3rd part of stochastic calculus lectures non-linear SDE from SDE to PDE for PDF Maxwell demon Thermal ratchets Brownian motors Parrondo paradox

1. Luc_Faucheux_2021
Stochastic Calculus – ITO – III
From SDEs to PDEs and back – part trois
Non-Linear SDE
1

2. Luc_Faucheux_2021
In this section
¨ We finally look at the non-linear SDE
¨ We will introduce the [𝛼] calculus in order to not get confused with the factor (1/2)
¨ We will look also at the backward and forward formalism for PDE
¨ Maybe if we have time some examples (OU canonical process)
¨ SDE: Stochastic Differential Equations
¨ SIE: Stochastic Integral Equations
¨ PDE: Partial Differential Equations
¨ PDF: Probability Distribution Function
2

3. Luc_Faucheux_2021
In this deck - II
¨ More importantly we will discuss at length (and maybe ad nauseam for some of you) the so
called “ITO-STRATANOVITCH controversy”
¨ That was something that for some reason was quite popular in the 90s
¨ Like a lot of things that were popular in the 90s, there was maybe not a good rationale for it,
and that kind of faded away
¨ So we will use it more as an example on how to use, and make sure that we understand
stochastic calculus
3

4. Luc_Faucheux_2021
In this deck - III
¨ If you Google “ITO-STRATANOVITCH controversy”, you can see that it is still creating a lot of
confusion out there, especially in the fields of Physics for non-homogeneous diffusion
coefficients (in the field of finance that would be for a volatility that is a function of the
stochastic underlier like equity for example)
¨ So that would be for modeling an SDE that would look something like that:
¨ dX(t)= a(X(t),t).dt+b(X(t),t).dW(t)
¨ We will show, that unlike all the previous examples in the deck II, writing something like the
above is actually not well defined.
¨ That is where the confusion comes from
¨ Interestingly enough it also comes from the fact that people started using digital computers
to simulate diffusion process, and in a nutshell (running the risk of oversimplifying):
¨ DIGITAL COMPUTERS LIVE IN AN ITO WORLD
¨ ANALOG COMPUTERS LIVE IN A STRATANOVITCH WORLD
4

5. Luc_Faucheux_2021
In this deck - IV
¨ To quote Manella:
¨ ” The Itô versus Stratonovich controversy, about the "correct" calculus to use for integration
of Langevin equations, was settled to general satisfaction some 30 years ago. Recently,
however, it has started to re-emerge, following the advent of new experimental techniques.”
¨ That was written in 2012 or so.
¨ But a lot of people are still confused by it.
¨ I get also confused all the time, and then I have a panic attack, then I take a couple of deep
breathes, and redo a bunch of pages of derivations to make sure that I still understand it.
¨ The older I get the more frequent that happens.
5

6. Luc_Faucheux_2021
In this deck - V
6

7. Luc_Faucheux_2021
dX= a(X(t),t).dt+b(X(t),t).dW
“Cent deux fois sur le métier remettez votre ouvrage”
Nicolas Boileau
7

8. Luc_Faucheux_2021
Finally
¨ We are almost there, so just before we tackle that one, because the factor (1/2) are
confusing (is that coming from the middle point in STRATO, or is is coming from the second
term in a Taylor expansion?), we generalize a little the formalism
8

9. Luc_Faucheux_2021
Introducing the [𝛼]calculus
9

10. Luc_Faucheux_2021
Introducing the [𝛼] calculus
¨ Here is where we take a pause to avoid being confused with factors 2
¨ Ito when defining the Ito integral of a function took the left-side
¨ Strato took the middle point
¨ Hence the existence of a factor (1/2) in the conversion between Ito and Strato
¨ HOWEVER, now that we are dealing with PDE, there is another factor (1/2) that crops up a
lot (from the integration by part of the second moment 𝑥!)
¨ Those 2 factors are NOT related (although in way they are, Strato chose middle point
specifically so that the rules of usual calculus would be formally conserved, so essentially he
solved for which point to use to offset the (1/2) term in Ito lemma)
¨ So they will be related, but in the mean-time we could get confused between the two
¨ Some authors (Gleeson, Arovas) have pointed out that it is somewhat easier to carry
derivations ”in the [𝛼] calculus”, and then setting the value of 𝛼, as opposed to duplicating
everything in Ito then Strato and carrying (1/2) factors
10

11. Luc_Faucheux_2021
Introducing the [𝛼] calculus - II
¨ The ITO integral is defined as:
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ([). 𝑑𝑋(𝑡) = lim
&→(
{∑)#*
)#&
𝑓(𝑋(𝑡))). [𝑋(𝑡)+*) − 𝑋(𝑡))]}
¨ The Stratonovitch integral is defined as:
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . (∘). 𝑑𝑋(𝑡) = lim
&→(
{∑)#*
)#&
𝑓 [𝑋(𝑡) + 𝑋(𝑡)+*)]/2). [𝑋(𝑡)+*) − 𝑋(𝑡))]}
¨ We can define the [𝛼] integral as:
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ([𝛼]). 𝑑𝑋(𝑡) =
lim
&→(
{∑)#*
)#&
𝑓(𝑋(𝑡)) + 𝛼. [𝑋(𝑡)+*) − 𝑋(𝑡))]). [𝑋(𝑡)+*) − 𝑋(𝑡))]}
¨ ITO will be the case 𝛼 = 0
¨ STRATO will be the case 𝛼 = 1/2
11

