𝜉. ∘ . 𝑑𝑊
¨ Remember this is really writing:
¨ 𝜉 𝑡. − 𝜉 𝑡* = ∫&/&*
&/&.
𝑑𝜉 𝑡 = ∫&/&*
&/&.
(−2𝑘𝜉,). 𝑑𝑡 + ∫&/&*
&/&.
𝜎 2𝑚𝜉. (∘). 𝑑𝑊(𝑡)
¨ This has the form: 𝑑𝜉 𝑡 = 6𝑎 𝜉 𝑡 , 𝑡 . 𝑑𝑡 + 7𝑏 𝜉 𝑡 , 𝑡 . (∘). 𝑑𝑊
¨ With : 6𝑎 𝜉, 𝑡 = (−2𝑘𝜉,)
¨ 7𝑏 𝜉, 𝑡 = 𝜎 2𝑚𝜉
¨ And:
!
!2
7𝑏 𝜉, 𝑡 = 𝜎 2𝑚𝜉.
+
,
. 𝜉4+
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18. Luc_Faucheux_2020
A useful example – Langevin - XIV
¨ This implies that the PDF for the energy follows the FORWARD STRATO Kolmogorov PDE
¨
!"(2,&|2!,&!)
!&
= −
!
!2
9
:
6𝑎 𝜉, 𝑡 . 𝑝 𝑥, 𝑡 𝜉*, 𝑡* +
+
,
. 7𝑏 𝜉, 𝑡 .
!
!2
7𝑏 𝜉, 𝑡 . 𝑝 𝑥, 𝑡 𝜉*, 𝑡* −
!
!2
[
+
,
. [7𝑏(𝜉, 𝑡), . 𝑝(𝜉, 𝑡|𝜉*, 𝑡*)]
¨
!"
!&
= −
!
!2
6𝑎𝑝 +
+
,
. 7𝑏.
!
!2
7𝑏. 𝑝 −
!
!2
(
+
,
7𝑏, . 𝑝)
¨
!"
!&
= −
!
!2
−2𝑘𝜉, 𝑝 +
+
,
. 𝜎 2𝑚𝜉.
!
!2
(𝜎 2𝑚𝜉). 𝑝 −
!
!2
(
+
,
𝜎, 2𝑚𝜉
,
. 𝑝)
¨
!"
!&
= −
!
!2
−2𝑘𝜉, 𝑝 +
+
,
. 𝜎, 2𝑚 𝜉.
!
!2
( 𝜉). 𝑝 −
!
!2
(𝑚𝜎, 𝜉𝑝)
¨
!"
!&
= −
!
!2
−2𝑘𝜉, 𝑝 +
+
,
. 𝜎, 𝑚. 𝑝 −
!
!2
(𝑚𝜎, 𝜉𝑝)
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19. Luc_Faucheux_2020
A useful example – Langevin - XV
¨ Now by construction the STRATO PDF and the ITO PDF should be the same
¨ ITO:
!"
!&
= −
!
!2
(−2𝑘𝜉, +
+
,
𝑚𝜎,). 𝑝 −
!
!2
[𝑚𝜉𝜎,. 𝑝
¨ STRATO:
!"
!&
= −
!
!2
−2𝑘𝜉, 𝑝 +
+
,
. 𝜎, 𝑚. 𝑝 −
!
!2
(𝑚𝜎, 𝜉𝑝)
¨ There is no surprise there, we are getting the same result. If we start with a well defined
equation (using one convention and sticking with it), we are free to apply non-linear
transformations to the variable.
¨ We are just proving that the results are consistent.
¨ There is an advantage in using STRATO in that the formal rules of calculus are formally
preserved, so we should be able to map the PDF for the velocity into the PDF for the energy.
¨ This is a neat trick that we can apply from time to time (Van Kampen page 194), put yourself
in Stratonovitch calculus, where you have convinced yourself that you can apply (formally)
the usual rules of calculus, and derive the PDE
19

20. Luc_Faucheux_2020
Auto correlation function
20

21. Luc_Faucheux_2020
Langevin Auto Correlation function
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ Let’s define 7𝑉 𝑡 = exp 𝑘𝑡 . 𝑉(𝑡)
¨ 𝑑 7𝑉 =
!5(
!(
. [ . 𝑑𝑉 +
+
,
.
!"5(
!(" . 𝑑𝑉, +
!5(
!&
. [ . 𝑑𝑡
¨
!5(
!&
= 𝑘𝑡. 𝑉 𝑡
¨
!5(
!(
= exp(𝑘𝑡)
¨
!"5(
!(" = 0
¨ 𝑑 7𝑉 = exp 𝑘𝑡 . ([). 𝑑𝑉 + 𝑘𝑡. 𝑉 𝑡 . ([). 𝑑𝑡
¨ 𝑑 7𝑉 = exp 𝑘𝑡 . ([). (−𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊) + 𝑘𝑡. 𝑉 𝑡 . ([). 𝑑𝑡 = exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊
21

22. Luc_Faucheux_2020
Langevin Auto Correlation function - II
¨ 𝑑 7𝑉 = exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊
¨ Which we should really write as an SIE anyways:
¨ 7𝑉 𝑡. − 7𝑉 𝑡* = ∫&/&*
&/&.
𝑑 7𝑉 𝑡 = ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊(𝑡)
¨ exp 𝑘𝑡. . 𝑉 𝑡. − exp 𝑘𝑡* . 𝑉 𝑡* = ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑉 𝑡. = exp −𝑘𝑡. . {exp 𝑘𝑡* . 𝑉 𝑡* + ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
¨ In stochastic processes, and especially regarding physical systems, it is quite useful to define
and use the autocorrelation function: If it includes the variance it is also referred to as the
auto-covariance function. If normalized by the variance, it is the auto-correlation.
¨ 𝐶 𝑡., 𝑡* = < 𝑉 𝑡. − < 𝑉 𝑡. > . 𝑉 𝑡* − < 𝑉 𝑡* > >
¨ 𝐶 𝑡., 𝑡* = 𝔼{[𝑉 𝑡. − 𝔼{𝑉 𝑡. }]. { 𝑉 𝑡* − 𝔼 𝑉 𝑡* }
22

23. Luc_Faucheux_2020
Langevin Auto Correlation function -III
¨ When 𝑡. = 𝑡* this is the expectation of the second moment:
¨ 𝐶 𝑡., 𝑡. = < 𝑉 𝑡. − < 𝑉 𝑡. > , >
¨ From the deck on PDE:
¨
!"(6,&)
!&
= −
!
!6
[𝑀+ 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 −
!
!6
[𝑀, 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 ]]
¨ Looking at a random process 𝑋(𝑡) such that 𝑝 𝑥, 𝑡 = 𝛿(𝑥 − 𝑋 𝑡 )
¨ < ∆𝑋 > = 𝐸 ∆𝑋 =< 𝑥 >&7∆& −< 𝑥 >&= 𝐹+ 𝑋 𝑡 , 𝑡 . ∆𝑡 (drift term)
¨ < ∆𝑋,> = 𝐸 ∆𝑋, =< (𝑥−< 𝑥 >&7∆&),>&7∆& = 𝐹, 𝑋 𝑡 , 𝑡 . ∆𝑡 (diffusion term)
¨ We showed that 𝐹+ 𝑋 𝑡 , 𝑡 = 𝑀+ 𝑋 𝑡 , 𝑡 and 𝐹, 𝑋 𝑡 , 𝑡 = 2. 𝑀, 𝑋 𝑡 , 𝑡
¨
!"($,&|$9,&9)
!&
= −
!
!$
[𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡|𝑣𝑜, 𝑡𝑜 −
!
!$
[𝑀, 𝑣, 𝑡 . 𝑝 𝑥, 𝑡|𝑣𝑜, 𝑡𝑜 ]]
23

24. Luc_Faucheux_2020
Langevin Auto Correlation function -IV
¨ We know that the velocity follows the following PDE:
¨
!"($,&|$9,&9)
!&
= −
!
!$
(−𝑘𝑣). 𝑝 𝑣, 𝑡 𝑣𝑜, 𝑡𝑜 −
!
!$
[
+
,
. [𝜎, . 𝑝(𝑣, 𝑡|𝑣𝑜, 𝑡𝑜)]
¨ So:
¨ 𝐶 𝑡 + ∆𝑡, 𝑡 + ∆𝑡 = < 𝑉 𝑡 + ∆𝑡 − < 𝑉 𝑡 + ∆𝑡 > , > = 2. 𝑀, 𝑣 𝑡 , 𝑡 . ∆𝑡 = 𝜎,. ∆𝑡
¨ Since:
¨ 𝑉 𝑡. = exp −𝑘𝑡. . {exp 𝑘𝑡* . 𝑉 𝑡* + ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
¨ 𝐶 𝑡., 𝑡* = < 𝑉 𝑡. − < 𝑉 𝑡. > . 𝑉 𝑡* − < 𝑉 𝑡* > >
¨ 𝐶 𝑡., 𝑡* = 𝔼{[𝑉 𝑡. − 𝔼{𝑉 𝑡. }]. { 𝑉 𝑡* − 𝔼 𝑉 𝑡* }
¨ 𝔼 𝑉 𝑡. = 𝔼{exp −𝑘𝑡. . {exp 𝑘𝑡* . 𝑉 𝑡* + ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }}
¨ 𝔼 𝑉 𝑡. = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 𝑉 𝑡* since the ITO integral is a martingale
24

25. Luc_Faucheux_2020
Langevin Auto Correlation function -V
¨ 𝐶 𝑡., 𝑡* = 𝔼{[𝑉 𝑡. − 𝔼{𝑉 𝑡. }]. { 𝑉 𝑡* − 𝔼 𝑉 𝑡* }
¨ 𝐶 𝑡., 𝑡* = 𝔼{[𝑉 𝑡. − 𝔼{𝑉 𝑡* }]. { 𝑉 𝑡* − 𝔼 𝑉 𝑡* }
¨ 𝐶 𝑡., 𝑡* = 𝔼 𝑉 𝑡. . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡* − 𝔼 𝑉 𝑡. . 𝔼 𝑉 𝑡* +
𝔼 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡*
¨ 𝐶 𝑡., 𝑡* = 𝔼 𝑉 𝑡. . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡. +
𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡*
¨ 𝐶 𝑡., 𝑡* = 𝔼 𝑉 𝑡. . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡. +
𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡*
¨ 𝐶 𝑡., 𝑡* = 𝔼 𝑉 𝑡. . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡.
¨ 𝑉 𝑡. = exp −𝑘𝑡. . {exp 𝑘𝑡* . 𝑉 𝑡* + ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
¨ 𝔼 𝑉 𝑡. . 𝑉 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 𝑉 𝑡* . 𝑉 𝑡*
25

26. Luc_Faucheux_2020
Langevin Auto Correlation function -VI
¨ 𝔼 𝑉 𝑡. . 𝑉 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 𝑉 𝑡* . 𝑉 𝑡*
¨ 𝔼 𝑉 𝑡. = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 𝑉 𝑡*
¨ 𝐶 𝑡., 𝑡* = 𝔼 𝑉 𝑡. . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡.
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . {𝔼 𝑉 𝑡* . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡* }
¨ 𝔼 𝑉 𝑡. = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 𝑉 𝑡*
¨ So as 𝑡. → ∞, 𝔼 𝑉 𝑡. → 0
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . {𝔼 𝑉 𝑡* . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡* }
¨ 𝐶 𝑡, 𝑡 = {𝔼 𝑉 𝑡 . 𝑉 𝑡 − 𝔼 𝑉 𝑡 . 𝔼 𝑉 𝑡 }
26

27. Luc_Faucheux_2020
Langevin Auto Correlation function -VII
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . {𝔼 𝑉 𝑡* . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡* }
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . [𝔼 𝑉 𝑡*
, − 𝔼 𝑉 𝑡*
,]
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 𝑉 𝑡*
, − 𝔼 𝑉 𝑡*
,
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 (𝑉 𝑡* − 𝔼 𝑉 𝑡* ),
¨ 𝐶 𝑡, 𝑡 = {𝔼 𝑉 𝑡 . 𝑉 𝑡 − 𝔼 𝑉 𝑡 . 𝔼 𝑉 𝑡 }
¨ 𝑉 𝑡. = exp −𝑘𝑡. . {exp 𝑘𝑡* . 𝑉 𝑡* + ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝐶 𝑡*, 𝑡*
¨ In particular to help the notation, with 𝑡* = 0 and 𝑡. = 𝑡
¨ 𝐶 𝑡, 0 = exp −𝑘𝑡 . 𝐶 0,0
¨ < 𝑉 𝑡 . 𝑉 0 > = < 𝑉(0), >. exp(−𝑘𝑡)
27

28. Luc_Faucheux_2020
Langevin Auto Correlation function -VIII
¨ Note that the Langevin process is a Markov process (no memory).
¨ HOWEVER, that does not mean zero correlation
¨ Markov really means that
¨ 𝑃( 𝑉 𝑡 + ℎ ≤ 𝑣 𝑉 𝑠 , 𝑠 ≤ 𝑡 = 𝑃( 𝑉 𝑡 + ℎ ≤ 𝑣 𝑉 𝑡
¨ For comparison, a Brownian process (Wiener) is such that:
¨ 𝔼 𝑊 𝑡. . 𝑊 𝑡* = min 𝑡., 𝑡*
¨ 𝔼 𝑊 𝑡 . 𝑊 𝑡 = 𝑡
¨ In some weird ways you can say that the Brownian motion is more strongly correlated than
the Langevin process.
28

29. Luc_Faucheux_2020
Langevin Auto Correlation function - X
¨ Another useful formula on the auto-correlation by expressly keeping the stochastic forcing
¨ 𝑉 𝑡. = exp −𝑘𝑡. . {exp 𝑘𝑡* . 𝑉 𝑡* + ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
¨ In particular:
¨ 𝑉 𝑡 = exp −𝑘𝑡 . {𝑉 0 + ∫:/;
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠 }
¨ 𝑉 𝑡 = exp −𝑘𝑡 . {exp −𝑘∞ . 𝑉 −∞ + ∫:/4<
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠 }
¨ 𝑉 𝑡 = exp −𝑘𝑡 . ∫:/4<
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠
¨ 𝑉 0 = ∫:/4<
:/;
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠
¨ 𝑉 𝑡 . 𝑉 0 = exp −𝑘𝑡 . ∫:/4<
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠 . ∫:=/4<
:=/;
exp 𝑘𝑠′ . 𝜎. ([). 𝑑𝑊 𝑠′
29

30. Luc_Faucheux_2020
Langevin Auto Correlation function - XI
¨ 𝑉 𝑡 . 𝑉 0 = exp −𝑘𝑡 . ∫:/4<
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠 . ∫:=/4<
:=/;
exp 𝑘𝑠′ . 𝜎. ([). 𝑑𝑊 𝑠′
¨ 𝑉 𝑡 . 𝑉 0 = exp −𝑘𝑡 . ∫:=/4<
:=/;
∫:/4<
:/&
exp 𝑘𝑠 . exp 𝑘𝑠′ . 𝜎,. ([). 𝑑𝑊 𝑠 . ([). 𝑑𝑊 𝑠′
¨ < 𝑉 𝑡 . 𝑉 0 >= exp −𝑘𝑡 . ∫:=/4<
:=/;
∫:/4<
:/&
exp 𝑘𝑠 . exp 𝑘𝑠= . 𝜎,. ([). < 𝑑𝑊 𝑠 . ([). 𝑑𝑊 𝑠= >
¨ < 𝑑𝑊 𝑠 . ([). 𝑑𝑊 𝑠= > = 𝑑𝑠. 𝛿(𝑠 − 𝑠=)
¨ < 𝑉 𝑡 . 𝑉 0 >= exp −𝑘𝑡 . ∫:/4<
:/;
exp 2𝑘𝑠 . 𝜎,. 𝑑𝑠
¨ < 𝑉 𝑡 . 𝑉 0 >= exp −𝑘𝑡 . ∫:/4<
:/;
exp 2𝑘𝑠 . 𝜎,. 𝑑𝑠
¨ < 𝑉 𝑡 . 𝑉 0 >= exp −𝑘𝑡 .
>"
,?
¨ And we had before: < 𝑉 𝑡 . 𝑉 0 > = < 𝑉(0), >. exp(−𝑘𝑡)
¨ So < 𝑉(0), > =
>"
,?
30