12. Luc_Faucheux_2021
Introducing the [𝛼] calculus - III
¨ We had the relation on the integrals:
¨ ∫
"#"$
"#"%
𝑓 𝑊 𝑡 . (∘). 𝑑𝑊(𝑡) = ∫
"#"$
"#"%
𝑓 𝑊 𝑡 . ([). 𝑑𝑊(𝑡) +
*
!
∫
"#"$
"#"%
𝑓′ 𝑊 𝑡 . 𝑑𝑡
¨ This becomes:
¨ ∫
"#"$
"#"%
𝑓 𝑊 𝑡 . ([𝛼]). 𝑑𝑊(𝑡) = ∫
"#"$
"#"%
𝑓 𝑊 𝑡 . ([). 𝑑𝑊(𝑡) + 𝛼. ∫
"#"$
"#"%
𝑓′ 𝑊 𝑡 . 𝑑𝑡
¨ Or:
¨ ∫
"#"$
"#"%
𝑓 𝑊 𝑡 . ([𝛼]). 𝑑𝑊(𝑡) = ∫
"#"$
"#"%
𝑓 𝑊 𝑡 . ([𝛼] = 0). 𝑑𝑊(𝑡) + 𝛼. ∫
"#"$
"#"%
𝑓′ 𝑊 𝑡 . 𝑑𝑡
12

13. Luc_Faucheux_2021
Introducing the [𝛼] calculus - IV
¨ For a more complicated stochastic process
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊
¨ We have:
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ∘ . 𝑑𝑊 𝑡 = ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) + ∫
"#"$
"#"% *
!
. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑓 𝑋(𝑡 . 𝑑𝑡
¨ This now becomes:
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . [𝛼] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) + ∫
"#"$
"#"%
𝛼. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑓 𝑋(𝑡 . 𝑑𝑡
13

14. Luc_Faucheux_2021
Introducing the [𝛼] calculus - V
¨ For the SDE we had the following mapping between ITO and STRATO
¨ The ITO SDE:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ Has the same solution (is the same) as the STRATO SDE in STRATO calculus:
¨ 𝑑𝑋 𝑡 = [𝑎 𝑡, 𝑋 𝑡 −
*
!
. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡 ]. 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ The STRATO SDE
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ Has the same solution (is the same) as the ITO SDE in ITO calculus
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 +
*
!
. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
14

15. Luc_Faucheux_2021
Introducing the [𝛼] calculus - VI
¨ This now becomes:
¨ The ITO SDE:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ Has the same solution (is the same) as the [𝛼] SDE in [𝛼] calculus:
¨ 𝑑𝑋 𝑡 = [𝑎 𝑡, 𝑋 𝑡 − 𝛼. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡 ]. 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ The [𝛼] SDE
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ Has the same solution (is the same) as the ITO SDE in ITO calculus
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
15

16. Luc_Faucheux_2021
Introducing the [𝛼] calculus - VII
¨ The ITO lemma (chain rule) reads:
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([). 𝑑𝑋(𝑡) +
*
!
∫
"#"$
"#"% ,!/
,0! (𝑋 𝑡 ). 𝑏 𝑡, 𝑋 𝑡
!
𝑑𝑡
¨ In the ”limit” of small time increments, this can be written formally as the Ito lemma:
¨ 𝛿𝑓 =
,.
,-
. 𝛿𝑋 +
*
!
.
,!/
,0! . 𝑏!𝛿𝑡
¨ The STRATO lemma reads:
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. (∘). 𝑑𝑋(𝑡)
¨ In the ”limit” of small time increments, this can be written formally as the Strato lemma:
¨ 𝛿𝑓 =
,.
,-
. ∘ . 𝛿𝑋
16

17. Luc_Faucheux_2021
Introducing the [𝛼] calculus - VIII
¨ In the [𝛼] calculus the [𝛼] lemma (chain rule) now reads :
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([𝛼]). 𝑑𝑋(𝑡) +
*
!
− 𝛼 . ∫
"#"$
"#"% ,!/
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ In the ”limit” of small time increments, this can be written formally as the [𝛼] lemma:
¨ 𝛿𝑓 =
,.
,-
. 𝛿𝑋 +
*
!
− 𝛼 .
,!/
,0! . 𝑏!. 𝛿𝑡
¨ NOTE: you can convince yourselves by redoing the derivation we had on pages 55-60
¨ This actually highlights why STRATO took the middle point 𝛼 = 1/2 , as this is the point
that cancels out the (1/2) coming from the Taylor expansion of 𝑓 𝑋 𝑡% from the left point.
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = lim
&→(
∑)#*
)#&
{𝑓(𝑋(𝑡))) − 𝑓(𝑋(𝑡)1*))}
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = lim
&→(
∑)#*
)#&
{
,.
,-
. ([). 𝛿𝑋 +
*
!
.
,!/
,0! . ([). (𝛿𝑋)!}
17

18. Luc_Faucheux_2021
Introducing the [𝛼] calculus - IX
¨ This is kind of nice, because we only have to do one derivation, and we do not get confused
where the factor (1/2) comes from in STRATO, and whether or not it is the same factor (1/2)
that comes from the integration by part of the second moment.
¨ Factors (1/2) are very confusing, so whenever possible it is better to keep them as variable
18

19. Luc_Faucheux_2021
Introducing the [𝛼] calculus - X
¨ As an illustrated example of [𝛼] calculus, we calculate
¨ 𝐴[𝛼] = 𝔼{∫
"#"$
"#"%
𝑊(𝑡). ([𝛼]). 𝑑𝑊(𝑡)} and with 𝑓 𝑊 𝑡 = 𝑊(𝑡)
¨ 𝐴[𝛼] = 𝔼{ lim
&→(
∑)#*
)#&
𝑓(𝑊(𝑡)) + 𝛼. [𝑊(𝑡)+*) − 𝑊(𝑡))]). [𝑊(𝑡)+*) − 𝑊(𝑡))]}
¨ 𝐴[𝛼] = 𝔼{ lim
&→(
∑)#*
)#&
(𝑊(𝑡)) + 𝛼. [𝑊(𝑡)+*) − 𝑊(𝑡))]). [𝑊(𝑡)+*) − 𝑊(𝑡))]}
¨ 𝔼{𝑊 𝑡 . 𝑊(𝑡2)} = min(𝑡, 𝑡2)
¨ 𝔼 𝑊 𝑡 . 𝑊 𝑡2 − 𝑊 𝑡22 = min 𝑡, 𝑡2 − min 𝑡, 𝑡22 = 0 when 𝑡 < 𝑡2 < 𝑡′′
¨ 𝔼 𝑊 𝑡 − 𝑊 𝑡2 !
= 𝑡 + 𝑡2 − 2. min 𝑡, 𝑡2 = |𝑡 − 𝑡′|
¨ 𝐴[𝛼] = 𝔼{ lim
&→(
∑)#*
)#&
𝛼. [𝑡)+* − 𝑡)]} = 𝛼. [𝑡% − 𝑡$]
¨ Remember if you could integrate X? Worth revisiting within the [𝛼] calculus
19

20. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XI
¨ We have the following conventions:
¨ 𝛼 = 0 ITO
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . [𝛼] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) + ∫
"#"$
"#"%
𝛼. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑓 𝑋(𝑡 . 𝑑𝑡
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . [0] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ ITO integral is a MARTINGALE
¨ ITO integral exhibits the ISOMETRY property
¨ ITO LEMMA
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([𝛼]). 𝑑𝑋(𝑡) +
*
!
− 𝛼 . ∫
"#"$
"#"% ,!/
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([). 𝑑𝑋(𝑡) +
*
!
. ∫
"#"$
"#"% ,!/
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
20

21. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XII
¨ 𝛼 = 1/2 STRATONOVITCH
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . [𝛼] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) + ∫
"#"$
"#"%
𝛼. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑓 𝑋(𝑡 . 𝑑𝑡
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . [1/2] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ STRATO integral is NOT a MARTINGALE
¨ STRATO integral does NOT exhibits the ISOMETRY property
¨ STRATO LEMMA, usual regular Newtonian Leibniz calculus chain rule is FORMALLY respected
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([𝛼]). 𝑑𝑋(𝑡) +
*
!
− 𝛼 . ∫
"#"$
"#"% ,!/
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. (∘). 𝑑𝑋(𝑡)
21

22. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XIII
¨ 𝛼 = 1 KLIMONTOVITCH (reverse ITO, or OTI)
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . [𝛼] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) + ∫
"#"$
"#"%
𝛼. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑓 𝑋(𝑡 . 𝑑𝑡
¨ ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . [1] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"%
𝑓 𝑋 𝑡 . (]). 𝑑𝑊(𝑡)
¨ KLIMONTOVITCH integral is NOT a MARTINGALE
¨ KLIMONTOVITCH integral does NOT exhibits the ISOMETRY property
¨ KLIMONTOVITCH LEMMA, does NOT formally looks like the regular Leibniz rule
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([𝛼]). 𝑑𝑋(𝑡) +
*
!
− 𝛼 . ∫
"#"$
"#"% ,!/
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ] . 𝑑𝑋 𝑡 −
*
!
. ∫
"#"$
"#"% ,!/
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
22

23. Luc_Faucheux_2021
Introducing the [𝛼] calculus – XIII a
¨ No idea what the advantages of the KLIMONTOVITCH calculus since, the integral is not a
martingale like ITO, the usual rules of calculus are not respected like STRATO
¨ But it is in the literature out there so thought I would include it
¨ Might be useful when you do backward numerical simulations on a digital computer, but not
sure, will look into this further
23

24. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XIV
¨ 𝐴[𝛼] = 𝔼{∫
"#"$
"#"%
𝑊(𝑡). ([𝛼]). 𝑑𝑊(𝑡)} and with 𝑓 𝑊 𝑡 = 𝑊(𝑡)
¨ 𝐴[𝛼] = 𝔼{ lim
&→(
∑)#*
)#&
𝛼. [𝑡)+* − 𝑡)]} = 𝛼. [𝑡% − 𝑡$]
¨ ITO 𝛼 = 0
¨ 𝔼{∫
"#"$
"#"%
𝑊(𝑡). ([). 𝑑𝑊(𝑡)} = 0
¨ No surprise there as the ITO integral is a MARTINGALE
¨ STRATO 𝛼 = 1/2
¨ 𝔼{∫
"#"$
"#"%
𝑊(𝑡). (∘). 𝑑𝑊(𝑡)} = (1/2). [𝑡% − 𝑡$]
¨ KLIMONTOVITCH 𝛼 = 1
¨ 𝔼{∫
"#"$
"#"%
𝑊(𝑡). (]). 𝑑𝑊(𝑡)} = [𝑡% − 𝑡$]
24

25. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XV
¨ Let’s look at the integration before taking the expectation:
¨ ∫
"#"$
"#"%
𝑊(𝑡). (∘). 𝑑𝑊(𝑡) =
*
!
(𝑊 𝑡%
! − 𝑊 𝑡$
!)
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([). 𝑑𝑋(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ∘ . 𝑑𝑋 𝑡
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ] . 𝑑𝑋 𝑡 −
*
!
. ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,-
. ∘ . 𝑑𝑋 𝑡 = ∫
"#"$
"#"% ,.
,-
. ([). 𝑑𝑋(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,-
. ] . 𝑑𝑋 𝑡 = ∫
"#"$
"#"% ,.
,-
. ([). 𝑑𝑋(𝑡) + ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
25

26. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XVI
¨ ∫
"#"$
"#"% ,.
,-
. ∘ . 𝑑𝑋 𝑡 = ∫
"#"$
"#"% ,.
,-
. ([). 𝑑𝑋(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,-
. ] . 𝑑𝑋 𝑡 = ∫
"#"$
"#"% ,.
,-
. ([). 𝑑𝑋(𝑡) + ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,3
. ∘ . 𝑑𝑊 𝑡 = ∫
"#"$
"#"% ,.
,3
. ([). 𝑑𝑊(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,3! 𝑤 𝑡 . 𝑏 𝑡, 𝑊 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,3
. ] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"% ,.
,3
. ([). 𝑑𝑊(𝑡) + ∫
"#"$
"#"% ,!.
,3! 𝑤 𝑡 . 𝑏 𝑡, 𝑊 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"%
𝑊(𝑡). (∘). 𝑑𝑊(𝑡) =
*
!
(𝑊 𝑡%
! − 𝑊 𝑡$
!)
¨ With 𝑓 𝑊 𝑡 = 𝑊(𝑡) we have
,.
,4
𝑊 𝑡 = 𝑊(𝑡)
¨
,.
,3
𝑤 = 𝑤
¨
,!.
,3! = 1
26

27. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XVII
¨ ∫
"#"$
"#"%
𝑊(𝑡). (∘). 𝑑𝑊(𝑡) =
*
!
(𝑊 𝑡%
! − 𝑊 𝑡$
!)
¨ ∫
"#"$
"#"% ,.
,-
. ∘ . 𝑑𝑋 𝑡 = ∫
"#"$
"#"% ,.
,-
. ([). 𝑑𝑋(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,3
. ∘ . 𝑑𝑊 𝑡 = ∫
"#"$
"#"% ,.
,3
. ([). 𝑑𝑊(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,3! 𝑤 𝑡 . 𝑏 𝑡, 𝑊 𝑡
!
. 𝑑𝑡
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ 𝑑𝑋 𝑡 = 𝑑𝑊(𝑡)
¨ 𝑎 𝑡, 𝑋 𝑡 = 0
¨ 𝑏 𝑡, 𝑋 𝑡 = 0
¨
,.
,3
𝑤 = 𝑤 and
,!.
,3! = 1
27

28. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XVIII
¨ ∫
"#"$
"#"% ,.
,3
. ∘ . 𝑑𝑊 𝑡 = ∫
"#"$
"#"% ,.
,3
. ([). 𝑑𝑊(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,3! 𝑤 𝑡 . 𝑏 𝑡, 𝑊 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,3
. ∘ . 𝑑𝑊 𝑡 = ∫
"#"$
"#"% ,.
,3
. ([). 𝑑𝑊(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,3! 𝑤 𝑡 . 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,3
. ([). 𝑑𝑊(𝑡) = ∫
"#"$
"#"% ,.
,3
. ∘ . 𝑑𝑊 𝑡 −
*
!
. ∫
"#"$
"#"% ,!.
,3! 𝑤 𝑡 . 𝑑𝑡
¨
,.
,3
𝑤 = 𝑤 and
,!.
,3! = 1
¨ ∫
"#"$
"#"%
𝑊(𝑡). (∘). 𝑑𝑊(𝑡) =
*
!
(𝑊 𝑡%
! − 𝑊 𝑡$
!)
¨ ∫
"#"$
"#"%
𝑊(𝑡). ([). 𝑑𝑊(𝑡) = ∫
"#"$
"#"%
𝑊(𝑡). ∘ . 𝑑𝑊 𝑡 −
*
!
. ∫
"#"$
"#"%
1. 𝑑𝑡
¨ ∫
"#"$
"#"%
𝑊(𝑡). ([). 𝑑𝑊(𝑡) =
*
!
(𝑊 𝑡%
! − 𝑊 𝑡$
!) −
*
!
. (𝑡% − 𝑡$)
28

29. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XIX
¨ ∫
"#"$
"#"%
𝑊(𝑡). (∘). 𝑑𝑊(𝑡) =
*
!
(𝑊 𝑡%
! − 𝑊 𝑡$
!)
¨ ∫
"#"$
"#"%
𝑊(𝑡). ([). 𝑑𝑊(𝑡) =
*
!
(𝑊 𝑡%
! − 𝑊 𝑡$
!) −
*
!
. (𝑡% − 𝑡$)
¨ ∫
"#"$
"#"% ,.
,3
. ] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"% ,.
,3
. ([). 𝑑𝑊(𝑡) + ∫
"#"$
"#"% ,!.
,3! 𝑤 𝑡 . 𝑏 𝑡, 𝑊 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"% ,.
,3
. ] . 𝑑𝑊 𝑡 = ∫
"#"$
"#"% ,.
,3
. (∘). 𝑑𝑊(𝑡) +
*
!
. ∫
"#"$
"#"% ,!.
,3! 𝑤 𝑡 . 𝑏 𝑡, 𝑊 𝑡
!
. 𝑑𝑡
¨ ∫
"#"$
"#"%
𝑊(𝑡). ] . 𝑑𝑊 𝑡 =
*
!
𝑊 𝑡%
! − 𝑊 𝑡$
! +
*
!
. (𝑡% − 𝑡$)
29

30. Luc_Faucheux_2021
Introducing the [𝛼] calculus - XX
¨ Making it a little simpler and easier on the eye with:
¨ 𝑡$ = 0 and 𝑡% = 𝑡
¨ 𝑊 𝑡$ = 0 and 𝑊 𝑡% = 𝑊 𝑡 = 𝑊
¨ ITO: ∫
5
"
𝑊. ([). 𝑑𝑊 =
4 " !
!
−
"
!
𝔼 ∫
5
"
𝑊. ([). 𝑑𝑊 = 0
¨ STRATONOVITCH: ∫
5
"
𝑊. (∘). 𝑑𝑊 =
4 " !
!
𝔼 ∫
5
"
𝑊. ([). 𝑑𝑊 =
"
!
¨ KLIMONTOVITCH: ∫
5
"
𝑊. (]). 𝑑𝑊 =
4 " !
!
+
"
!
𝔼 ∫
5
"
𝑊. ([). 𝑑𝑊 = 𝑡
¨ Remember of course that 𝔼 𝑊 = 0 and 𝔼 𝑊! = 𝑡
30

31. Luc_Faucheux_2021
A very quick summary
31

32. Luc_Faucheux_2021
A very quick summary
¨ In the deck II on stochastic calculus, we reviewed the following equations in increasing order
of complexity:
¨ 𝑑𝑋 𝑡 = 𝑎. 𝑑𝑡
¨ 𝑑𝑋 𝑡 = 𝑏. 𝑑𝑊
¨ 𝑑𝑋 𝑡 = 𝑎. 𝑑𝑡 + 𝑏. 𝑑𝑊
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡 . 𝑑𝑡 + 𝑏. 𝑑𝑊
¨ 𝑑𝑋 𝑡 = 𝑏 𝑡 . 𝑑𝑊
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡 . 𝑑𝑡 + 𝑏 𝑡 . 𝑑𝑊
¨ In all cases we were justified to write “.”, and this was the ”usual” product that we are used
to. There was no confusion. This is because the drift and diffusion coefficients were NOT
functions of the stochastic variables
32