31. Luc_Faucheux_2020
Langevin Auto Correlation function - X
¨ Let’s also now look at < 𝑉 𝑡 . 𝑉 𝑡 >
¨ 𝑉 𝑡 = exp −𝑘𝑡 . {𝑉 0 + ∫:/;
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠 }
¨ 𝑉 𝑡 . 𝑉 𝑡 = exp −2𝑘𝑡 . {𝑉 0 + ∫:/;
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠 }
,
¨ 𝑉 𝑡 . 𝑉 𝑡 = exp −2𝑘𝑡 . {𝑉 0 , +
2. 𝑉 0 . ∫:/;
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠 + ∫:/;
:/&
𝑒?:. 𝜎. ([). 𝑑𝑊 𝑠 . ∫:/;
:/&
𝑒?:. 𝜎. ([). 𝑑𝑊 𝑠 }
¨ 𝑉 𝑡 . 𝑉 𝑡 = exp −2𝑘𝑡 . {𝑉 0 , +
2. 𝑉 0 . ∫:/;
:/&
𝑒?:. 𝜎. ([). 𝑑𝑊 𝑠 + ∫:/;
:/&
∫:=/;
:=/&
𝑒?:#
. 𝑒?:. 𝜎,. ([). 𝑑𝑊 𝑠 . ([). 𝑑𝑊 𝑠′ }
¨ We then take the average and use the fact that the ITO integral is a martingale
¨ < ∫:/;
:/&
exp 𝑘𝑠 . 𝜎. ([). 𝑑𝑊 𝑠 > = 0
31

32. Luc_Faucheux_2020
Langevin Auto Correlation function - XI
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . {< 𝑉 0 , >
+ ∫:/;
:/&
∫:=/;
:=/&
𝑒?:#
. 𝑒?:. 𝜎,. ([). < 𝑑𝑊 𝑠 . ([). 𝑑𝑊 𝑠= >}
¨ And we also use the fact that for a Wiener (Brownian process) the increments are
independents:
¨ < 𝑑𝑊 𝑠 . ( . 𝑑𝑊 𝑠= > = 𝛿 𝑠 − 𝑠= . 𝑑𝑠
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . {< 𝑉 0 , > + ∫:/;
:/&
𝑒?:. 𝑒?:. 𝜎,. 𝑑𝑠}
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . {< 𝑉 0 , > + ∫:/;
:/&
𝑒,?:. 𝜎,. 𝑑𝑠}
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . {< 𝑉 0 , > +𝜎,.
@"$%
,? :/;
:/&
}
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . {< 𝑉 0 , > +𝜎,.
@"$&4+
,?
}
32

33. Luc_Faucheux_2020
Langevin Auto Correlation function - XII
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . {< 𝑉 0 , > +𝜎,.
@"$&4+
,?
}
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . {< 𝑉 0 , >} + 𝜎,.
+4@'"$&
,?
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . < 𝑉 0 , > +
>"
,?
. [1 − 𝑒4,?&]
¨ Or alternatively
¨ < 𝑉 𝑡 . 𝑉 𝑡 > = exp −2𝑘𝑡 . [< 𝑉 0 , > −
>"
,?
] +
>"
,?
¨ < 𝑉 ∞ . 𝑉 ∞ > =
>"
,?
¨ < 𝑉 𝑡 , > = < 𝑉 ∞ , > + exp −2𝑘𝑡 . [< 𝑉 0 , > −< 𝑉 ∞ , >]
33

34. Luc_Faucheux_2020
From the Langevin equation
to the particle diffusion
34

35. Luc_Faucheux_2020
From Langevin (velocity) to particle diffusion (position)
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . {𝔼 𝑉 𝑡* . 𝑉 𝑡* − 𝔼 𝑉 𝑡* . 𝔼 𝑉 𝑡* }
¨ 𝐶 𝑡., 𝑡* = exp −𝑘. 𝑡. . exp 𝑘. 𝑡* . 𝐶 𝑡*, 𝑡*
¨ Let us now tie this with the diffusion of a particle:
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ The position of the particle is then given by :
¨ 𝑋 𝑡 − 𝑋 𝑡* = ∫&#/&*
&#/&
𝑉 𝑡= . 𝑑𝑡′ with 𝑉 𝑡 =
A
A&
. 𝑋(𝑡)
¨ Let’s calculate
¨ 𝐴 =
A
A&
< (𝑋 𝑡 − 𝑋 𝑡* ),> =
A
A&
𝔼 (𝑋 𝑡 − 𝑋 𝑡* ),
¨ 𝐴 =
A
A&
< 2.
A
A&
𝑋 𝑡 − 𝑋 𝑡* . (𝑋 𝑡 − 𝑋 𝑡* ) >
35

36. Luc_Faucheux_2020
From Langevin (velocity) to particle diffusion (position) - II
¨ 𝐴 = < 2.
A
A&
𝑋 𝑡 − 𝑋 𝑡* . (𝑋 𝑡 − 𝑋 𝑡* ) >
¨ 𝐴 = < 2. 𝑉(𝑡). (𝑋 𝑡 − 𝑋 𝑡* ) >
¨ 𝐴 = < 2. 𝑉(𝑡). ∫&#/&*
&#/&
𝑉 𝑡= . 𝑑𝑡′ >
¨ 𝐴 = < ∫&#/&*
&#/&
2. 𝑉 𝑡 . 𝑉 𝑡= . 𝑑𝑡′ >
¨ 𝐴 = ∫&#/&*
&#/&
< 2. 𝑉 𝑡 . 𝑉 𝑡= >. 𝑑𝑡′
¨
A
A&
< (𝑋 𝑡 − 𝑋 𝑡* ),> = ∫&#/&*
&#/&
< 2. 𝑉 𝑡 . 𝑉 𝑡= >. 𝑑𝑡′
¨ And from the Langevin auto correlation function:
¨ 𝐶 𝑡., 𝑡* = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝐶 𝑡*, 𝑡*
36

37. Luc_Faucheux_2020
From Langevin (velocity) to particle diffusion (position) - III
¨ This is where the physics comes in (note that we are just glancing over it).
¨ We could:
¨ Derive the SIE for 𝑋(𝑡)
¨ Derive the PDE, solve for the PDF (in the other deck we do it through the neat trick of
Fourier transform)
¨ Look at overdamped – underdamped regime
¨ Take the steady state limit
¨ Show that this converges indeed towards the usual diffusion equation.
¨ But we can also ”define” the diffusion coefficient in space as:
¨
A
A&
< (𝑋 𝑡 − 𝑋 𝑡* ),> = ∫&#/&*
&#/&
< 2. 𝑉 𝑡 . 𝑉 𝑡= >. 𝑑𝑡′ = 2𝐷3
¨ Note: not to be confused with the diffusion coefficient in the velocity space
37

38. Luc_Faucheux_2020
From Langevin (velocity) to particle diffusion (position) - IV
¨
A
A&
< (𝑋 𝑡 − 𝑋 𝑡* ),> = ∫&#/&*
&#/&
< 2. 𝑉 𝑡 . 𝑉 𝑡= >. 𝑑𝑡′ = 2𝐷3
¨ With 𝐷3 = 𝐷3(𝑡, 𝑡*, . . )
¨ In the steady-state limit we assume that 𝐷3 is a constant
¨ ∫&#/4<
&#/&
< 2. 𝑉 𝑡 . 𝑉 𝑡= >. 𝑑𝑡′ = 2𝐷3
¨ 𝐷3 = ∫&#/4<
&#/&
< 2. 𝑉 𝑡 . 𝑉 𝑡= >. 𝑑𝑡′ = ∫:/;
:/<
< 𝑉 𝑠 . 𝑉 0 >. 𝑑𝑠 using 𝑠 = 𝑡 − 𝑡′
¨ 𝐷3 = ∫:/;
:/<
< 𝑉 𝑠 . 𝑉 0 >. 𝑑𝑠
¨ and we had:
¨ < 𝑉 𝑠 . 𝑉 0 > = < 𝑉(0), >. exp −𝑘𝑠 =
>"
,?
. exp −𝑘𝑠
38

39. Luc_Faucheux_2020
From Langevin (velocity) to particle diffusion (position) - V
¨ 𝐷3 = ∫:/;
:/<
< 𝑉 𝑠 . 𝑉 0 >. 𝑑𝑠
¨ 𝐷3 = ∫:/;
:/< >"
,?
. exp −𝑘𝑠 . 𝑑𝑠 =
>"
,?
.
+
?
=< 𝑉(0), >.
+
?
¨ In the steady state of the physical process that is diffusion of a particle in a thermal bath:
¨
+
,
𝑀 < 𝑉(0), >=
+
,
𝐾B. 𝑇
¨ Where (and for now we can just take those as almost formal definitions):
¨ 𝑀 is the mass of the particle
¨ 𝐾B is a constant (the Boltzmann constant)
¨ 𝑇 is the temperature of the surrounding fluid
39

40. Luc_Faucheux_2020
From Langevin (velocity) to particle diffusion (position) - VI
¨ So we get if we want the Langevin equation to accurately describe the diffusion of a particle,
at least in the steady state limit
¨ (note, in another deck we will go through the actual full derivation of the PDF for the
particle diffusion from the Langevin, and justify the steady state limit as the correct
approximation)
¨ 𝐷3 =
C(D
E?
=
>"
,?"
¨ This is an illustration of the celebrated fluctuation-dissipation theorem
¨ If we choose for the viscous damping the Stokes equation:
¨ 𝑘 =
FGHI
E
, where the particle is a sphere of radius 𝑅 in a fluid of steady state viscosity 𝜂
¨ We then obtain the Einstein (1905) equation: 𝐷3 =
C(D
E?
=
C(D
FGHI
¨ That was verified experimentally by the illustrious Frenchman Jean Perrin in 1908
40

41. Luc_Faucheux_2020
From Langevin (velocity) to particle diffusion (position) - VII
¨ This section did pack a lot and did not go into the details of actually deriving the PDF for the
particle position from the PDF from the velocity, or from any other way (from SIE or PDE).
¨ That will be for another deck
¨ This section was more to illustrate how central is the Langevin equation in Physics
¨ The same way that it should be in Finance, as the underlying dynamics for Black-Sholes, the
GBM (Geometric Brownian motion) suffers from not only allowing only positive security
prices, but also exhibits unstable dynamics (higher moments will diverge).
¨ For many securities (in particular rates, which are already the derivative of something like
the velocity is to the particle position), a Langevin approach is more favored (or should be).
¨ Salomon Brothers in the 1970 had already a Langevin approach using more than one factor
(hence the name 2+), with factor correlation and a skew function famously known as IRMA.
They were quite ahead of their time, as most of the market kept on using multiple tweaks on
Black-Sholes to try to make it work in a satisfactory manner
41

42. Luc_Faucheux_2020
Langevin equation
Dynamics of moments from the PDF
42

43. Luc_Faucheux_2020
Langevin equation – dynamics of moments
¨ In the deck (II), we looked at moments from the FP:
¨
!"($,&)
!&
= −
!
!$
[𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 −
!
!$
[𝑀, 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 ]]
¨ Really in terms of notation to highlight the fact that this is a FORWARD FP:
¨
!"($,&|$;,&;)
!&
= −
!
!$
[𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡|𝑣0, 𝑡0 −
!
!$
[𝑀, 𝑣, 𝑡 . 𝑝 𝑣, 𝑡|𝑣𝑜, 𝑡𝑜 ]]
¨ 𝑚J 𝑣, 𝑡 =< 𝑉J >&= ∫4<
7<
𝑝 𝑣, 𝑡 . 𝑣J. 𝑑𝑣
¨ We showed in deck II that by integration by part:
¨ 𝐼, 𝑛 = ∫4<
7<
𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 𝑛. 𝑣J4+. 𝑑𝑣 + ∫4<
7<
𝑀, 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 𝑛. (𝑛 − 1). 𝑣J4,. 𝑑𝑣
¨
A
A&
. 𝑚J 𝑣, 𝑡 = 𝐼, 𝑛
¨
A
A&
. 𝑚; 𝑣, 𝑡 = 0 since the probability 𝑚; 𝑣, 𝑡 = ∫4<
7<
𝑝 𝑣, 𝑡 . 𝑑𝑣 is conserved
43

44. Luc_Faucheux_2020
Langevin equation – dynamics of moments - II
¨ For the Langevin case:
¨
!"($,&|$;,&;)
!&
= −
!
!$
[𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡|𝑣0, 𝑡0 −
!
!$
[𝑀, 𝑣, 𝑡 . 𝑝 𝑣, 𝑡|𝑣𝑜, 𝑡𝑜 ]]
¨
!"($,&|(!,&!)
!&
= −
!
!$
(−𝑘𝑣). 𝑝 𝑣, 𝑡 𝑉*, 𝑡* −
!
!$
[
+
,
. [𝜎, . 𝑝(𝑣, 𝑡|𝑉*, 𝑡*)]
¨ 𝑀+ 𝑣, 𝑡 = −𝑘𝑣
¨ 𝑀, 𝑣, 𝑡 =
+
,
. 𝜎,
¨
A
A&
. 𝑚+ 𝑣, 𝑡 = 𝐼, 1 = ∫4<
7<
𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 𝑑𝑣 = ∫4<
7<
(−𝑘𝑣). 𝑝 𝑣, 𝑡 . 𝑑𝑣
¨
A
A&
. 𝑚+ 𝑣, 𝑡 = ∫4<
7<
−𝑘𝑣 . 𝑝 𝑣, 𝑡 . 𝑑𝑣 = −𝑘. ∫4<
7<
𝑣. 𝑝 𝑣, 𝑡 . 𝑑𝑣 = −𝑘. 𝑚+ 𝑣, 𝑡
44

45. Luc_Faucheux_2020
Langevin equation – dynamics of moments - III
¨
A
A&
. 𝑚+ 𝑣, 𝑡 = −𝑘. 𝑚+ 𝑣, 𝑡
¨ 𝑚+ 𝑣, 𝑡 = 𝑚+ 𝑣, 0 . exp(−𝑘𝑡)
¨ < 𝑉 >& = < 𝑉 𝑡 > = < 𝑉 0 >. exp(−𝑘𝑡)
¨ This is also what we had from an explicit formulation of the Langevin equation:
¨ 𝔼 𝑉 𝑡. = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 𝑉 𝑡*
¨ So as 𝑡. → ∞, 𝔼 𝑉 𝑡. → 0
¨ From:
¨ 𝑉 𝑡. = exp −𝑘𝑡. . {exp 𝑘𝑡* . 𝑉 𝑡* + ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
45

46. Luc_Faucheux_2020
Langevin equation – dynamics of moments - IV
¨
A
A&
. 𝑚, 𝑣, 𝑡 = 𝐼, 2
¨ 𝐼, 𝑛 = ∫4<
7<
𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 𝑛. 𝑣J4+. 𝑑𝑣 + ∫4<
7<
𝑀, 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 𝑛. (𝑛 − 1). 𝑣J4,. 𝑑𝑣
¨
A
A&
. 𝑚, 𝑣, 𝑡 = ∫4<
7<
𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 2𝑣. 𝑑𝑣 + ∫4<
7<
𝑀, 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 2. 𝑑𝑣
¨ 𝑀+ 𝑣, 𝑡 = −𝑘𝑣
¨ 𝑀, 𝑣, 𝑡 =
+
,
. 𝜎,
¨
A
A&
. 𝑚, 𝑣, 𝑡 = ∫4<
7<
(−𝑘𝑣). 𝑝 𝑣, 𝑡 . 2𝑣. 𝑑𝑣 + ∫4<
7<
(
+
,
. 𝜎, ). 𝑝 𝑣, 𝑡 . 2. 𝑑𝑣
¨
A
A&
. 𝑚, 𝑣, 𝑡 = −2𝑘 ∫4<
7<
𝑝 𝑣, 𝑡 . 𝑣,. 𝑑𝑣 + 𝜎,
∫4<
7<
𝑝 𝑣, 𝑡 . 𝑑𝑣
¨
A
A&
. 𝑚, 𝑣, 𝑡 = −2𝑘. 𝑚, 𝑣, 𝑡 + 𝜎,. 𝑚; 𝑣, 𝑡 = −2𝑘. 𝑚, 𝑣, 𝑡 + 𝜎,
46

47. Luc_Faucheux_2020
Langevin equation – dynamics of moments - V
¨
A
A&
. 𝑚, 𝑣, 𝑡 = −2𝑘. 𝑚, 𝑣, 𝑡 + 𝜎,
¨
A
A&
. 𝑚, 𝑣, 𝑡 + 2𝑘. 𝑚, 𝑣, 𝑡 = 𝜎,
¨ We choose to write using the separation of variables method:
¨ 𝑚, 𝑣, 𝑡 = 𝑛, 𝑡 . exp(−2𝑘𝑡)
¨
A
A&
𝑛, 𝑡 . exp −2𝑘𝑡 + 𝑛, 𝑡 . −2𝑘 . 𝑒𝑥𝑝 −2𝑘𝑡 = −2𝑘. 𝑛, 𝑡 . exp(−2𝑘𝑡) + 𝜎,
¨
A
A&
𝑛, 𝑡 . exp −2𝑘𝑡 = 𝜎,
¨
A
A&
𝑛, 𝑡 = 𝜎,. exp 2𝑘𝑡
¨ 𝑛, 𝑡 =
+
,?
𝜎,. exp 2𝑘𝑡 + 𝐶
¨ 𝑛, 𝑡 =
+
,?
𝜎,. exp 2𝑘𝑡 + 𝑛, 0 −
+
,?
𝜎,
47

48. Luc_Faucheux_2020
Langevin equation – dynamics of moments - VI
¨ 𝑛, 𝑡 =
+
,?
𝜎,. exp 2𝑘𝑡 + 𝑛, 0 −
+
,?
𝜎,
¨ 𝑚, 𝑡 = 𝑛, 𝑡 . exp −2𝑘𝑡 =
>
,?
,
+ 𝑛, 0 −
+
,?
𝜎, . exp −2𝑘𝑡
¨ And 𝑛, 0 = 𝑚, 0
¨ 𝑚, 𝑡 =
>
,?
,
+ 𝑚, 0 −
+
,?
𝜎, . exp −2𝑘𝑡
¨ < 𝑉&
, >& = < 𝑉 𝑡 , > = < 𝑉 0 , > −
>"
,?
. exp −2𝑘𝑡 +
>"
,?
¨ < 𝑉 𝑡 , > converges to
>"
,?
when 𝑡 → ∞
¨ This is also what we obtained when explicitly calculating the autocorrelation function.
48

49. Luc_Faucheux_2020
Langevin equation – dynamics of moments - VII
¨ We can also by recurrence get the higher moments:
¨ 𝐼, 𝑛 = ∫4<
7<
𝑀+ 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 𝑛. 𝑣J4+. 𝑑𝑣 + ∫4<
7<
𝑀, 𝑣, 𝑡 . 𝑝 𝑣, 𝑡 . 𝑛. (𝑛 − 1). 𝑣J4,. 𝑑𝑣
¨
A
A&
. 𝑚J 𝑣, 𝑡 = 𝐼, 𝑛
¨ 𝑀+ 𝑣, 𝑡 = −𝑘𝑣
¨ 𝑀, 𝑣, 𝑡 =
+
,
. 𝜎,
¨
A
A&
. 𝑚J 𝑣, 𝑡 = ∫4<
7<
(−𝑘𝑣). 𝑝 𝑣, 𝑡 . 𝑛. 𝑣J4+. 𝑑𝑣 + ∫4<
7<
(
>"
,
). 𝑝 𝑣, 𝑡 . 𝑛. (𝑛 − 1). 𝑣J4,. 𝑑𝑣
¨
A
A&
. 𝑚J 𝑣, 𝑡 = −𝑘. 𝑛. ∫4<
7<
𝑝 𝑣, 𝑡 . 𝑣J . 𝑑𝑣 +
>"
,
. ∫4<
7<
𝑝 𝑣, 𝑡 . 𝑛. (𝑛 − 1). 𝑣J4,. 𝑑𝑣
¨
A
A&
. 𝑚J 𝑡 = −𝑘. 𝑛. 𝑚J 𝑡 +
>"
,
. 𝑛. 𝑛 − 1 . 𝑚J4, 𝑡
49

50. Luc_Faucheux_2020
Langevin equation – dynamics of moments - VIII
¨
A
A&
. 𝑚J 𝑡 = −𝑘. 𝑚J 𝑡 +
>"
,
. 𝑛. 𝑛 − 1 . 𝑚J4, 𝑡
¨ We can solve this by recurrence using the method of “variations of parameters” first
originated by Joseph-Henri Lagrange (on the left) for ODE, then extended to PDE by Jean-
Marie Duhamel on the right (born in Saint-Malo !)
50

51. Luc_Faucheux_2020
Jean-Marie Duhamel was born in Saint-Malo !
¨ Saint Malo is just awesome. Many reasons why. In random order
¨ It is the location for a #1 New York Times bestseller
51

52. Luc_Faucheux_2020
Saint Malo is awesome - II
¨ The Surcouf family is from Saint Malo. Robert was a renowned “corsair” (French pirate) who
gave a lot of grief to the Beefeaters. The whole family were essentially pirates.
52

53. Luc_Faucheux_2020
Saint Malo is awesome - III
¨ Duguay-Trouin is also from Saint Malo. He was also a French corsair giving grief to the Brits
(there is a pattern there)
53

54. Luc_Faucheux_2020
Saint Malo is awesome - IV
¨ Pierre Louis Maupertuis is from Saint Malo. He invented the “least action principle” in
Physics. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian
equations of motion, and is fundamental to general relativity and Quantum mechanics
¨ In the drawing below he is wearing appropriate attire for an expedition in Lapland (Sapmi)
54

55. Luc_Faucheux_2020
Saint Malo is awesome - V
¨ Jean-Baptiste Benard de la Harpe is from Saint Malo. He discovered Little Rock, Arkansas.
We forgive him.
55

56. Luc_Faucheux_2020
Saint Malo is awesome - VI
¨ Jacques Cartier is from Saint Malo. He discovered Canada. Take that Jean-Baptiste Benard
de la Harpe.
56

57. Luc_Faucheux_2020
Saint Malo is awesome - VII
¨ Colin Clive is also from Saint Malo. He was the doctor Frankenstein (on the right)
57

58. Luc_Faucheux_2020
Saint Malo is awesome - VIII
¨ It is a great place to watch a storm.
58

59. Luc_Faucheux_2020
Variation of parameters method for first-order ODE
¨ In the general case:
¨ 𝑦= + 𝑝 𝑥 . 𝑦 = 𝑞(𝑥)
¨ If 𝑞 𝑥 = 0 we then have: 𝑦= + 𝑝 𝑥 . 𝑦 = 0
¨
AK
A6
= −𝑝 𝑥 . 𝑦
¨
AK
K
= 𝑑 ln 𝑦 = −𝑝 𝑥 . 𝑑𝑥
¨ ln 𝑦 = − ∫ 𝑝 𝑥 . 𝑑𝑥 + 𝐶
¨ Similar to the SIE formulation we write it as :
¨ ln 𝑦 𝑥. − ln 𝑦 𝑥* = − ∫6/6*
6/6.
𝑝 𝑥 . 𝑑𝑥
¨ 𝑦 𝑥. = 𝑦 𝑥* . exp(− ∫6/6*
6/6.
𝑝 𝑥 . 𝑑𝑥)
59

60. Luc_Faucheux_2020
Variation of parameters method for first-order ODE - II
¨ If 𝑞 𝑥 <> 0 we then have: 𝑦= + 𝑝 𝑥 . 𝑦 = 𝑞 𝑥
¨ We choose: 𝑦 𝑥 = 𝐶(𝑥). exp(− ∫;
6
𝑝 𝑠 . 𝑑𝑠)
¨ 𝑦= 𝑥 = 𝐶= 𝑥 . exp − ∫;
6
𝑝 𝑠 . 𝑑𝑠 + 𝐶 𝑥 . −𝑝 𝑥 . exp(− ∫;
6
𝑝 𝑠 . 𝑑𝑠)
¨ 𝑦= 𝑥 = 𝐶= 𝑥 . exp − ∫;
6
𝑝 𝑠 . 𝑑𝑠 − 𝑝 𝑥 . 𝑦(𝑥)
¨ So we get: 𝐶= 𝑥 . exp − ∫;
6
𝑝 𝑠 . 𝑑𝑠 = 𝑞 𝑥
¨ 𝐶= 𝑥 = exp ∫;
6
𝑝 𝑠 . 𝑑𝑠 . 𝑞 𝑥
¨ 𝐶 𝑥. − 𝐶 𝑥* = ∫6/6*
6/6.
exp ∫;
6
𝑝 𝑠 . 𝑑𝑠 . 𝑞 𝑥 . 𝑑𝑥
¨ You can see that for ODE of higher orders we can see a nesting of integrals that is started to
rear its ugly head
60

61. Luc_Faucheux_2020
Variation of parameters method for first-order ODE - III
¨ 𝐶 𝑥. − 𝐶 𝑥* = ∫6/6*
6/6.
exp ∫;
6
𝑝 𝑠 . 𝑑𝑠 . 𝑞 𝑥 . 𝑑𝑥
¨ 𝑦 𝑥. = 𝐶(𝑥.). exp(− ∫6)
6)
𝑝 𝑠 . 𝑑𝑠)
¨ 𝑦 𝑥. = 𝐶 𝑥* . exp − ∫6)
6)
𝑝 𝑠 . 𝑑𝑠 + exp − ∫6)
6)
𝑝 𝑠 . 𝑑𝑠 . ∫6/6*
6/6.
exp ∫;
6
𝑝 𝑠 . 𝑑𝑠 . 𝑞 𝑥 . 𝑑𝑥
¨ Jean-Mari Duhamel extended this to ODE of degree 𝑛
¨ He also essentially invented the record player (vibroscope)
61

62. Luc_Faucheux_2020
Langevin equation – dynamics of moments - IX
¨ Back to the ODE at stake here:
¨
A
A&
. 𝑚J 𝑡 = −𝑘. 𝑛. 𝑚J 𝑡 +
>"
,
. 𝑛. 𝑛 − 1 . 𝑚J4, 𝑡
¨ 𝑦= + 𝑝 𝑡 . 𝑦 = 𝑞(𝑡)
¨ With 𝑝 𝑡 = 𝑘𝑛 and 𝑞 𝑡 =
>"
,
. 𝑛. 𝑛 − 1 . 𝑚J4, 𝑡
¨ 𝑦 𝑡. = 𝐶 𝑡* . exp − ∫&!
&)
𝑝 𝑠 . 𝑑𝑠 + exp − ∫&!
&)
𝑝 𝑠 . 𝑑𝑠 . ∫&/&*
&/&.
exp ∫;
&
𝑝 𝑠 . 𝑑𝑠 . 𝑞 𝑡 . 𝑑𝑡
¨ 𝑚J 𝑡. = 𝐶 𝑡* . exp − ∫&!
&)
(𝑘𝑛). 𝑑𝑠 + exp − ∫&!
&)
(𝑘𝑛). 𝑑𝑠 . ∫&/&*
&/&.
exp ∫;
&
(𝑘𝑛). 𝑑𝑠 . 𝑞 𝑡 . 𝑑𝑡
¨ 𝑚J 𝑡. = 𝐶 𝑡* . exp −𝑛𝑘(𝑡. − 𝑡*) + exp −𝑛𝑘(𝑡. − 𝑡*) . ∫&/&*
&/&.
exp ∫;
&
(𝑘𝑛). 𝑑𝑠 . 𝑞 𝑡 . 𝑑𝑡
62

63. Luc_Faucheux_2020
Langevin equation – dynamics of moments - X
¨ 𝑚J 𝑡. = 𝐶 𝑡* . exp −𝑛𝑘(𝑡. − 𝑡*) + exp −𝑛𝑘(𝑡. − 𝑡*) . ∫&/&*
&/&.
exp ∫;
&
(𝑘𝑛). 𝑑𝑠 . 𝑞 𝑡 . 𝑑𝑡
¨ Choosing 𝑡* = 0 for sake of clarity:
¨ 𝑚J 𝑡. = 𝐶 0 . exp −𝑛𝑘𝑡. + exp −𝑛𝑘𝑡. . ∫&/;
&/&.
exp ∫;
&
(𝑘𝑛). 𝑑𝑠 . 𝑞 𝑡 . 𝑑𝑡
¨ 𝑞 𝑡 =
>"
,
. 𝑛. 𝑛 − 1 . 𝑚J4, 𝑡
¨ exp ∫;
&
(𝑘𝑛). 𝑑𝑠 = 𝑘𝑛𝑡
¨ 𝑚J 𝑡. = 𝐶 0 . exp −𝑛𝑘𝑡. + exp −𝑛𝑘𝑡. . ∫&/;
&/&.
exp 𝑛𝑘𝑡 . 𝑞 𝑡 . 𝑑𝑡
¨ 𝑚J 𝑡. = 𝑚J 0 . exp −𝑛𝑘𝑡. +
>"
,
. 𝑛. 𝑛 − 1 . exp −𝑛𝑘𝑡. . ∫&/;
&/&.
exp 𝑛𝑘𝑡 . 𝑚J4, 𝑡 . 𝑑𝑡
¨ 𝑚J 𝑡 = 𝑚J 0 . exp −𝑛𝑘𝑡 +
>"
,
. 𝑛. 𝑛 − 1 . exp −𝑛𝑘𝑡 . ∫:/;
:/&
exp 𝑛𝑘𝑠 . 𝑚J4, 𝑠 . 𝑑𝑠
63

64. Luc_Faucheux_2020
Langevin equation – dynamics of moments - XI
¨ 𝑚J 𝑡 = 𝑚J 0 . exp −𝑛𝑘𝑡 +
>"
,
. 𝑛. 𝑛 − 1 . exp −𝑛𝑘𝑡 . ∫:/;
:/&
exp 𝑛𝑘𝑠 . 𝑚J4, 𝑠 . 𝑑𝑠
¨ 𝑚J 𝑡 = 𝑚J 0 . exp −𝑛𝑘𝑡 +
>"
,
. 𝑛. 𝑛 − 1 . ∫:/;
:/&
exp −𝑛𝑘(𝑡 − 𝑠) . 𝑚J4, 𝑠 . 𝑑𝑠
¨ For 𝑛 = 2 we recover:
¨ 𝑚, 𝑡 = 𝑚, 0 . exp −2𝑘𝑡 +
>"
,
. 2. 2 − 1 . ∫:/;
:/&
exp −2𝑘(𝑡 − 𝑠) . 𝑚; 𝑠 . 𝑑𝑠
¨ And 𝑚; 𝑠 = 1
¨ 𝑚, 𝑡 = 𝑚, 0 . exp −2𝑘𝑡 + 𝜎,. ∫:/;
:/&
exp −2𝑘(𝑡 − 𝑠) . 𝑑𝑠
¨ 𝑚, 𝑡 = 𝑚, 0 . exp −2𝑘𝑡 + 𝜎,. exp −2𝑘𝑡 [
+
,?
exp(2𝑘𝑠)]:/;
:/&
¨ 𝑚, 𝑡 = 𝑚, 0 . exp −2𝑘𝑡 + 𝜎,. exp −2𝑘𝑡 . [
+
,?
exp 2𝑘𝑡 −
+
,?
]
64

65. Luc_Faucheux_2020
Langevin equation – dynamics of moments - XII
¨ 𝑚, 𝑡 = 𝑚, 0 . exp −2𝑘𝑡 + 𝜎,. exp −2𝑘𝑡 . [
+
,?
exp 2𝑘𝑡 −
+
,?
]
¨ 𝑚, 𝑡 =
>"
,?
+ exp −2𝑘𝑡 . [𝑚, 0 −
>"
,?
]
¨ What we had before was:
¨ 𝑚, 𝑡 =
>
,?
,
+ 𝑚, 0 −
+
,?
𝜎, . exp −2𝑘𝑡
¨ So it works !
65

66. Luc_Faucheux_2020
Langevin equation – dynamics of moments - XIII
¨ Steady-state values of the moments (noted for 𝑡 = ∞)
¨ 𝑚; 𝑡 = 1 so 𝑚; ∞ = 1
¨
A
A&
. 𝑚+ 𝑣, 𝑡 = −𝑘. 𝑚+ 𝑣, 𝑡 = 0 so 𝑚+ ∞ = 0
¨ 𝑚, 𝑡 =
>"
,?
+ exp −2𝑘𝑡 . [𝑚, 0 −
>"
,?
] converges when 𝑡 → ∞ to
>"
,?
, so 𝑚, ∞ =
>"
,?
¨ Also,
A
A&
. 𝑚, 𝑣, 𝑡 = −2𝑘. 𝑚, 𝑣, 𝑡 + 𝜎,= 0 so 𝑚, ∞ =
>"
,?
¨ 𝑚J 𝑡 = 𝑚J 0 . exp −𝑛𝑘𝑡 +
>"
,
. 𝑛. 𝑛 − 1 . ∫:/;
:/&
exp −𝑛𝑘(𝑡 − 𝑠) . 𝑚J4, 𝑠 . 𝑑𝑠
¨ The first term converges to 0 so we are left for 𝑡 → ∞ with:
¨ 𝑚J ∞ = lim
&→<
>"
,
. 𝑛. 𝑛 − 1 . ∫:/;
:/&
exp −𝑛𝑘(𝑡 − 𝑠) . 𝑚J4, 𝑠 . 𝑑𝑠
66