33. Luc_Faucheux_2021
A very quick summary - II
¨ HOWEVER, if there is one thing that you should remember from this deck, is that you
CANNOT write something like:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊(𝑡)
¨ The “.” is meaningless if you want in this formulation
¨ As soon as the diffusion coefficient becomes inhomogeneous (depends on the position if
you thinking about a physical diffusion process), or the volatility depends on the level of the
underlier in Finance, you need to pick a convention (Ito, Strato, Klimontovitch, or anything
else) to explain what you mean by “.” in the term 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊(𝑡)
¨ Stochastic calculus is a very strange world as we saw in deck I, and just because you are
writing something that looks familiar does not mean that you are allowed to do it and use it.
¨ You should just go the extra mile and really write:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑊
¨ Just to know that you cannot use “.” in the diffusive term
33

34. Luc_Faucheux_2021
A very quick summary - III
¨ Note that the drift term is ok
¨ You do not need to specify:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑊(𝑡)
¨ This has to do with the definition of the integral
¨ Remember you cannot differentiate in stochastic calculus (or at least if you do you should be
very careful), always better to integrate and write SIE instead of SDE
¨ For all the previous examples, and even for a non-homogeneous drift term, the point you
pick inside the bin when you define the integral as the limit of a sum, does not matter, tey all
converge to the same usual Riemann integral.
¨ HOWEVER for non-homogeneous diffusion, the point you pick does matter
34

35. Luc_Faucheux_2021
A very quick summary - IV
¨ You pick the left point -> ITO
¨ which in the [𝛼] calculus means 𝛼 = 0
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼 = 0). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
35

36. Luc_Faucheux_2021
A very quick summary - V
¨ You pick the middle point -> STRATANOVITCH
¨ which in the [𝛼] calculus means 𝛼 = 1/2
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼 = 1/2). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
36

37. Luc_Faucheux_2021
A very quick summary - VI
¨ You pick the right point -> KLIMONTOVITCH
¨ which in the [𝛼] calculus means 𝛼 = 1
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (𝛼 = 1). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (]). 𝑑𝑊(𝑡)
37

38. Luc_Faucheux_2021
A very quick summary - VII
¨ ALWAYS go back to the SIE:
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([𝛼]). 𝑑𝑋(𝑡) +
*
!
− 𝛼 . ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ ONLY in the STRATNOVITCH case of 𝛼 = 1/2 does the above reduces to:
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([𝛼]). 𝑑𝑋(𝑡)
¨ Which FORMALLY looks like the usual rules of calculus
¨ But think of this more as a coincidence than something that you can use willy nilly with
utmost confidence.
¨ A little like Malliavin derivation operator has been designed to formally follow the usual
chain rule, it is more of a coincidence than a given, so always be super careful
38

39. Luc_Faucheux_2021
A very quick summary - VIII
¨ Also another quick note
¨ For sake of simplicity a lot of books or also in this deck we write:
,.
,-
¨ What we mean really is:
¨ 𝑓 is a regular function of say variable 𝑥
¨ Partial derivatives
,.
,0
and
,!.
,0! are well defined quantities
¨ We can evaluate those quantities for 𝑥 = 𝑋(𝑡)
¨ We would then write something like
,.
,0
|0#-(") and
,!.
,0! |0#-(")
¨ But sometimes because we are lazy or want to keep the notations simple we just write:
¨
,.
,-
and
,!.
,-!
39

40. Luc_Faucheux_2021
A very quick summary - IX
¨
,.
,-
and
,!.
,-! are really not well defined quantity
¨ Remember that we said right at the beginning that one of the defining charaterictic of a
stochastic process is that it was NOT differentiable
¨ So really what a lot of textbooks (like Hull for example) mean (or in this deck also), is that we
are trying to not make the equations too cumbersome, so really when we write:
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,0
. ([𝛼]). 𝑑𝑋(𝑡) +
*
!
− 𝛼 . ∫
"#"$
"#"% ,!.
,0! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ or
¨ 𝑓 𝑋 𝑡% − 𝑓 𝑋 𝑡$ = ∫
"#"$
"#"% ,.
,-
. ([𝛼]). 𝑑𝑋(𝑡) +
*
!
− 𝛼 . ∫
"#"$
"#"% ,!.
,-! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
40

41. Luc_Faucheux_2021
A very quick summary - X
¨ We really should be writing:
¨ 𝑓 𝑋 𝑡" − 𝑓 𝑋 𝑡# = ∫
$%$#
$%$" &'())
&)
|)%+($). ([𝛼]). 𝑑𝑋(𝑡) +
,
-
− 𝛼 . ∫
$%$#
$%$" &!'())
&)! |)%+($). 𝑏 𝑡, 𝑋 𝑡
-
. 𝑑𝑡
¨ That is getting a little cumbersome
¨ Baxter is one of the few textbooks that actually bothers to be fully rigorous in his notations.
¨ So many apologies in those deck if depending on how much space I have on the line, but sometimes
I will use one of those notations or another, but what I mean really is, going from the most exact and
rigorous to the simplest in notation but least rigorous
¨
,.(0)
,0
|0#-(") =
,.(0#-("))
,0
=
,.(-("))
,0
=
,.(-("))
,-
=
,.
,-
¨ Same for the other higher order of derivatives
41

42. Luc_Faucheux_2021
The McKean and Goodman derivation of the
link between SDE and PDE for the PDF, a
first taste of the Feynmann-Kac theorem
42