67. Luc_Faucheux_2020
Langevin equation – dynamics of moments - XIV
¨ We already have:
¨ 𝑚; ∞ = 1
¨ 𝑚+ ∞ = 0
¨ 𝑚, ∞ =
>"
,?
¨ 𝑚J ∞ = lim
&→<
>"
,
. 𝑛. 𝑛 − 1 . ∫:/;
:/&
exp −𝑛𝑘(𝑡 − 𝑠) . 𝑚J4, 𝑠 . 𝑑𝑠
¨ 𝑚J ∞ =
>"
,
. 𝑛. 𝑛 − 1 .
+
J?
𝑚J4, ∞ = 𝑚, ∞ . 𝑛 − 1 . 𝑚J4, ∞
¨ So for all odd n numbers, 𝑚J ∞ = 0
¨ For all even numbers: 𝑚J ∞ = 𝑛 − 1 . 𝑛 − 3 … 3.1. (𝑚, ∞ )J/,
67

68. Luc_Faucheux_2020
Langevin equation – dynamics of moments - XV
¨ In the deck on Bachelier, we calculated all the moments for the Gaussian distribution:
¨ < 𝑥,J > = (𝜎, 𝑡)J. 2𝑛 − 1 ‼ and < 𝑥,J7+ > = 0
¨ For the regular Gaussian ℎ 𝑥, 𝑡 =
+
,G>"&
. exp(
46"
,>"&
)
¨ 𝑛! = ∏N/+
N/J
𝑗 is the usual factorial
¨ 𝑛!! = ∏N/+
N/J
𝑗 is called the “double factorial” and only includes in the product the terms that
have the SAME parity as 𝑘
¨ Here for the Langevin equation we have:
¨ 𝑚,J ∞ = 2𝑛 − 1 ‼ (𝑚, ∞ )J and 𝑚,J7+ ∞ = 0
¨ 𝑚, ∞ =
>"
,?
68

69. Luc_Faucheux_2020
Langevin equation – dynamics of moments - XVI
¨ This suggests, that whatever the PDF for the Langevin equation (that we have not solved
yet), it might converge to:
¨ ℎ 𝑣, 𝑡 → ∞ =
+
,G.-" <
. exp(
4$"
,-" <
)
¨ ℎ 𝑣, 𝑡 → ∞ =
+
,G.
*"
"$
. exp(
4$"
,
*"
"$
)
¨ ℎ 𝑣, 𝑡 → ∞ =
?
G >" . exp(
4?$"
>" )
¨ We also have of course:
¨ 𝑚,J ∞ = ∫$/4<
$/7<
𝑣J. ℎ 𝑣, 𝑡 → ∞ . 𝑑𝑣
¨ We will use that when trying to guess / derive the PDF for the Langevin equation.
69

70. Luc_Faucheux_2020
Langevin equation
Dynamics of moments from the SDE
70

71. Luc_Faucheux_2020
Langevin equation – dynamics of moments from SDE
¨ We can also derive the dynamics of moments from the SDE as opposed to the PDF, using the
ITO lemma
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ ITO lemma for a function 𝑓(𝑉)
¨ 𝑓 𝑉 𝑡. − 𝑓 𝑉 𝑡* = ∫&/&*
&/&. !0
!(
. ([). 𝑑𝑉(𝑡) + ∫&/&*
&/&. +
,
.
!"1
!(" . ([). (𝛿𝑉),
¨ In the ”limit” of small time increments, this can be written formally as the Ito lemma:
¨ 𝛿𝑓 =
!0
!(
. 𝛿𝑉 +
+
,
.
!"0
!(" . (𝛿𝑉), and we choose 𝑓 𝑉 = 𝑉J
¨
!0
!(
= 𝑛. 𝑉J4+
¨
!"0
!(" = 𝑛. 𝑛 − 1 . 𝑉J4,
71

72. Luc_Faucheux_2020
Langevin equation – dynamics of moments from SDE - II
¨ 𝑉 𝑡.
J − 𝑉 𝑡*
J = ∫&/&*
&/&.
𝑛. 𝑉J4+. ([). 𝑑𝑉(𝑡) + ∫&/&*
&/&. +
,
𝑛. 𝑛 − 1 . 𝑉J4,. ([). (𝛿𝑉),
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ 𝑉 𝑡.
J − 𝑉 𝑡*
J = −𝑘 ∫&/&*
&/&.
𝑛𝑉J 𝑑𝑡 + ∫&/&*
&/&. +
,
𝑛 𝑛 − 1 𝑉J4, 𝜎, 𝑑𝑡 + ∫&/&*
&/&.
𝑛. 𝑉J4+. 𝜎. ([). 𝑑𝑊
¨ 𝑉 𝑡.
J = 𝑉 𝑡*
J − 𝑘 ∫&/&*
&/&.
𝑛𝑉J 𝑑𝑡 + ∫&/&*
&/&. +
,
𝑛 𝑛 − 1 𝑉J4, 𝜎, 𝑑𝑡 + ∫&/&*
&/&.
𝑛. 𝑉J4+. 𝜎. ([). 𝑑𝑊
¨ 𝑚J 𝑡. =< 𝑉J >&)
= ∫4<
7<
𝑝 𝑣, 𝑡. . 𝑣J. 𝑑𝑣
¨ 𝑚J 𝑡. = 𝔼(𝑉J)
¨ Since the ITO integral is a martingale,
¨ 𝔼 ∫&/&*
&/&.
𝑛. 𝑉J4+. 𝜎. ([). 𝑑𝑊 = 0
72

73. Luc_Faucheux_2020
Langevin equation – dynamics of moments from SDE - III
¨ 𝔼(𝑉 𝑡.
J) = 𝔼(𝑉 𝑡*
J) − 𝑘 ∫&/&*
&/&.
𝑛𝔼(𝑉J)𝑑𝑡 + ∫&/&*
&/&. +
,
𝑛 𝑛 − 1 𝔼(𝑉J4,)𝜎, 𝑑𝑡
¨ 𝑚J 𝑡. = 𝑚J 𝑡* − 𝑘 ∫&/&*
&/&.
𝑛. 𝑚J 𝑡 𝑑𝑡 + ∫&/&*
&/&. +
,
𝑛 𝑛 − 1 𝑚J 𝑡 𝜎, 𝑑𝑡
¨ Or in differential form:
¨
A
A&
. 𝑚J 𝑡 = −𝑘. 𝑛. 𝑚J 𝑡 +
>"
,
. 𝑛. 𝑛 − 1 . 𝑚J4, 𝑡
¨ This is the same exact formula we obtained when getting the dynamics from the PDE
(forward PDE) when integrating by parts
¨ Here we obtained it directly from ITO lemma and using the martingale property
¨ No surprise there, as we saw before the correspondence between the Forward and
Backward PDEs using the integration by parts.
73

74. Luc_Faucheux_2020
A quick note on averaging
74

75. Luc_Faucheux_2020
A quick note on averaging
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ One is tempted to write
¨ < 𝑑𝑉 𝑡 > = < −𝑘𝑉 𝑡 >. 𝑑𝑡 +< 𝜎. 𝑑𝑊 >
¨ 𝑑 < 𝑉 𝑡 > = −𝑘 < 𝑉 𝑡 >. 𝑑𝑡
¨ And so < 𝑉 𝑡 > =< 𝑉 𝑡 >. exp(−𝑘𝑡)
¨ This is exactly what we had from either solving specifically for a solution of the SDE, or using
the moments:
¨ 𝑚+ 𝑣, 𝑡 = 𝑚+ 𝑣, 0 . exp(−𝑘𝑡)
¨ < 𝑉 >& = < 𝑉 𝑡 > = < 𝑉 0 >. exp(−𝑘𝑡)
¨ 𝔼 𝑉 𝑡. = exp −𝑘𝑡. . exp 𝑘𝑡* . 𝔼 𝑉 𝑡*
¨ 𝑉 𝑡. = exp −𝑘𝑡. . {exp 𝑘𝑡* . 𝑉 𝑡* + ∫&/&*
&/&.
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
75

76. Luc_Faucheux_2020
A quick note on averaging - II
¨ One is tempted to do the same for the second moment by multiplying by 𝑉 on both sides
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ 𝑉 𝑡 . 𝑑𝑉 𝑡 = −𝑘𝑉 𝑡 𝑉 𝑡 . 𝑑𝑡 + 𝜎. 𝑉 𝑡 . ([). 𝑑𝑊
¨ 𝑑[
("
,
] = −𝑘𝑉 𝑡 𝑉 𝑡 . 𝑑𝑡 + 𝜎. 𝑉 𝑡 . ([). 𝑑𝑊
¨ And then take the average
¨ < 𝑑
("
,
> = < −𝑘𝑉 𝑡 𝑉 𝑡 . 𝑑𝑡 >+< 𝜎. 𝑉 𝑡 . ([). 𝑑𝑊 >
¨ 𝑑 <
("
,
> = −𝑘 < 𝑉, >. 𝑑𝑡 > + < 𝜎. 𝑉 𝑡 . ([). 𝑑𝑊 >
¨ And then say : < 𝜎. 𝑉 𝑡 . ([). 𝑑𝑊 > = 0
¨ So: < 𝑉, 𝑡 > =< 𝑉,(0) >. exp(−2𝑘𝑡)
76

77. Luc_Faucheux_2020
A quick note on averaging - III
¨ Previous slide is wrong because we cannot rely on usual rules of calculus.
¨ We know that the previous slide is wrong because the actual result is:
¨ < 𝑉&
, >& = < 𝑉 𝑡 , > = < 𝑉 0 , > −
>"
,?
. exp −2𝑘𝑡 +
>"
,?
¨ So we can either stay in ITO calculus so that we can use: < 𝜎. 𝑉 𝑡 . ([). 𝑑𝑊 > = 0
¨ OR we formally use the usual rules of calculus, but in that case we have to rely on the
STRATO convention for the integral and in this case: < 𝜎. 𝑉 𝑡 . (∘). 𝑑𝑊 > ≠ 0
¨ This was motivated by a footnote by Van Kampen on page 221
77

78. Luc_Faucheux_2020
A quick note on averaging - IV
¨ Let’s do it right in ITO
¨ 𝑑
("
,
=
!
!(
("
,
. ([). 𝑑𝑉 +
+
,
.
!"
!("
("
,
. ([). 𝑑𝑉. ([). 𝑑𝑉 +
!
!&
("
,
. 𝑑𝑡
¨ 𝑑
("
,
= 𝑉. ([). 𝑑𝑉 +
+
,
. 1. ([). 𝑑𝑉. ([). 𝑑𝑉 + 0. 𝑑𝑡
¨ 𝑑
("
,
= 𝑉. ([). {−𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊} +
+
,
. 𝜎, 𝑑𝑡
¨ 𝑑
("
,
= −2𝑘
("
,
. 𝑑𝑡 + 𝜎. 𝑉. ([). 𝑑𝑊 +
+
,
. 𝜎, 𝑑𝑡
¨ We then take the average:
¨ 𝑑 <
("
,
> = −2𝑘 <
("
,
>. 𝑑𝑡 + < 𝜎. 𝑉. ([). 𝑑𝑊 > +
+
,
. 𝜎, 𝑑𝑡
¨ We can then use the property that the ITO integral is a martingale
78

79. Luc_Faucheux_2020
A quick note on averaging - V
¨ < 𝜎. 𝑉. ([). 𝑑𝑊 > = 0
¨ 𝑑 <
("
,
> = −2𝑘 <
("
,
>. 𝑑𝑡 +
+
,
. 𝜎, 𝑑𝑡
¨ That is now a closed equation which we can write as:
¨ With:
¨
A
A&
. 𝑚, 𝑣, 𝑡 + 2𝑘. 𝑚, 𝑣, 𝑡 = 𝜎,
¨ This is exactly the equation we got for the moment so we will get the same solution
¨ < 𝑉&
, >& = < 𝑉 𝑡 , > = < 𝑉 0 , > −
>"
,?
. exp −2𝑘𝑡 +
>"
,?
79

80. Luc_Faucheux_2020
A quick note on averaging - VI
¨ If we do it the correct way in STRATO:
¨ 𝑑
("
,
=
!
!(
("
,
. (∘). 𝑑𝑉 +
!
!&
("
,
. 𝑑𝑡
¨ 𝑑
("
,
= 𝑉 𝑡 . ∘ . 𝑑𝑉 = 𝑉 𝑡 . ∘ . {−𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊}
¨ 𝑑
("
,
= 𝑉 𝑡 . ∘ . 𝑑𝑉 = −2𝑘
("
,
. 𝑑𝑡 + 𝜎. 𝑉(𝑡). (∘). 𝑑𝑊
¨ We then take the average:
¨ 𝑑 <
("
,
> = −2𝑘 <
("
,
>. 𝑑𝑡 + < 𝜎. 𝑉. (∘). 𝑑𝑊 >
¨ This is NOT closed equation since the STRATO integral is NOT a martingale
¨ < 𝜎. 𝑉. ∘ . 𝑑𝑊 > ≠ 0
80

81. Luc_Faucheux_2020
A quick note on averaging - V
¨ In fact comparing the two equations we get:
¨ 𝑑 <
("
,
> = −2𝑘 <
("
,
>. 𝑑𝑡 +
+
,
. 𝜎, 𝑑𝑡
¨ 𝑑 <
("
,
> = −2𝑘 <
("
,
>. 𝑑𝑡 + < 𝜎. 𝑉. (∘). 𝑑𝑊 >
¨ SO:
¨ < 𝜎. 𝑉. ∘ . 𝑑𝑊 > =
+
,
. 𝜎, 𝑑𝑡
¨ We could also derive this explicitly, in a couple of different ways
81

82. Luc_Faucheux_2020
A quick note on averaging - VI
¨ We could use the relation between the ITO and STRATO integrals.
¨ For a stochastic process
¨ 𝑑𝑉 𝑡 = 𝑎 𝑡, 𝑉 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑉 𝑡 . 𝑑𝑊
¨ We have:
¨ ∫&/&*
&/&.
𝑓 𝑉 𝑡 . ∘ . 𝑑𝑊 𝑡 = ∫&/&*
&/&.
𝑓 𝑉 𝑡 . ([). 𝑑𝑊(𝑡) + ∫&/&*
&/&. +
,
. 𝑏 𝑡, 𝑉 𝑡 .
!
!(
𝑓 𝑉(𝑡 . 𝑑𝑡
¨ 𝑓 𝑉 𝑡 = 𝜎. 𝑉(𝑡)
¨
!
!(
𝑓 𝑉(𝑡 = 𝜎
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ 𝑏 𝑡, 𝑉 𝑡 = 𝜎
82

83. Luc_Faucheux_2020
A quick note on averaging - VII
¨ ∫&/&*
&/&.
𝜎. 𝑉(𝑡) . ∘ . 𝑑𝑊 𝑡 = ∫&/&*
&/&.
𝜎. 𝑉(𝑡) . ([). 𝑑𝑊(𝑡) + ∫&/&*
&/&. +
,
. 𝜎. 𝜎. 𝑑𝑡
¨ In the limit of small time increment:
¨ 𝜎. 𝑉 𝑡 . ∘ . 𝑑𝑊 𝑡 = 𝜎. 𝑉(𝑡) . ([). 𝑑𝑊 𝑡 +
+
,
. 𝜎. 𝜎. 𝑑𝑡
¨ We then take the average:
¨ <𝜎. 𝑉 𝑡 . ∘ . 𝑑𝑊 𝑡 > = < 𝜎. 𝑉(𝑡) . ([). 𝑑𝑊 𝑡 > + <
+
,
. 𝜎. 𝜎. 𝑑𝑡 >
¨ And we use the fact that the ITO integral is a martingale
¨ <𝑉 𝑡 . ∘ . 𝑑𝑊 𝑡 > = < 𝑉(𝑡) . ([). 𝑑𝑊 𝑡 > +
+
,
. 𝜎, 𝑑𝑡
¨ <𝑉 𝑡 . ∘ . 𝑑𝑊 𝑡 > =
+
,
. 𝜎, 𝑑𝑡
83