43. Luc_Faucheux_2021
Non-linear SDE – backward equation
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ We are in the general case now 𝑎 𝑡, 𝑋 𝑡 and 𝑏 𝑡, 𝑋 𝑡
¨ 𝑋 𝑡% − 𝑋 𝑡$ = ∫
"#"$
"#"%
𝑑𝑋 𝑡 = ∫
"#"$
"#"%
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡) + ∫
"#"$
"#"%
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ We follow here the derivation from McKean and Goodman, but might actually also re-derive
it from first principle after just to convince ourselves
¨ We are in ITO calculus, and will use the fact that the ITO integral of a trading strategy is a
martingale.
¨ For any function 𝐶(𝑋 𝑡 , 𝑡) (hint: we are using 𝐶 notation like in a call option, so people who
have derived before the Black-Sholes equation should be in familiar territory).
¨ The Ito lemma reads:
43

44. Luc_Faucheux_2021
Non-linear SDE – backward equation - II
¨ 𝐶 𝑋 𝑡% , 𝑡% − 𝐶 𝑋 𝑡$ , 𝑡$ = ∫
"#"$
"#"% ,8
,-
. ([). 𝑑𝑋(𝑡) +
*
!
∫
"#"$
"#"% ,!8
,-! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡 +
∫
"#"$
"#"% ,8
,"
(𝑋 𝑡 ). 𝑑𝑡
¨ In the ”limit” of small time increments, this can be written formally as the Ito lemma:
¨ 𝛿𝐶 =
,8
,-
. 𝛿𝑋 +
*
!
.
,!8
,-! . 𝛿𝑋! +
,8
,"
. 𝛿𝑡
¨ Note here the factor (1/2) coming from the Taylor expansion
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ 𝑑𝑋 𝑡 ! = 𝑏 𝑡, 𝑋 𝑡
!
. 𝑑𝑡
¨ 𝛿𝐶 =
,8
,-
. 𝛿𝑋 +
*
!
.
,!8
,-! . 𝑏!𝛿𝑡 +
,8
,"
. 𝛿𝑡
44

45. Luc_Faucheux_2021
Non-linear SDE – backward equation - III
¨ 𝛿𝐶 =
,8
,-
. 𝛿𝑋 +
*
!
.
,!8
,-! . 𝑏!𝛿𝑡 +
,8
,"
. 𝛿𝑡
¨ 𝛿𝐶 =
,8
,"
+
*
!
.
,!8
,-! . 𝑏! +
,8
,-
. 𝑎 . 𝛿𝑡 +
,8
,-
. 𝑎. 𝛿𝑊
¨ 𝐶 𝑋 𝑡% , 𝑡% − 𝐶 𝑋 𝑡$ , 𝑡$ = ∫
"#"$
"#"% ,8
,-
. 𝑏 𝑋 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡)
+ ∫
"#"$
"#"% ,8(- " ,")
,"
+
*
!
.
,!8(- " ,")
,-! . 𝑏(𝑋 𝑡 , 𝑡)! +
,8(- " ,")
,-
. 𝑎(𝑋 𝑡 , 𝑡) . 𝑑𝑡
¨ It would be nice if the term of first order in time would disappear.
¨ This is what the Feynman-Kac theorem is about
¨ IF the function 𝐶(𝑋 𝑡 , 𝑡) is such that:
¨
,8(- " ,")
,"
+
*
!
. 𝑏(𝑋 𝑡 , 𝑡)!.
,!8(- " ,")
,-! + 𝑎 𝑋 𝑡 , 𝑡 .
,8(- " ,")
,-
= 0
45

46. Luc_Faucheux_2021
Non-linear SDE – backward equation - IV
¨ Then:
¨ 𝐶 𝑋 𝑡% , 𝑡% − 𝐶 𝑋 𝑡$ , 𝑡$ = ∫
"#"$
"#"% ,8
,-
. 𝑏 𝑋 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡)
¨ Fixing 𝑡% = 𝑇 maturity of the option (boundary condition)
¨ And the conditional expectation:
¨ 𝔼{𝐶(𝑋 𝑡 , 𝑡)|𝑋 𝑡$ = 𝑋$}
¨ 𝔼 𝐶 𝑋 𝑡% , 𝑡% 𝑋 𝑡$ = 𝑋$ = 𝔼 𝐶 𝑋 𝑡$ , 𝑡$ 𝑋 𝑡$ = 𝑋$ +
𝔼 ∫
"#"$
"#"% ,8
,-
. 𝑏 𝑋 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡) 𝑋 𝑡$ = 𝑋$
¨ 𝔼 𝐶 𝑋 𝑡$ , 𝑡$ 𝑋 𝑡$ = 𝑋$ = 𝐶 𝑋 𝑡$ , 𝑡$
¨ 𝔼 ∫
"#"$
"#"% ,8
,-
. 𝑏 𝑋 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡) 𝑋 𝑡$ = 𝑋$ = 0
¨ because this is an ITO integral (note that this would NOT be the case in STRATO, so here it
actually pays to work in ITO)
46

47. Luc_Faucheux_2021
Non-linear SDE – backward equation - V
¨ 𝐶 𝑋 𝑡$ , 𝑡$ = 𝔼 𝐶 𝑋 𝑡% , 𝑡% 𝑋 𝑡$ = 𝑋$
¨ This is essentially the Feynman-Kac formula
¨ IF 𝐶(𝑥, 𝑡) follows the BACKWARD Kolmogorov (FP) equation:
¨
,8(0,")
,"
+
*
!
. 𝑏(𝑥, 𝑡)!.
,!8(0,")
,0! + 𝑎 𝑥, 𝑡 .
,8(0,")
,0
= 0
¨ Then 𝐶(𝑥, 𝑡) can be written as a conditional expectation:
¨ 𝐶 𝑥, 𝑡 = 𝔼 𝐶 𝑋 𝑡% , 𝑡% 𝑋 𝑡 = 𝑥
¨ Under the probability measure such that 𝑋 𝑡 is an ITO process with the ITO SDE:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ 𝑑𝑊 is a Wiener process (Brownian motion) under that probability measure, and the initial
condition is 𝑋 𝑡 = 𝑥
47