84. Luc_Faucheux_2020
A quick note on averaging - VIII
¨ <𝑉 𝑡 . ∘ . 𝑑𝑊 𝑡 > =
+
,
. 𝜎, 𝑑𝑡
¨ We can plug this back into 𝑑 <
("
,
> = −2𝑘 <
("
,
>. 𝑑𝑡 + < 𝜎. 𝑉. (∘). 𝑑𝑊 >
¨ To recover:
¨ 𝑑 <
("
,
> = −2𝑘 <
("
,
>. 𝑑𝑡 +
+
,
. 𝜎, 𝑑𝑡
¨ And then solve again and get:
¨ < 𝑉&
, >& = < 𝑉 𝑡 , > = < 𝑉 0 , > −
>"
,?
. exp −2𝑘𝑡 +
>"
,?
¨ So again ITO and STRATO are equivalent, we will obtain the same solutions, as long as we do
not mix and match.
¨ ITO integral is a martingale but the rules of calculus are NOT the usual one
¨ STRATO integral is NOT a martingale but we can formally use the usual rules of calculus
84

85. Luc_Faucheux_2020
PDF for the Langevin
85

86. Luc_Faucheux_2020
PDF for the Langevin equation
¨ From the dynamics of the moments, it looks like the steady state solution could be
something like:
¨ ℎ 𝑣, 𝑡 → ∞ =
+
,G.-" <
. exp(
4$"
,-" <
)
¨ We also have:
¨ 𝑚; ∞ = 1
¨ 𝑚+ ∞ = 0
¨ 𝑚, ∞ =
>"
,?
¨ 𝑚+ 𝑣, 𝑡 = 𝑚+ 𝑣, 0 . exp(−𝑘𝑡)
¨ 𝑚, 𝑡 =
>"
,?
+ exp −2𝑘𝑡 . [𝑚, 0 −
>"
,?
]
86

87. Luc_Faucheux_2020
PDF for the Langevin equation - II
¨ We also know when looking at the SDE of the type (deck ITO – II):
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡 . 𝑑𝑡 + 𝑏(𝑡). ([). 𝑑𝑊
¨ 𝑋 𝑡. − 𝑋 𝑡* = ∫&/&*
&/&.
𝑑𝑋 𝑡 = ∫&/&*
&/&.
𝑎(𝑡). 𝑑𝑡) + ∫&/&*
&/&.
𝑏(𝑡). ([). 𝑑𝑊(𝑡)
¨ We had shown that the PDF: 𝑝 𝑥, 𝑡 =
+
,G(*P(&)
. 𝑒𝑥𝑝(−
(64I(&))"
,(*P(&)
)
¨ With
¨ 𝑅 𝑡 = 𝑋 𝑡 = 𝑋; + ∫&/&;
&
𝑎(𝑠). 𝑑𝑠 so 𝑅= 𝑡 = 𝑎(𝑡)
¨ 𝑉𝑎𝑟 𝑡 = 𝑉 𝑡 = p𝑏(𝑡),. 𝑡 = ∫:/;
:/&
𝑏 𝑠 ,. 𝑑𝑠 so 𝑉𝑎𝑟= 𝑡 = 𝑏 𝑡 ,
¨ Followed the PDE (ITO FORWARD):
¨
!
!&
𝑝 = −
!
!6
𝑎(𝑡). 𝑝 −
. & "
,
!
!6
𝑝 = −
!
!6
𝐽1 + 𝐽Q
87

88. Luc_Faucheux_2020
PDF for the Langevin equation - III
¨
!
!&
𝑝 = −
!
!6
𝑎(𝑡). 𝑝 −
. & "
,
!
!6
𝑝 = −
!
!6
𝐽1 + 𝐽Q
¨ 𝐽1 = 𝑎 𝑡 . 𝑝(𝑥, 𝑡) Forcing / drift current
¨ 𝐽Q = −
. & "
,
!
!6
𝑝 𝑥, 𝑡 = −
> & "
,
!
!6
𝑝 𝑥, 𝑡 = −𝐷(𝑡)
!
!6
𝑝(𝑥, 𝑡) Diffusion current
¨ In the Langevin case:
¨ 𝑑𝑉 𝑡 = 𝑎 𝑡, 𝑉 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑉 𝑡 . 𝑑𝑊
¨ 𝑎 𝑡, 𝑉 𝑡 = −𝑘𝑉
¨ 𝑏 𝑡, 𝑉 𝑡 = 𝜎
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
88

89. Luc_Faucheux_2020
PDF for the Langevin equation - IV
¨ 𝑅 𝑡 = 𝑉 𝑡 = 𝑉; + ∫&/&;
&
𝑎(𝑠). 𝑑𝑠 so 𝑅= 𝑡 = 𝑎(𝑡)
¨ 𝑉𝑎𝑟 𝑡 = 𝑉𝑎𝑟 𝑡 = p𝑏(𝑡),. 𝑡 = ∫:/;
:/&
𝑏 𝑠 ,. 𝑑𝑠 so 𝑉𝑎𝑟= 𝑡 = 𝑏 𝑡 ,
¨ 𝑎 𝑡, 𝑉 𝑡 = −𝑘𝑉
¨ 𝑏 𝑡, 𝑉 𝑡 = 𝜎
¨ 𝑉𝑎𝑟 𝑡 = 𝑉𝑎𝑟 𝑡 = p𝑏(𝑡),. 𝑡 = ∫:/;
:/&
𝜎,. 𝑑𝑠 = 𝜎,. 𝑡 so 𝑉= 𝑡 = 𝜎,
¨ HOWEVER, for 𝑅 𝑡 = 𝑋 𝑡 = 𝑋; + ∫&/&;
&
𝑎(𝑠). 𝑑𝑠, we are stuck because we only looked at
the case 𝑎 𝑡, 𝑉 𝑡 , not 𝑎 𝑉 𝑡
¨ So this is going to be a little tricky, but based on what we think is the steady state solution,
we could try to be as lucky as Bachelier in 1900 and maybe guess something like
¨ 𝑝 𝑣, 𝑡 =
+
,G-" &
. 𝑒𝑥𝑝(−
($4-+ & )"
,-" &
)
89

90. Luc_Faucheux_2020
PDF for the Langevin equation - V
¨ Let’s try indeed:
¨ 𝑝 𝑣, 𝑡 =
+
,G-" &
. 𝑒𝑥𝑝(−
($4-+ & )"
,-" &
)
¨ BUT for now let’s not equate 𝑚+ 𝑡 and 𝑚, 𝑡 to the functions:
¨ 𝑚+ 𝑡 = 𝑚+ 0 . exp(−𝑘𝑡)
¨ 𝑚, 𝑡 =
>"
,?
+ exp −2𝑘𝑡 . [𝑚, 0 −
>"
,?
]
¨ We could try from the get-go and see if 𝑝 𝑣, 𝑡 verifies:
¨
!"($,&|(!,&!)
!&
= −
!
!$
(−𝑘𝑣). 𝑝 𝑣, 𝑡 𝑉*, 𝑡* −
!
!$
[
+
,
. [𝜎, . 𝑝(𝑣, 𝑡|𝑉*, 𝑡*)]
¨ Or keep 𝑚+ 𝑡 and 𝑚, 𝑡 for a little longer to try to simplify the derivation
90

91. Luc_Faucheux_2020
PDF for the Langevin equation - VI
¨ 𝑝 𝑣, 𝑡 =
+
,G-" &
. 𝑒𝑥𝑝(−
($4-+ & )"
,-" &
)
¨
!" $,&
!$
=
+
,G-" &
.
!
!$
𝑒𝑥 𝑝 −
$4-+ &
"
,-" &
=
+
,G-" &
.
4, $4-+ &
,-" &
. 𝑒𝑥 𝑝 −
$4-+ &
"
,-" &
¨
!" $,&
!$
=
+
,G-" &
. 𝑒𝑥 𝑝 −
$4-+ &
"
,-" &
.
4 $4-+ &
-" &
=
4 $4-+ &
-" &
. 𝑝 𝑣, 𝑡
¨
!"" $,&
!R" =
!
!$
!
!$
𝑝 𝑣, 𝑡 =
!
!$
4 $4-+ &
-" &
. 𝑝 𝑣, 𝑡 =
4 $4-+ &
-" &
.
!" $,&
!$
+ 𝑝 𝑣, 𝑡 .
!
!$
4 $4-+ &
-" &
¨
!"" $,&
!R" =
4 $4-+ &
-" &
.
4 $4-+ &
-" &
. 𝑝 𝑣, 𝑡 +
4+
-" &
. 𝑝 𝑣, 𝑡
91

92. Luc_Faucheux_2020
PDF for the Langevin equation - VII
¨ 𝑝 𝑣, 𝑡 =
+
,G-" &
. 𝑒𝑥𝑝(−
($4-+ & )"
,-" &
)
¨
!" $,&
!$
=
+
,G-" &
. 𝑒𝑥 𝑝 −
$4-+ &
"
,-" &
.
4 $4-+ &
-" &
=
4 $4-+ &
-" &
. 𝑝 𝑣, 𝑡
¨
!"
!$
=
4 $4-+
-"
. 𝑝
¨
!"" $,&
!R" =
4 $4-+ &
-" &
.
4 $4-+ &
-" &
. 𝑝 𝑣, 𝑡 +
4+
-" &
. 𝑝 𝑣, 𝑡
¨
!""
!R" =
4 $4-+
-"
.
4 $4-+
-"
. 𝑝 +
4+
-"
. 𝑝
¨ Dropping the explicit dependencies for sake of notation
92

93. Luc_Faucheux_2020
PDF for the Langevin equation - VIII
¨ 𝑝 𝑣, 𝑡 =
+
,G-" &
. 𝑒𝑥𝑝(−
($4-+ & )"
,-" &
)
¨
!" $,&
!&
=
!
!&
+
,G-" &
. 𝑒𝑥 𝑝 −
$4-+ &
"
,-" &
+
+
,G-" &
!
!&
𝑒𝑥 𝑝 −
$4-+ &
"
,-" &
¨
!" $,&
!&
=
4+
,
+
,G-"
+
,G-"
. 2𝜋𝑚,
=
. 𝑒𝑥 𝑝 −
$4-+
"
,-"
+
+
,G-"
𝑒𝑥 𝑝 −
$4-+
"
,-"
!
!&
[
4 $4-+
"
,-"
]
¨
!" $,&
!&
=
4+
,
-"
#
-"
. 𝑝 + 𝑝.
!
!&
4 $4-+
"
,-"
=
4+
,
-"
#
-"
. 𝑝 + 𝑝.
7 $4-+
"
,-"-"
𝑚,
= + 2 𝑣 − 𝑚+ . 𝑚+
= +
,-"
¨
!" $,&
!&
=
4+
,
-"
#
-"
. 𝑝 + 𝑝.
7 $4-+
"
,-"-"
𝑚,
= + 𝑣 − 𝑚+ . 𝑚+
= .
+
-"
93

94. Luc_Faucheux_2020
PDF for the Langevin equation - IX
¨
!" $,&
!&
=
4+
,
-"
#
-"
. 𝑝 + 𝑝.
7 $4-+
"
,-"-"
𝑚,
= + 𝑣 − 𝑚+ . 𝑚+
= .
+
-"
¨
!"
!$
=
4 $4-+
-"
. 𝑝
¨
!""
!R" =
4 $4-+
-"
.
4 $4-+
-"
. 𝑝 +
4+
-"
. 𝑝
¨ At this point, we can either plug in those equations the formulas for 𝑚+ 𝑡 and 𝑚, 𝑡 :
¨ 𝑚+ 𝑡 = 𝑚+ 0 . exp(−𝑘𝑡)
¨ 𝑚+
=
𝑡 = −𝑘. 𝑚+ 𝑡 = −𝑘. 𝑚+ 0 . exp(−𝑘𝑡)
¨ 𝑚, 𝑡 =
>"
,?
+ exp −2𝑘𝑡 . [𝑚, 0 −
>"
,?
]
¨ 𝑚,
= 𝑡 = −2𝑘. exp −2𝑘𝑡 . 𝑚, 0 −
>"
,?
= −2𝑘. exp −2𝑘𝑡 . [𝑚, 0 − 𝑚, ∞ ]
94

95. Luc_Faucheux_2020
PDF for the Langevin equation - X
¨ We know that we are after something like:
¨
!"($,&|(!,&!)
!&
= −
!
!$
(−𝑘𝑣). 𝑝 𝑣, 𝑡 𝑉*, 𝑡* −
!
!$
[
+
,
. [𝜎, . 𝑝(𝑣, 𝑡|𝑉*, 𝑡*)]
¨
!"
!&
= −
!
!$
(−𝑘𝑣). 𝑝 −
!
!$
[
+
,
. [𝜎, . 𝑝
¨ We see a bunch of terms that appear as orders of 𝑣 − 𝑚+
¨ So let’s rewrite above as:
¨
!"
!&
= −
!
!$
(−𝑘𝑣). 𝑝 −
!
!$
[
+
,
. [𝜎, . 𝑝 =
!
!$
[𝑘𝑣𝑝] +
>"
,
!""
!R"
¨
!"
!&
=
!
!$
[𝑘 𝑣 − 𝑚+ 𝑝 + 𝑘𝑚+ 𝑝] +
>"
,
!""
!R"
¨
!"
!&
= 𝑘 𝑣 − 𝑚+
!"
!$
+ 𝑘𝑝 + 𝑘𝑚+
!"
!$
+
>"
,
!""
!R"
95

96. Luc_Faucheux_2020
PDF for the Langevin equation - XI
¨ Plugging the expressions we derived:
¨
!"
!&
= 𝑘 𝑣 − 𝑚+
!"
!$
+ 𝑘𝑝 + 𝑘𝑚+
!"
!$
+
>"
,
!""
!R"
¨ Left Hand Side
¨
!" $,&
!&
=
4+
,
-"
#
-"
. 𝑝 + 𝑝.
7 $4-+
"
,-"-"
𝑚,
=
+ 𝑣 − 𝑚+ . 𝑚+
=
.
+
-"
¨ Right Hand Side
¨ 𝑘 𝑣 − 𝑚+ .
4 $4-+
-"
. 𝑝 + 𝑘𝑝 + 𝑘𝑚+.
4 $4-+
-"
. 𝑝 +
>"
,
. [
4 $4-+
-"
.
4 $4-+
-"
. 𝑝 +
4+
-"
. 𝑝]
96

97. Luc_Faucheux_2020
PDF for the Langevin equation - XII
¨ Ordering the terms in order of 𝑣 − 𝑚+
¨ Left Hand Side
¨
!" $,&
!&
=
4+
,
-"
#
-"
. 𝑝 + 𝑝. 𝑣 − 𝑚+ . 𝑚+
= .
+
-"
+ 𝑝.
$4-+
"
,-"-"
𝑚,
=
¨ Right Hand Side
¨ 𝑘𝑝 +
>"
,
.
4+
-"
. 𝑝 + 𝑘𝑚+.
4 $4-+
-"
. 𝑝 + 𝑝. 𝑣 − 𝑚+
,. [−𝑘
+
-"
+
>"
,-"-"
]
97

98. Luc_Faucheux_2020
PDF for the Langevin equation - XII
¨ Terms in 𝑣 − 𝑚+
;:
¨
4+
,
-"
#
-"
. 𝑝 = 𝑘𝑝 +
>"
,
.
4+
-"
. 𝑝
¨ Terms in 𝑣 − 𝑚+
+:
¨ 𝑝. 𝑣 − 𝑚+ . 𝑚+
=
.
+
-"
= 𝑘𝑚+.
4 $4-+
-"
. 𝑝
¨ Terms in 𝑣 − 𝑚+
,:
¨ 𝑝.
$4-+
"
,-"-"
𝑚,
=
= 𝑝. 𝑣 − 𝑚+
,. [−𝑘
+
-"
+
>"
,-"-"
]
98

99. Luc_Faucheux_2020
PDF for the Langevin equation - XIII
¨ Terms in 𝑣 − 𝑚+
;:
¨
+
,
𝑚,
=
= −𝑘𝑚, +
>"
,
¨ Terms in 𝑣 − 𝑚+
+:
¨ 𝑚+
=
= −𝑘𝑚+
¨ Terms in 𝑣 − 𝑚+
,:
¨
+
,
𝑚,
=
= −𝑘𝑚, +
>"
,
¨ If all those equations are verified, then our guess will indeed forward PDE for the Langevin
PDF.
¨ Note that the set of 3 equations actually reduces to only 2. I do not have much intuition
why it is, but again we only had 2 moments 𝑚+ 𝑡 and 𝑚, 𝑡 , so maybe if our guess was
incorrect we would have gotten inconsistent equations, meaning that we needed a 3rd
moment in our guess ?
99

100. Luc_Faucheux_2020
PDF for the Langevin equation - XIV
¨ 𝑚+
=
= −𝑘𝑚+
¨ Turns out that this is EXACTLY the equation we had derived from the SDE.
¨ So obviously if we plug into that equation the formula : 𝑚+ 𝑡 = 𝑚+ 0 . exp(−𝑘𝑡), we
verify the ODE or alternatively we can solve it and we will recover the above formula
¨
+
,
𝑚,
= = −𝑘𝑚, +
>"
,
¨ Turns out again (boy oh boy aren’t we lucky!) that this is the same ODE for 𝑚, 𝑡 that we
had derived from the SDE, or from the dynamics of the moments section (from the PDE).
¨ So we can solve and recover the formula, or apply the formula in the ODE to convince
ourselves, but we have:
¨ 𝑚, 𝑡 =
>"
,?
+ exp −2𝑘𝑡 . [𝑚, 0 −
>"
,?
]
100

101. Luc_Faucheux_2020
PDF for the Langevin equation - XV
¨ So we finally have a solution for the Langevin PDF and it looks like this:
¨ 𝑝 𝑣, 𝑡 = 𝑝 𝑣, 𝑡|𝑉; = 𝑚+ 0 , 𝑡 = 0 =
+
,G-" &
. 𝑒𝑥𝑝(−
($4-+ & )"
,-" &
)
¨ 𝑚+ 𝑡 = 𝑚+ 0 . exp(−𝑘𝑡)
¨ 𝑚, 𝑡 =
>"
,?
+ exp −2𝑘𝑡 . [𝑚, 0 −
>"
,?
]
¨ 𝑚, 𝑡 = 𝑚, ∞ + exp −2𝑘𝑡 . [𝑚, 0 − 𝑚, ∞ ]
¨ 𝑚, ∞ =
>"
,?
101

102. Luc_Faucheux_2020
PDF for the Langevin equation - XVI
¨ 𝑝 𝑣, 𝑡 = 𝑝 𝑣, 𝑡|𝑉; = 𝑚+ 0 , 𝑡 = 0 =
+
,G-" &
. 𝑒𝑥𝑝(−
($4-+ & )"
,-" &
)
¨ SO we also need to impose the condition:
¨ 𝑝 𝑣, 𝑡 = 0|𝑉; = 𝑚+ 0 , 𝑡 = 0 = 𝛿(𝑣 − 𝑚+ 0 )
¨ 𝑚, 𝑡 = 𝑚, ∞ + exp −2𝑘𝑡 . [𝑚, 0 − 𝑚, ∞ ]
¨ 𝑚, ∞ =
>"
,?
= 𝐷/𝑘 to simplify somewhat with the usual Diffusion coefficient 𝐷 =
>"
,
¨ 𝑚, 𝑡 =
Q
?
[1 − exp −2𝑘𝑡 ] + 𝑚, 0 . exp −2𝑘𝑡
¨ When 𝑡 → 0, 𝑚, 𝑡 =
Q
?
. 1 − 1 − 2𝑘𝑡 + 𝕆 𝑡, + 𝑚, 0 . 1 − 2𝑘𝑡 + 𝕆 𝑡,
¨ 𝑚, 𝑡 = 𝑚, 0 + 2𝐷𝑡 + 𝕆 𝑡,
¨ 𝑚+ 𝑡 = 𝑚+ 0 . exp −𝑘𝑡 = 𝑚+ 0 + 𝕆 𝑡
102

103. Luc_Faucheux_2020
PDF for the Langevin equation - XVII
¨ So if 𝑚, 0 <> 0,
¨ 𝑝 𝑣, 𝑡 = 0 = 𝑝 𝑣, 𝑡 = 0|𝑉; = 𝑚+ 0 , 𝑡 = 0 =
+
,G-" ;
. 𝑒𝑥𝑝(−
($4-+ ; )"
,-" ;
)
¨ That is a Gaussian centered around 𝑚, 0 of width 𝑚, 0
¨ It is still normalized but does not converge to the Dirac peak 𝛿(𝑣 − 𝑚+ 0 )
¨ So we have to enforce 𝑚, 0 = 0
¨ 𝑚, 𝑡 =
Q
?
[1 − exp −2𝑘𝑡 ] + 𝑚, 0 . exp −2𝑘𝑡 =
Q
?
[1 − exp −2𝑘𝑡 ]
¨ We can rewrite the PDF as:
¨ 𝑝 𝑣, 𝑡|𝑚+ 0 , 𝑡 = 0 =
?
,GQ.(+4STU 4,?& )
. 𝑒𝑥𝑝(−𝑘
($4-+ ; .STU 4?& )"
,Q.(+4STU 4,?& )
)
103

104. Luc_Faucheux_2020
PDF for the Langevin equation - XVIII
¨ After much calculation, this is the celebrated Langevin PDF:
¨ 𝑝 𝑣, 𝑡|𝑚+ 0 , 𝑡 = 0 =
?
,GQ.(+4STU 4,?& )
. 𝑒𝑥𝑝(−𝑘
($4-+ ; .STU 4?& )"
,Q.(+4STU 4,?& )
)
¨ SMALL TIME LIMIT
¨ IF 𝑡 → 0
?
Q.(+4STU 4,?& )
=
+
,Q&
+ 𝕆 𝑡,
¨ 𝑝 𝑣, 𝑡|𝑚+ 0 , 𝑡 = 0 →
+
VGQ&
. 𝑒𝑥𝑝(−
($4-+ ; )"
VQ&
)
¨ At short time scales (underdamped regime), the Langevin diffuses as a regular diffusion
process
¨ 𝑑𝑉 𝑡 = −𝑘𝑉 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊 → 𝜎. 𝑑𝑊
104

105. Luc_Faucheux_2020
PDF for the Langevin equation - XIX
¨ SMALL 𝑘 limit
¨ IF 𝑘 → 0
?
Q.(+4STU 4,?& )
=
+
,Q&
+ 𝕆 𝑘,
¨ 𝑝 𝑣, 𝑡|𝑚+ 0 , 𝑡 = 0 →
+
VGQ&
. 𝑒𝑥𝑝(−
($4-+ ; )"
VQ&
)
¨ This is expected since when 𝑘 → 0 we should recover the usual diffusion:
¨ 𝑑𝑉 𝑡 = −𝑘𝑉 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊 → 𝜎. 𝑑𝑊
105

106. Luc_Faucheux_2020
PDF for the Langevin equation - XX
¨ STEADY STATE LIMIT
¨ IF 𝑡 → ∞
?
Q.(+4STU 4,?& )
=
?
Q
+ 𝕆 𝑡4+
¨ 𝑝 𝑣, 𝑡|𝑚+ 0 , 𝑡 = 0 =
?
,GQ.(+4STU 4,?& )
. 𝑒𝑥𝑝(−𝑘
($4-+ ; .STU 4?& )"
,Q.(+4STU 4,?& )
)
¨ 𝑝 𝑣, 𝑡 → ∞|𝑚+ 0 , 𝑡 = 0 =
?
,GQ
. 𝑒𝑥𝑝(−𝑘
$"
,Q
)
¨ This is referred to as the “invariant Gaussian distribution”
106

107. Luc_Faucheux_2020
PDF for the Langevin equation - XXI
¨ In the case where 𝑘 → 0, the SDE becomes :
¨ 𝑑𝑉 𝑡 = −𝑘𝑉 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊 = 𝜎. 𝑑𝑊
¨ And we should recover the usual Brownian diffusion
¨ 𝑚+ 𝑡 = 𝑚+ 0 . exp −𝑘𝑡 → 𝑚+ 0
¨ 𝑚, 𝑡 = 𝑚, ∞ + exp −2𝑘𝑡 . 𝑚, 0 − 𝑚, ∞ → 𝑚, 0
¨ 𝑝 𝑣, 𝑡 = 𝑝 𝑣, 𝑡|𝑉; = 𝑚+ 0 , 𝑡 = 0 =
+
,G-" &
. 𝑒𝑥𝑝(−
($4-+ & )"
,-" &
)
¨ 𝑝 𝑣, 𝑡 = 𝑝 𝑣, 𝑡|𝑉; = 𝑚+ 0 , 𝑡 = 0 =
+
,G-" ;
. 𝑒𝑥𝑝(−
($4-+ ; )"
,-" ;
)
107

108. Luc_Faucheux_2020
PDF for the Langevin equation - XXII
¨ 𝑝 𝑣, 𝑡|𝑉 𝑡* , 𝑡* =
?
,GQ.(+4@'"$(&'&!))
. 𝑒𝑥𝑝(−𝑘
($4-+ &! .@'$(&'&!))"
,Q(+4@'"$(&'&!))
)
¨ The Langevin process is Gaussian (the PDF can be expressed as a Gaussian function)
¨ The Langevin process is Markov (the PDF only depends on 𝑉 𝑡* , 𝑡* and not on the entire
history before)
¨ 𝑝 𝑣, 𝑡|{𝑉 𝑠 , 𝑠 ≤ 𝑡*} = 𝑝 𝑣, 𝑡|𝑉 𝑡* , 𝑡*
¨ The Langevin process is stationary (only depends on (𝑡 − 𝑡*))
¨ 𝑝 𝑣, 𝑡 + ℎ|𝑉 𝑡* + ℎ = 𝑉*, 𝑡* + ℎ = 𝑝 𝑣, 𝑡|𝑉*, 𝑡*
¨ The increments of the Langevin process are NOT independents. Indeed the increments are
not even uncorrelated (as opposed to a Wiener process)
¨ The correlation function decays as an exponential. In some textbooks they base the
definition of the process on the knowledge of the auto-correlation function, as an
equivalent starting point
108

109. Luc_Faucheux_2020
Langevin PDF
Via the Distribution Function
109

110. Luc_Faucheux_2020
Langevin PDF via the Distribution function
¨ So there I have somewhat of a confession to make, I was already a couple hundred pages
into writing those notes (deck on Bachelier, Black-Sholes, binomial trees, ITO lemma, Risk
management,…) when I bought the book below. I was tempted to throw my notes in the
trash because this book is awesome and has pretty much all you want, and more…
110

111. Luc_Faucheux_2020
Langevin PDF via the Distribution function – II
¨ In particular, on page 31, the author goes through a derivation of the Langevin PDF that is
truly awesome using the distribution function:
¨ PDF Probability Density Function: 𝑝((𝑣, 𝑡)
¨ Distribution function : 𝑉(𝑣, 𝑡)
¨ 𝑃( 𝑣, 𝑡 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑉 ≤ 𝑣, 𝑡 = ∫K/4<
K/$
𝑝( 𝑦, 𝑡 . 𝑑𝑦
¨ 𝑝((𝑣, 𝑡) =
!
!$
𝑃( 𝑣, 𝑡
¨ 𝑝( 𝑣, 𝑡 = 𝑝 𝑣, 𝑡 = 𝑝 𝑣, 𝑡|𝑉; = 𝑚+ 0 , 𝑡 = 0
111

112. Luc_Faucheux_2020
Langevin PDF via the Distribution function – III
¨ We know the distribution function for the Brownian motion 𝑊(𝑡)
¨ [𝑊 𝑡. − 𝑊(𝑡*)] is 𝑁(0, 𝑡. − 𝑡* )
¨ [𝑊 𝑡. − 𝑊(𝑡*)] is normally distributed according to the Gaussian function:
¨ ℎ 𝑥, 𝑡 =
+
,G&
. exp(
46"
,&
)
¨ 𝑃W 𝑤, 𝑡|𝑊 𝑡* , 𝑡* = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑊(𝑡) ≤ 𝑤, 𝑡|𝑊 𝑡* , 𝑡* = ∫K/4<
K/X
𝑝W 𝑦, 𝑡 . 𝑑𝑦
¨ 𝑃W 𝑤, 𝑡|𝑊 𝑡* , 𝑡* = ∫K/4<
K/X +
,G(&4&!)
. exp(
4(K4W &! )"
,(&4&!)
) . 𝑑𝑦
¨ Sometimes for ease of notation, choosing 𝑊 𝑡* = 0 and 𝑡* = 0
¨ 𝑃W 𝑤, 𝑡|0,0 = ∫K/4<
K/X +
,G&
. exp
46"
,&
. 𝑑𝑦
¨ 𝑝W 𝑤, 𝑡 =
!
!X
𝑃W 𝑤, 𝑡 =
+
,G&
. exp(
4X"
,&
)
112

113. Luc_Faucheux_2020
Langevin PDF via the Distribution function – IV
¨ Define now the Langevin process as :
¨ 𝑉 𝑡 =
>"
?
. exp −𝑘𝑡 . 𝑊(exp 2𝑘𝑡 )
¨ So in some ways if you already have a Brownian motion {𝑊(𝑡Y)}, for example on a computer
simulation, you can simulate a Langevin process {𝑉(𝑡Y)} by mapping:
¨ 𝑖 → 𝑗 so that 𝑡N = exp(2𝑘𝑡Y)
¨ Pick the value of {𝑊(𝑡N)}
¨ Multiply by
>"
?
. exp −𝑘𝑡Y
¨ That would be a way to replicate a Langevin process from a given Brownian process
113

114. Luc_Faucheux_2020
Langevin PDF via the Distribution function – V
¨ 𝑉 𝑡 =
>"
?
. exp −𝑘𝑡 . 𝑊(exp 2𝑘𝑡 )
¨ 𝑃( 𝑣, 𝑡|𝑉 𝑡* , 𝑡* = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑉(𝑡) ≤ 𝑣, 𝑡|𝑉 𝑡* , 𝑡* = 𝑃𝑟𝑜𝑏 𝑉(𝑡) ≤ 𝑣, 𝑡|𝑉 𝑡* , 𝑡*
¨ 𝑃( 𝑣, 𝑡|𝑉 𝑡* , 𝑡* = 𝑃𝑟𝑜𝑏
>"
?
. exp −𝑘𝑡 . 𝑊(exp 2𝑘𝑡 ) ≤ 𝑣, 𝑡|𝑉 𝑡* , 𝑡*
¨ 𝑃( 𝑣, 𝑡|𝑉 𝑡* , 𝑡* = 𝑃𝑟𝑜𝑏
>"
?
𝑒4?&. 𝑊(𝑒,?&) ≤ 𝑣, 𝑡|𝑉 𝑡* = 𝑉*, 𝑡*
¨ 𝑃( 𝑣, 𝑡|𝑉 𝑡* , 𝑡* = 𝑃𝑟𝑜𝑏
>"
?
𝑒4?&. 𝑊(𝑒,?&) ≤ 𝑣, 𝑡|
>"
?
𝑒4?&!. 𝑊(𝑒,?&!) = 𝑉*, 𝑡*
114

115. Luc_Faucheux_2020
Langevin PDF via the Distribution function – VI
¨ 𝑃( 𝑣, 𝑡|𝑉 𝑡* , 𝑡* = 𝑃𝑟𝑜𝑏 𝑊(𝑒,?&) ≤
?
>" 𝑣𝑒?&, 𝑡|𝑊(𝑒,?&!) =
?
>" 𝑉* 𝑒?&!, 𝑡*
¨ 𝑃 = 𝑃𝑟𝑜𝑏 𝑊(𝑒,?&) ≤
?
>" 𝑣𝑒?&, 𝑡|𝑊(𝑒,?&!) =
?
>" 𝑉* 𝑒?&!, 𝑡*
¨ 𝑃 = 𝑃W 𝑊(𝑒,?&) ≤
?
>" 𝑣𝑒?&, 𝑡|𝑊(𝑒,?&!) =
?
>" 𝑉* 𝑒?&!
¨ 𝑃 = ∫K/4<
K/
$
*"$@$&
+
,G(@"$&4@"$&!)
. exp(
4(K4
$
*"(!@$&!)"
,(@"$&4@"$&!)
) . 𝑑𝑦
115

116. Luc_Faucheux_2020
Langevin PDF via the Distribution function – VII
¨ 𝑃 = ∫K/4<
K/
$
*"$@$&
+
,G(@"$&4@"$&!)
. exp(
4(K4
$
*"(!@$&!)"
,(@"$&4@"$&!)
) . 𝑑𝑦
¨ Changing to the variable: 𝑦 =
?
>" 𝑒?&. 𝜌 with 𝑑𝑦 =
?
>" 𝑒?&. 𝑑𝜌
¨ 𝑃 = ∫Z/4<
Z/$ +
,G(@"$&4@"$&!)
. exp(
4(Z
$
*"@$&4
$
*"(!@$&!)"
,(@"$&4@"$&!)
) . 𝑑𝜌
?
>" 𝑒?&
¨ 𝑃 = ∫Z/4<
Z/$ @$&
,G(@"$&4@"$&!)
. exp(
4(Z
$
*"@$&4
$
*"(!@$&!)"
,(@"$&4@"$&!)
) .
?
>" . 𝑑𝜌
116