48. Luc_Faucheux_2021
Non-linear SDE – backward equation - VI
¨ Note that it is called BACKWARD because it concerns the expectation of something that is
fixed in the future at time 𝑡% = 𝑇 (Boundary condition)
¨ Just to be in familiar territory, if the boundary condition is a call payoff
¨ 𝐶 𝑋 𝑡% , 𝑡% = 𝐶 𝑋 𝑇 , 𝑇 = 𝑀𝐴𝑋(𝑋 𝑇 − 𝐾, 0)
¨ 𝐶 𝑥, 𝑡 = 𝔼 𝑀𝐴𝑋(𝑋 𝑇 − 𝐾, 0) 𝑋 𝑡 = 𝑥
¨
,8(0,")
,"
+
*
!
. 𝑏(𝑥, 𝑡)!.
,!8(0,")
,0! + 𝑎 𝑥, 𝑡 .
,8(0,")
,0
= 0
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ From the deck on numeraire, using the Money Market account:
¨ 𝑎 𝑡, 𝑋 𝑡 = 𝑟𝑆
¨ 𝑏 𝑡, 𝑋 𝑡 = 𝜎𝑆
¨ 𝑑𝑆 𝑡 = 𝑟𝑆. 𝑑𝑡 + 𝜎𝑆. ([). 𝑑𝑊
48

49. Luc_Faucheux_2021
Non-linear SDE – backward equation - VII
¨
,8(0,")
,"
+
*
!
. 𝑏(𝑥, 𝑡)!.
,!8(0,")
,0! + 𝑎 𝑥, 𝑡 .
,8(0,")
,0
= 0
¨
,8(:,")
,"
+
*
!
. (𝜎𝑆)!.
,!8(:,")
,:! + 𝑟𝑆.
,8(:,")
,:
= 0
¨ This is exactly the Black-Sholes equation for 𝐶 𝑆, 𝑡 = 𝐶5(𝑆, 𝑡)/𝑒1;(<1")
¨ Backward equations are for conditional expectations (we know the final value, we want to
evaluate the value of a derivative or a claim now)
¨ Forward equations like the one we have been working on for the PDF tend to be for
probability distributions (we know the starting point, usually a Dirac peak, and we look at
the evolution in time)
¨ Note that in the BACKWARD equation, the terms 𝑏(𝑥, 𝑡) and 𝑎(𝑥, 𝑡) are OUTSIDE the partial
derivatives (because that came from using Ito lemma)
¨ Note that in the FORWARD equation, the terms will be INSIDE
¨ Note that the PDF will follow BOTH the backward and forward (PDF is expectation of a Delta
peak payoff, following Dupire, we will do it again)
49

50. Luc_Faucheux_2021
Non-linear SDE – forward equation -
¨ OK, so we still do not know what equation does the PDF follow.
¨ We almost there
¨ 𝐶 𝑋 𝑡% , 𝑡% − 𝐶 𝑋 𝑡$ , 𝑡$ = ∫
"#"$
"#"% ,8
,-
. 𝑏 𝑋 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡)
+ ∫
"#"$
"#"% ,8(- " ,")
,"
+
*
!
.
,!8(- " ,")
,-! . 𝑏(𝑋 𝑡 , 𝑡)! +
,8(- " ,")
,-
. 𝑎(𝑋 𝑡 , 𝑡) . 𝑑𝑡
¨ Let’s not assume that the term in time vanishes
¨ Let’s actually assume that 𝐶 𝑋 𝑡 , 𝑡 can be ANY function, and in particular we can choose
¨ 𝐶 𝑋 𝑡 = ∞ , 𝑡 = ∞ = 0
¨ 𝐶 𝑋 𝑡 = 𝑡$ , 𝑡 = 𝑡$ = 0
¨ 0 = ∫
"#".
"#( ,8
,-
. 𝑏 𝑋 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡)
+ ∫
"#".
"#( ,8(- " ,")
,"
+
*
!
.
,!8(- " ,")
,-! . 𝑏(𝑋 𝑡 , 𝑡)! +
,8(- " ,")
,-
. 𝑎(𝑋 𝑡 , 𝑡) . 𝑑𝑡
50

51. Luc_Faucheux_2021
Non-linear SDE – forward equation - II
¨ Let us look again at the conditional expectation, but this time we will explicitly write it as an
integral over the possible outcomes
¨ 𝔼 𝐵 𝑋 𝑡 , 𝑡 𝑋 𝑡$ = 𝑋$ = ∫
0#1(
0#+(
𝐵 𝑥, 𝑡 . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$) . 𝑑𝑥
¨ We are obviously after what kind of equation could 𝑃(𝑥, 𝑡|𝑋$, 𝑡$) verify
¨ Now again, because we are in ITO calculus:
¨ 𝔼 ∫
"#".
"#< ,8
,-
. 𝑏 𝑋 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡) 𝑋 𝑡$ = 𝑋$ = 0
¨ In particular:
¨ 𝔼 ∫
"#".
"#+( ,8
,-
. 𝑏 𝑋 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡) 𝑋 𝑡$ = 𝑋$ = 0
51

52. Luc_Faucheux_2021
Non-linear SDE – forward equation - III
¨ 0 =
𝔼 ∫
"#".
"#( ,8(- " ,")
,"
+
*
!
.
,!8(- " ,")
,-! . 𝑏(𝑋 𝑡 , 𝑡)! +
,8(- " ,")
,-
. 𝑎(𝑋 𝑡 , 𝑡) . 𝑑𝑡 𝑋 𝑡$ = 𝑋$
¨ 0 = 𝔼 ∫
"#".
"#( ,8
,"
+
*
!
. 𝑏(𝑋 𝑡 , 𝑡)!.
,!8
,-! + 𝑎 𝑋 𝑡 , 𝑡 .
,8
,-
. 𝑑𝑡 𝑋 𝑡$ = 𝑋$
¨ 0 = ∫
"#".
"#(
∫
0#1(
0#+( ,8
,"
+
*
!
. 𝑏(𝑥, 𝑡)!.
,!8
,0! + 𝑎 𝑥, 𝑡 .
,8
,0
. 𝑝(𝑥, 𝑡|𝑋$, 𝑡$) . 𝑑𝑥. 𝑑𝑡
¨ Remember that this is true for ANY function 𝐶(𝑋 𝑡 , 𝑡) that we can choose arbitrarily so
that the boundary values, as well as the derivatives, vanishes
¨ We now integrate by part the above integral (we are now in the world of regular calculus, so
the usual rules of calculus apply, none of this ITO / STRATO issue)
¨ 0 = ∫
"#".
"#(
∫
0#1(
0#+(
−𝐶
,=
,"
+ 𝐶.
*
!
.
,!
,0! [𝑏(𝑥, 𝑡)!. 𝑝] − 𝐶.
,
,0
[𝑎 𝑥, 𝑡 . 𝑝] . 𝑑𝑥. 𝑑𝑡
52