117. Luc_Faucheux_2020
Langevin PDF via the Distribution function – VIII
¨ 𝑃 = ∫Z/4<
Z/$ ?
,G>"(+4@'"$(&'&!))
. exp(
4(Z4(!.@'$(&'&!))"
,(>"/?).(+4@'"$(&'&!))
) . 𝑑𝜌
¨ 𝑃 = 𝑃( 𝑣, 𝑡|𝑉 𝑡* , 𝑡*
¨ 𝑝((𝑣, 𝑡) =
!
!$
𝑃( 𝑣, 𝑡
¨ 𝑝( 𝑣, 𝑡 𝑉 𝑡* , 𝑡* =
?
,G>"(+4@'"$(&'&!))
. exp(
4(Z4(!.@'$(&'&!))"
,(>"/?).(+4@'"$(&'&!))
)
¨ Using 𝐷 =
>"
,
¨ 𝑝 𝑣, 𝑡|𝑉 𝑡* , 𝑡* =
?
,GQ.(+4@'"$(&'&!))
. 𝑒𝑥𝑝(−𝑘
($4-+ &! .@'$(&'&!))"
,Q(+4@'"$(&'&!))
)
¨ This is EXACTLY the same PDF we have already arrived at !!
117

118. Luc_Faucheux_2020
Langevin PDF via the Distribution function – IX
¨ So we know that
¨ 𝑉 𝑡 =
>"
?
. exp −𝑘𝑡 . 𝑊(exp 2𝑘𝑡 )
¨ Is the Langevin process following the SDE: 𝑑𝑉 𝑡 = −𝑘𝑉 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ Deriving the PDF was surprisingly easy (I broke it down to make it very explicit, but Pavliotis
does it in 6 lines on page 31
¨ It is also avoiding pages and pages of algebra using the ansatz (guess) method.
¨ This is quite elegant
118

119. Luc_Faucheux_2020
Langevin Auto correlation through the Distribution function
¨ 𝑉 𝑡 =
>"
?
. exp −𝑘𝑡 . 𝑊(exp 2𝑘𝑡 )
¨ So:
¨ 𝔼 𝑉 𝑡. . 𝑉 𝑡* =
>"
?
.
>"
?
. exp −𝑘𝑡. . exp −𝑘𝑡* . 𝔼 𝑊(exp 2𝑘𝑡* ). 𝑊(exp 2𝑘𝑡. )
¨ 𝔼 𝑉 𝑡. . 𝑉 𝑡* =
>"
?
. exp −𝑘𝑡. . exp −𝑘𝑡* . min(exp 2𝑘𝑡* , exp 2𝑘𝑡. )
¨ With 𝑡. > 𝑡*
¨ 𝔼 𝑉 𝑡. . 𝑉 𝑡* =
>"
?
. exp 𝑘𝑡. . exp −𝑘𝑡* and 𝔼 𝑉 𝑡* . 𝑉 𝑡* =
>"
?
¨ 𝔼 𝑉 𝑡. . 𝑉 𝑡* = 𝔼 𝑉 𝑡* . 𝑉 𝑡* . exp 𝑘𝑡. . exp −𝑘𝑡*
¨ Again, so quick and elegant !
119

120. Luc_Faucheux_2020
Langevin versus GBM
Geometric Brownian motion
Dynamics of moments
120

121. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion)
¨ Let’s redo the analysis on the dynamics of the moments for the GBM (Geometric Brownian
Motion).
¨ GBM was introduced to model stock prices. It is the first process you see in textbooks when
they go on deriving Black-Sholes
¨ However, recently a lot more people woke up to the advantages of the Langevin approach
(also called OU or Ornstein-Uhlenbeck)
¨ The Langevin has a lot of advantages that the GBM does not possess
¨ In particular we are going to show that the higher order moments of the GBM do not always
converge (as the OU-Langevin do).
¨ In the 1970, Salomon Brothers developed a 3-factor OU (Langevin) model with mean
reversion and correlation, as well as their own skew distribution, well ahead of their time.
¨ This model slowly percolated through the industry and is sometimes called the “2+” or “2+
IRMA”.
¨ Ask anyone who worked on rates options and this model is quite famous
121

122. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - II
¨ The Langevin equation was written with the particle velocity 𝑉 𝑡 as the stochastic variable
¨ We usually write the GBM with the stock (security) 𝑆(𝑡) as the stochastic variable or also
sometimes with just the usual stochastic notation 𝑋(𝑡)
¨ The canonical GBM is given by:
¨ 𝑑𝑋 𝑡 = 𝜇. 𝑋 𝑡 . 𝑑𝑡 + 𝜎. 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ Note that this is of the form:
¨ 𝑑𝑋 𝑡 = 𝑎(𝑋, 𝑡). 𝑑𝑡 + 𝑏(𝑋, 𝑡). ([). 𝑑𝑊(𝑡)
¨ With
¨ 𝑎 𝑋, 𝑡 = 𝜇. 𝑋 𝑡
¨ 𝑏 𝑋, 𝑡 = 𝜎. 𝑋 𝑡
¨ So we have to be a little careful about ITO versus STRATANOVITCH
122

123. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - III
¨ ITO SDE : 𝑑𝑋 𝑡 = 𝑎(𝑋, 𝑡). 𝑑𝑡 + 𝑏(𝑋, 𝑡). ([). 𝑑𝑊(𝑡)
¨ ITO SDE: 𝑑𝑋 𝑡 = 𝜇. 𝑋 𝑡 . 𝑑𝑡 + 𝜎. 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ The corresponding STRATO SDE is: 𝑑𝑋 𝑡 = 6𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 7𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ With: 𝑎 𝑋, 𝑡 = 𝜇. 𝑋 𝑡 and 𝑏 𝑋, 𝑡 = 𝜎. 𝑋 𝑡
¨ 6𝑎 𝑡, 𝑋 𝑡 = 𝑎 𝑋 𝑡 , 𝑡 −
+
,
. 𝑏 𝑡, 𝑋 𝑡 .
!
!3
𝑏 𝑡, 𝑋 𝑡 = 𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎
¨ 7𝑏 𝑡, 𝑋 𝑡 = 𝑏 𝑋 𝑡 , 𝑡 = 𝜎. 𝑋 𝑡
¨ STRATO SDE: 𝑑𝑋 𝑡 = [𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎]. 𝑑𝑡 + 𝜎. 𝑋 𝑡 . (∘). 𝑑𝑊
123

124. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - IV
¨ ITO SDE: 𝑑𝑋 𝑡 = 𝜇. 𝑋 𝑡 . 𝑑𝑡 + 𝜎. 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ ITO SIE:
¨ 𝑋 𝑡. − 𝑋 𝑡* = ∫&/&*
&/&.
𝑑𝑋 𝑡 = ∫&/&*
&/&.
𝜇. 𝑋 𝑡 . 𝑑𝑡) + ∫&/&*
&/&.
𝜎. 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ STRATO SDE: 𝑑𝑋 𝑡 = [𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎]. 𝑑𝑡 + 𝜎. 𝑋 𝑡 . (∘). 𝑑𝑊
¨ STRATO SIE:
¨ 𝑋 𝑡. − 𝑋 𝑡* = ∫&/&*
&/&.
[𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎]. 𝑑𝑡) + ∫&/&*
&/&.
𝜎. 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
124

125. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - V
¨ ITO SDE: 𝑑𝑋 𝑡 = 𝜇. 𝑋 𝑡 . 𝑑𝑡 + 𝜎. 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ ITO lemma on 𝑓 𝑋 𝑡 = ln(𝑋 𝑡 )
¨ 𝑑𝑓 𝑋 𝑡 = 𝑑(ln 𝑋 𝑡 ) =
!0
!3
. [ . 𝑑𝑋 +
+
,
!"0
!3" . [ . 𝑑𝑋 , +
!0
!3
. 𝑑𝑡
¨ 𝑑𝑓 𝑋 𝑡 = 𝑑(ln 𝑋 𝑡 ) =
+
3
. [ . 𝑑𝑋 +
4+
,
+
3" . [ . 𝜎. 𝑋 𝑡
,
. 𝑑𝑡
¨ 𝑑𝑓 𝑋 𝑡 =
+
3
. [ . {𝜇. 𝑋 𝑡 . 𝑑𝑡 + 𝜎. 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)} +
4+
,
+
3" . [ . 𝜎. 𝑋 𝑡
,
. 𝑑𝑡
¨ 𝑑(ln 𝑋 𝑡 ) = 𝜇 −
>"
,
. 𝑑𝑡 + 𝜎. [ . 𝑑𝑊(𝑡)
¨ ITO SDE on ln 𝑋 𝑡 : 𝑑(ln 𝑋 𝑡 ) = 𝜇 −
>"
,
. 𝑑𝑡 + 𝜎. [ . 𝑑𝑊(𝑡)
125

126. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - VI
¨ ITO SDE on ln 𝑋 𝑡 : 𝑑(ln 𝑋 𝑡 ) = 𝜇 −
>"
,
. 𝑑𝑡 + 𝜎. [ . 𝑑𝑊(𝑡)
¨ Note that since 𝜎 is constant in that case 𝜎. [ . 𝑑𝑊 𝑡 = 𝜎. ∘ . 𝑑𝑊 𝑡 = 𝜎. 𝑑𝑊(𝑡)
¨ ITO SIE on ln 𝑋 𝑡
¨ 𝑙𝑛𝑋 𝑡. − 𝑙𝑛𝑋 𝑡* = ∫&/&*
&/&.
𝑑𝑙𝑛𝑋 𝑡 = ∫&/&*
&/&.
𝜇 −
>"
,
. 𝑑𝑡) + ∫&/&*
&/&.
𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑙𝑛𝑋 𝑡. − 𝑙𝑛𝑋 𝑡* = ∫&/&*
&/&.
𝑑𝑙𝑛𝑋 𝑡 = 𝜇 −
>"
,
. (𝑡. − 𝑡*) + 𝜎[𝑊 𝑡. − 𝑊(𝑡*)]
¨ ln[
3 &)
3 &!
] = 𝜇 −
>"
,
. (𝑡. − 𝑡*) + 𝜎[𝑊 𝑡. − 𝑊(𝑡*)]
¨ 𝑋 𝑡. = 𝑋 𝑡. . exp{ 𝜇 −
>"
,
. 𝑡. − 𝑡* + 𝜎 𝑊 𝑡. − 𝑊 𝑡* }
126

127. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - VII
¨ STRATO SDE: 𝑑𝑋 𝑡 = [𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎]. 𝑑𝑡 + 𝜎. 𝑋 𝑡 . (∘). 𝑑𝑊
¨ STRATO SIE:
¨ 𝑋 𝑡. − 𝑋 𝑡* = ∫&/&*
&/&.
[𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎]. 𝑑𝑡) + ∫&/&*
&/&.
𝜎. 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ STRATO lemma on 𝑓 𝑋 𝑡 = ln(𝑋 𝑡 )
¨ 𝑑𝑓 𝑋 𝑡 = 𝑑(ln 𝑋 𝑡 ) =
!0
!3
. ∘ . 𝑑𝑋 +
!0
!3
. 𝑑𝑡
¨ 𝑑𝑓 𝑋 𝑡 = 𝑑(ln 𝑋 𝑡 ) =
+
3
. ∘ . ([𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎]. 𝑑𝑡 + 𝜎. 𝑋 𝑡 . (∘). 𝑑𝑊)
¨ 𝑑𝑓 𝑋 𝑡 = 𝜇 −
>"
,
. 𝑑𝑡 + 𝜎. (∘). 𝑑𝑊(𝑡)
¨ 𝑑(ln 𝑋 𝑡 ) = 𝜇 −
>"
,
. 𝑑𝑡 + 𝜎. ∘ . 𝑑𝑊(𝑡)
127

128. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - VIII
¨ STRATO SDE on ln 𝑋 𝑡 : 𝑑(ln 𝑋 𝑡 ) = 𝜇 −
>"
,
. 𝑑𝑡 + 𝜎. ∘ . 𝑑𝑊(𝑡)
¨ Note that since 𝜎 is constant in that case 𝜎. [ . 𝑑𝑊 𝑡 = 𝜎. ∘ . 𝑑𝑊 𝑡 = 𝜎. 𝑑𝑊(𝑡)
¨ STRATO SIE on ln 𝑋 𝑡
¨ 𝑙𝑛𝑋 𝑡. − 𝑙𝑛𝑋 𝑡* = ∫&/&*
&/&.
𝑑𝑙𝑛𝑋 𝑡 = ∫&/&*
&/&.
𝜇 −
>"
,
. 𝑑𝑡) + ∫&/&*
&/&.
𝜎. (∘). 𝑑𝑊(𝑡)
¨ 𝑙𝑛𝑋 𝑡. − 𝑙𝑛𝑋 𝑡* = ∫&/&*
&/&.
𝑑𝑙𝑛𝑋 𝑡 = 𝜇 −
>"
,
. (𝑡. − 𝑡*) + 𝜎[𝑊 𝑡. − 𝑊(𝑡*)]
¨ ln[
3 &)
3 &!
] = 𝜇 −
>"
,
. (𝑡. − 𝑡*) + 𝜎[𝑊 𝑡. − 𝑊(𝑡*)]
¨ 𝑋 𝑡. = 𝑋 𝑡. . exp{ 𝜇 −
>"
,
. 𝑡. − 𝑡* + 𝜎 𝑊 𝑡. − 𝑊 𝑡*
¨ This is the same solution that we got from ITO (as expected but always worth checking)
128

129. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - IX
¨ ITO SDE: 𝑑𝑋 𝑡 = 𝜇. 𝑋 𝑡 . 𝑑𝑡 + 𝜎. 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ ITO SDE is: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD ITO Kolmogorov PDE
¨
!"(6,&|3!,&!)
!&
= −
!
!6
𝑎 𝑥 𝑡 , 𝑡 . 𝑝 𝑥, 𝑡 𝑋*, 𝑡* −
!
!6
[
+
,
. [𝑏(𝑥 𝑡 , 𝑡), . 𝑝(𝑥, 𝑡|𝑋*, 𝑡*)]
¨
!"(6,&|3!,&!)
!&
= −
!
!6
𝜇. 𝑥. 𝑝 𝑥, 𝑡 𝑋*, 𝑡* −
!
!6
[
+
,
. [(𝜎. 𝑥), . 𝑝(𝑥, 𝑡|𝑋*, 𝑡*)]
129

130. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - X
¨ STRATO SDE: 𝑑𝑋 𝑡 = [𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎]. 𝑑𝑡 + 𝜎. 𝑋 𝑡 . (∘). 𝑑𝑊
¨ STRATO SDE is: 𝑑𝑋 𝑡 = 6𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 7𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD STRATO Kolmogorov PDE
¨
!"(6,&|3!,&!)
!&
= −
!
!6
9
:
{6𝑎 𝑡, 𝑥 +
+
,
. 7𝑏 𝑡, 𝑥 .
!
!6
7𝑏 𝑡, 𝑥 }. 𝑝 𝑥, 𝑡 𝑋*, 𝑡* −
!
!6
[
+
,
. [7𝑏(𝑡, 𝑥), . 𝑝(𝑥, 𝑡|𝑋*, 𝑡*)]
¨ 6𝑎 𝑡, 𝑋 𝑡 = 𝑎 𝑋 𝑡 , 𝑡 −
+
,
. 𝑏 𝑡, 𝑋 𝑡 .
!
!3
𝑏 𝑡, 𝑋 𝑡 = 𝜇. 𝑋 𝑡 −
+
,
. 𝜎. 𝑋 𝑡 . 𝜎
¨ 7𝑏 𝑡, 𝑋 𝑡 = 𝑏 𝑡, 𝑋 𝑡 = 𝜎. 𝑋 𝑡
¨
!"(6,&|3!,&!)
!&
= −
!
!6
{𝜇. 𝑥}. 𝑝 𝑥, 𝑡 𝑋*, 𝑡* −
!
!6
[
+
,
. [(𝜎. 𝑥), . 𝑝(𝑥, 𝑡|𝑋*, 𝑡*)]
¨ Again, same as derived in ITO, but always worth checking
130

131. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - XI
¨ In the deck (II), we looked at moments from the FP:
¨
!"(6,&)
!&
= −
!
!6
[𝑀+ 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 −
!
!6
[𝑀, 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 ]]
¨ Really in terms of notation to highlight the fact that this is a FORWARD FP:
¨
!"(6,&|6;,&;)
!&
= −
!
!6
[𝑀+ 𝑥, 𝑡 . 𝑝 𝑥, 𝑡|𝑣0, 𝑡0 −
!
!6
[𝑀, 𝑥, 𝑡 . 𝑝 𝑥, 𝑡|𝑥𝑜, 𝑡𝑜 ]]
¨ 𝑚J 𝑥, 𝑡 =< 𝑋J >&= ∫4<
7<
𝑝 𝑥, 𝑡 . 𝑥J. 𝑑𝑣
¨ We showed in deck II that by integration by part:
¨ 𝐼, 𝑛 = ∫4<
7<
𝑀+ 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 . 𝑛. 𝑥J4+. 𝑑𝑣 + ∫4<
7<
𝑀, 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 . 𝑛. (𝑛 − 1). 𝑥J4,. 𝑑𝑣
¨
A
A&
. 𝑚J 𝑥, 𝑡 = 𝐼, 𝑛
¨
A
A&
. 𝑚; 𝑥, 𝑡 = 0 since the probability 𝑚; 𝑥, 𝑡 = ∫4<
7<
𝑝 𝑥, 𝑡 . 𝑑𝑣 is conserved
131

132. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - XII
¨ For the GBM case:
¨
!"(6,&|3!,&!)
!&
= −
!
!6
{𝜇. 𝑥}. 𝑝 𝑥, 𝑡 𝑋*, 𝑡* −
!
!6
[
+
,
. [(𝜎. 𝑥), . 𝑝(𝑥, 𝑡|𝑋*, 𝑡*)]
¨
!"(6,&)
!&
= −
!
!6
[𝑀+ 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 −
!
!6
[𝑀, 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 ]]
¨ 𝑀+ 𝑥, 𝑡 = 𝜇. 𝑥
¨ 𝑀, 𝑥, 𝑡 =
+
,
. (𝜎. 𝑥),
¨ 𝐼, 𝑛 = ∫4<
7<
𝑀+ 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 . 𝑛. 𝑥J4+. 𝑑𝑣 + ∫4<
7<
𝑀, 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 . 𝑛. (𝑛 − 1). 𝑥J4,. 𝑑𝑣
¨ 𝐼, 𝑛 = ∫4<
7<
𝜇. 𝑥. 𝑝 𝑥, 𝑡 . 𝑛. 𝑥J4+. 𝑑𝑣 + ∫4<
7< +
,
. (𝜎. 𝑥), . 𝑝 𝑥, 𝑡 . 𝑛. (𝑛 − 1). 𝑥J4,. 𝑑𝑣
¨ 𝐼, 𝑛 = 𝜇 ∫4<
7<
𝑝 𝑥, 𝑡 . 𝑛. 𝑥J. 𝑑𝑣 +
>"
,
. ∫4<
7<
𝑝 𝑥, 𝑡 . 𝑛. (𝑛 − 1). 𝑥J. 𝑑𝑣
132

133. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - XIII
¨ 𝐼, 𝑛 = 𝑛𝜇 ∫4<
7<
𝑝 𝑥, 𝑡 . 𝑥J . 𝑑𝑣 + 𝑛(𝑛 − 1)
>"
,
. ∫4<
7<
𝑝 𝑥, 𝑡 . 𝑥J . 𝑑𝑣
¨ 𝐼, 𝑛 = 𝑛𝜇 + 𝑛 𝑛 − 1
>"
,
. ∫4<
7<
𝑝 𝑥, 𝑡 . 𝑥J . 𝑑𝑣
¨ 𝐼, 𝑛 = 𝑛𝜇 + 𝑛 𝑛 − 1
>"
,
. 𝑚J 𝑥, 𝑡
¨
A
A&
. 𝑚J 𝑥, 𝑡 = 𝐼, 𝑛 = 𝑛𝜇 + 𝑛 𝑛 − 1
>"
,
. 𝑚J 𝑥, 𝑡
¨
A
A&
. 𝑚; 𝑥, 𝑡 = 0
¨
A
A&
. 𝑚+ 𝑡 = 𝐼, 1 = 𝜇𝑚+ 𝑡 so 𝑚+ 𝑡 = 𝑚+ 0 . exp(𝜇𝑡)
¨
A
A&
. 𝑚J 𝑥, 𝑡 = 𝐼, 𝑛 = 𝑛𝜇 + 𝑛 𝑛 − 1
>"
,
. 𝑚J 𝑥, 𝑡
133

134. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) - XIV
¨
A
A&
. 𝑚J 𝑥, 𝑡 = 𝐼, 𝑛 = 𝑛𝜇 + 𝑛 𝑛 − 1
>"
,
. 𝑚J 𝑥, 𝑡
¨ 𝑚J 𝑡 = 𝑚J 0 . exp[ 𝑛𝜇 + 𝑛 𝑛 − 1
>"
,
. 𝑡]
¨ 𝑚; 𝑥, 𝑡 = 1
¨ 𝑚+ 𝑡 = 𝑚+ 0 . exp(𝜇𝑡) diverges when 𝑡 → ∞ if 𝜇 > 0
¨ 𝑚J 𝑡 = 𝑚J 0 . exp[ 𝑛𝜇 + 𝑛 𝑛 − 1
>"
,
. 𝑡]
¨ That moment also diverges when 𝑡 → ∞ if 𝜇 + (𝑛 − 1) 𝜎, > 0
¨ SO there will ALWAYS be a value of n large enough (𝑛 > 1 −
,[
>") for which the moment will
diverge
¨ This is one of the drawback of the GBM, even if you start with a large negative value for 𝜇
there will always be a moment that will diverge (the dynamics is unstable)
134

135. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XV
¨ Igor Halperin (NYU machine learning professor)
¨ My pretty dramatic conclusion was that financial academics collectively missed all the
relevant development in physics starting from 1908 when Paul Langevin developed a
generalization of the theory of Brownian motion of Einstein, which describes a Brownian
particle moving in an external potential field. Einstein’s theory is mathematically equivalent
to the Bachelier model from 1900 for stock prices. In its turn, the Bachelier model was
reformulated as a model for a log-price (instead of the price itself) with a linear drift by Paul
Samuelson in 1964, resulting in his celebrated Geometric Brownian Motion (GBM) model.
¨ As the GBM model produces a poor fit to market data, financial engineers have since
modified or extended it in myriad ways, proposing various stochastic volatility, jump-
diffusion, Levy etc. models to ‘better match the market’.
135

136. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XVI
¨ I was quite shocked to find that a simple two-line comparison of two very famous equations,
namely the GBM model and the Langevin equation, shows that the GBM model (as well as
its multiple descends) describes a world with globally unstable dynamics, and thus does not
make sense from the point of view of physics – at best, it can only be used to describe small
market fluctuations over short period of time, but not dynamics that can proceed at
arbitrary long times.
¨ Though this observation is very basic, it appears that it has been overlooked since 1964
when the GBM model was proposed. I believe that if Samuelson was familiar with the
Langevin equation from 1908, he would not propose his GBM model – just because the latter
does not make sense!
¨ Paraphrasing a famous quote about string theory, I would say that most financial models
used by practitioners are ``not even wrong” - they are not about actual ‘physical’ markets,
but rather about something else (a pure math).
¨ https://www.rebellionresearch.com/blog/did-finance-oversleep-a-century-of-development-
in-physics-interview-with
¨ Salomon Brothers and their 2+ Langevin model from 1970 would also agree with Igor
Halperin
136

137. Luc_Faucheux_2020
PDF for the GBM
137

138. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XVII
¨ Just to check one more time, we can derive the PDF for the GBM using the distribution
functions
¨ PDF Probability Density Function: 𝑝3(𝑥, 𝑡)
¨ Distribution function : 𝑃3(𝑥, 𝑡)
¨ 𝑃3 𝑥, 𝑡 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑋 ≤ 𝑥, 𝑡 = ∫K/4<
K/6
𝑝3 𝑦, 𝑡 . 𝑑𝑦
¨ 𝑝3(𝑥, 𝑡) =
!
!6
𝑃3 𝑥, 𝑡
¨ 𝑙𝑛𝑋 𝑡. − 𝑙𝑛𝑋 𝑡* = ∫&/&*
&/&.
𝑑𝑙𝑛𝑋 𝑡 = 𝜇 −
>"
,
. (𝑡. − 𝑡*) + 𝜎[𝑊 𝑡. − 𝑊(𝑡*)]
¨ Right from the start you see that for the GBM we need to restrict ourselves to having:
¨ 𝑥 ∈ ]0 , +∞[
¨ That is another drawback of the GBM, it does not allow for negative prices for the stock
138

139. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XVIII
¨ 𝑃3 𝑥, 𝑡. = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦(𝑋 ≤ 𝑥|𝑋 𝑡* = 𝑥*)
¨ 𝑃3 𝑥, 𝑡. = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦(𝑙𝑛𝑋(𝑡.) ≤ 𝑙𝑛𝑥|𝑙𝑛𝑋 𝑡* = 𝑙𝑛𝑥*)
¨ 𝑃3 𝑥, 𝑡. = 𝑃(𝑙𝑛𝑋 𝑡* + 𝜇 −
>"
,
. (𝑡. − 𝑡*) + 𝜎[𝑊 𝑡. − 𝑊(𝑡*)] ≤ 𝑙𝑛𝑥|𝑙𝑛𝑋 𝑡* =
𝑙𝑛𝑥*)
¨ 𝑃3 𝑥, 𝑡. = 𝑃( 𝜇 −
>"
,
. (𝑡. − 𝑡*) + 𝜎[𝑊 𝑡. − 𝑊(𝑡*)] ≤ 𝑙𝑛𝑥 − 𝑙𝑛𝑋 𝑡* |𝑙𝑛𝑋 𝑡* =
𝑙𝑛𝑥*)
¨ 𝑃3 𝑥, 𝑡. = 𝑃(𝜎[𝑊 𝑡. − 𝑊(𝑡*)] ≤ 𝑙𝑛𝑥 − 𝑙𝑛𝑋 𝑡* − 𝜇 −
>"
,
. (𝑡. − 𝑡*)|𝑙𝑛𝑋 𝑡* =
𝑙𝑛𝑥*)
¨ 𝑃3 𝑥, 𝑡. = 𝑃([𝑊 𝑡. − 𝑊(𝑡*)] ≤
J64J3 &! 4 [4
*"
"
.(&)4&!)
>
|𝑙𝑛𝑋 𝑡* = 𝑙𝑛𝑥*)
139

140. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XIX
¨ 𝑃3 𝑥, 𝑡. = 𝑃([𝑊 𝑡. − 𝑊(𝑡*)] ≤
J64J3 &! 4 [4
*"
"
.(&)4&!)
>
|𝑙𝑛𝑋 𝑡* = 𝑙𝑛𝑥*)
¨ We have now brought this back to the distribution function on the Gaussian, since by
definition [𝑊 𝑡. − 𝑊(𝑡*)] is normally distributed with mean 0 and variance (𝑡. − 𝑡*)
¨ Math people sometimes write something like this:
¨ [𝑊 𝑡. − 𝑊(𝑡*)] is 𝑁(0, 𝑡. − 𝑡* )
¨ [𝑊 𝑡. − 𝑊(𝑡*)] is normally distributed according to the Gaussian function:
¨ ℎ 𝑥, 𝑡 =
+
,G&
. exp(
46"
,&
)
¨ Writing 𝜉 =
J64J3 &! 4 [4
*"
"
.(&)4&!)
>
140

141. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XX
¨ Writing 𝜉 =
J64J3 &! 4 [4
*"
"
.(&)4&!)
>
¨ 𝑃3 𝑥, 𝑡. = 𝑃([𝑊 𝑡. − 𝑊(𝑡*)] ≤
J64J3 &! 4 [4
*"
"
.(&)4&!)
>
|𝑙𝑛𝑋 𝑡* = 𝑙𝑛𝑥*)
¨ 𝑃3 𝑥, 𝑡. = ∫K/4<
K/2
ℎ 𝑦, 𝑡 . 𝑑𝑦
¨ 𝑃3 𝑥, 𝑡. = ∫K/4<
K/2 +
,G&
. exp(
4K"
,&
) . 𝑑𝑦
¨ Using the change of variable: 𝑦 =
J]4J3 &! 4 [4
*"
"
.(&)4&!)
>
¨ 𝑑𝑦 =
J]4J3 &! 4 [4
*"
"
.(&)4&!)
>
=
+
>]
𝑑𝑢
141

142. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XXI
¨ 𝑃3 𝑥, 𝑡. = ∫K/4<
K/2 +
,G(&)4&!)
. exp(
4K"
,(&)4&!)
) . 𝑑𝑦
¨ 𝜉 =
J64J3 &! 4 [4
*"
"
.(&)4&!)
>
¨ 𝑃3 𝑥, 𝑡. = ∫]/;
]/6 +
,G(&)4&!)
. exp(−
[J]4J3 &! 4 [4
*"
"
.(&)4&!)]"
,>"(&)4&!)
) .
A]
]>
¨ 𝑃3 𝑥, 𝑡. = ∫]/;
]/6 +
,G>"(&)4&!)
. exp(−
[J]4J3 &! 4 [4
*"
"
.(&)4&!)]"
,>"(&)4&!)
) .
A]
]
¨ Note that it is not (. 𝑑𝑢) but (.
A]
]
)
¨ Seems obvious but that little (
+
]
) can be tricky at times
142

143. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XXII
¨ 𝑃3 𝑥, 𝑡. = ∫]/;
]/6 +
,G>"(&)4&!)
. exp(−
[J]4J3 &! 4 [4
*"
"
.(&)4&!)]"
,>"(&)4&!)
) .
A]
]
¨ 𝑝3(𝑥, 𝑡) =
!
!6
𝑃3 𝑥, 𝑡
¨ 𝑝3(𝑥, 𝑡) =
+
,G>"(&)4&!)
. exp(−
[J64J3 &! 4 [4
*"
"
.(&)4&!)]"
,>"(&)4&!)
) .
+
6
¨ That is essentially all we need in order to calculate Black-Sholes through integration
¨ With the appropriate discounting (numeraire) being taken out of the integral (see the deck
on Black-Sholes Numeraire), and also Hull White p.
143

144. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XXIII
¨ 𝑝3(𝑥, 𝑡) =
+
,G>"(&)4&!)
. exp(−
[J64J3 &! 4 [4
*"
"
.(&)4&!)]"
,>"(&)4&!)
) .
+
6
¨ Sometimes to avoid forgetting the little
+
6
we write as a function of 𝑝J3(𝑙𝑛𝑥, 𝑡)
¨ If we have 𝑋′ 𝑡 = Φ(𝑋 𝑡 ) and 𝑋 𝑡 = 𝜑(𝑋′ 𝑡 )
¨ 𝑝3# 𝑥=, 𝑡 = 𝑝3 𝑥, 𝑡 .
!
!6# 𝜑 𝑥=
¨ 𝑝3# 𝑥=, 𝑡 = 𝑝3 𝑥, 𝑡 .
!
!6# 𝜑 𝑥= and noting 𝑥 = 𝜑 𝑥= and 𝑥′ = Φ(𝑥)
¨
!
!6# 𝜑 𝑥= =
A6
A6# =
A` 6#
A6#
¨ The density of probability {𝑝3# 𝑥=, 𝑡 . 𝑑𝑥′} = {𝑝3 𝑥, 𝑡 . 𝑑𝑥} is conserved
¨ If you integrate under the curve, then change the variable of integration, this is the usual
result
144

145. Luc_Faucheux_2020
Langevin versus GBM (geometric Brownian motion) – XXIV
¨ {𝑝3# 𝑥=, 𝑡 . 𝑑𝑥′} = {𝑝3 𝑥, 𝑡 . 𝑑𝑥} with 𝑥= = 𝑙𝑛𝑥
¨ {𝑝J3 𝑙𝑛𝑥, 𝑡 . 𝑑𝑙𝑛𝑥} = {𝑝3 𝑥, 𝑡 . 𝑑𝑥} with 𝑑𝑙𝑛𝑥 = 𝑑𝑥/𝑥
¨ 𝑝J3 𝑙𝑛𝑥, 𝑡 = 𝑝3 𝑥, 𝑡 . 𝑥
¨ 𝑝3(𝑥, 𝑡) =
+
,G>"(&)4&!)
. exp(−
[J64J3 &! 4 [4
*"
"
.(&)4&!)]"
,>"(&)4&!)
) .
+
6
¨ 𝑝J3(𝑙𝑛𝑥, 𝑡) =
+
,G>"(&)4&!)
. exp(−
[J64J3 &! 4 [4
*"
"
.(&)4&!)]"
,>"(&)4&!)
)
145

146. Luc_Faucheux_2020
Black-Sholes in the BGM
(because where else?)
146