53. Luc_Faucheux_2021
Non-linear SDE – forward equation - IV
¨ 0 = ∫
"#".
"#(
∫
0#1(
0#+(
−𝐶
,=
,"
+ 𝐶.
*
!
.
,!
,0! [𝑏(𝑥, 𝑡)!. 𝑝] − 𝐶.
,
,0
[𝑎 𝑥, 𝑡 . 𝑝] . 𝑑𝑥. 𝑑𝑡
¨ This is true for any function 𝐶(𝑥, 𝑡)
¨ And so:
¨ 0 = −
,=
,"
+
*
!
.
,!
,0! [𝑏(𝑥, 𝑡)!. 𝑝] −
,
,0
[𝑎 𝑥, 𝑡 . 𝑝]
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ +
*
!
.
,!
,0! [𝑏(𝑥, 𝑡)!. 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ This is the FORWARD Kolmogorov (FP) equation, because we know the initial condition and
we are looking at an advection/diffusion in time
¨ Note that in the FORWARD equation the terms 𝑎 𝑋 𝑡 , 𝑡 and 𝑏 𝑋 𝑡 , 𝑡 are INSIDE the
partial derivatives (because that came from integrating by parts the ITO lemma)
53

54. Luc_Faucheux_2021
Non-linear SDE – forward equation - V
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ = 𝑝 𝑥, 𝑡 subject to the initial condition: 𝑝(𝑥, 𝑡$) = 𝛿 𝑥 − 𝑋$
¨
,=(0,")
,"
= −
,
,0
[𝑀* 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 −
,
,0
[𝑀! 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 ]]
¨ < ∆𝑋 > = 𝐸 ∆𝑋 =< 𝑥 >"+∆" −< 𝑥 >"= 𝐹* 𝑋 𝑡 , 𝑡 . ∆𝑡 (drift term)
¨ < ∆𝑋!> = 𝐸 ∆𝑋! =< (𝑥−< 𝑥 >"+∆")!>"+∆"= 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
¨ We now have also shown that:
¨ 𝑀* 𝑥, 𝑡 = 𝑎 𝑥, 𝑡
¨ 𝑀! 𝑥, 𝑡 =
*
!
. 𝑏(𝑥, 𝑡)!
¨ Note that we do not always have an explicit solution for 𝑝 𝑥, 𝑡 at this point.
54

55. Luc_Faucheux_2021
Non-linear SDE – Summary
¨ As we saw with the Dupire approach, what is interesting is that the PDF follows BOTH a
FORWARD and a BACKWARD equation:
¨ In short in ITO: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,-
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ ] +
*
!
.
,!
,-! [. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ This is the FORWARD equation: given the starting point (𝑋$, 𝑡$), the conditional probability
𝑝(𝑥, 𝑡|𝑋$, 𝑡$) evolves forward in time
¨
,=(0,"|-.,".)
,".
= −𝑎 𝑋$, 𝑡$
,
,-.
𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
*
!
. 𝑏(𝑋$, 𝑡$)! ,!
,-.
! 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)
¨ This is the BACKWARD equation: given the end point (𝑥, 𝑡), the conditional probability can
also be expressed as an expectation (of the Delta peak), and will thus follow a BACKWARD FP
diffusing backward in time
55

56. Luc_Faucheux_2021
Non-linear SDE – Summary II
¨ 𝑝(𝑋%, 𝑡%|𝑋$, 𝑡$) = ∫
0#1(
0#+(
𝑝(𝑥, 𝑡%|𝑋$, 𝑡$). 𝛿(𝑥 − 𝑋%). 𝑑𝑥
¨ This is fairly obvious, and recast 𝑃(𝑋%, 𝑡%|𝑋$, 𝑡$) as an expectation of a terminal payoff,
which is the Delta peak 𝛿(𝑥 − 𝑋%), with the probability measure associated to
𝑝(𝑋%, 𝑡%|𝑋$, 𝑡$) .
¨ Just like the call option was the expectation of the terminal payoff 𝑀𝐴𝑋(𝑆2 − 𝐾, 0)
¨ So the idea is that we can apply ITO lemma to 𝑝(𝑋%, 𝑡%|𝑋$, 𝑡$) (this time on the “starting”
variables (𝑋$, 𝑡$)
¨ This s exactly what we just did, but because it can be confusing since in some ways, the
conditional probability is “two things”: it is the conditional probability that we use in order
to calculate the expectation (integral) of ANY derivative function, and hence will follow the
FORWARD equation, it is ALSO the expectation of a claim (mainly in an evident manner the
Delta peak), and so will follow the BACKWARD equation
¨ Note: only in ITO calculus can we zero out the integral as a martingale, otherwise we would
have to carry those terms in the derivation
56

57. Luc_Faucheux_2021
Why going through this trouble?
The Ito-Stratanovitch controversy
57

58. Luc_Faucheux_2021
Why did we go through all this trouble?
¨ Let’s recap:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊 in ITO calculus
¨ We have shown that in that case the PDF follows a FORWARD ITO Kolmogorov (FP)
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ < ∆𝑋 > = 𝐸 ∆𝑋 =< 𝑥 >"+∆" −< 𝑥 >"= 𝑎 𝑋 𝑡 , 𝑡 . ∆𝑡 (advection term)
¨ < ∆𝑋!> = 𝐸 ∆𝑋! =< (𝑥−< 𝑥 >"+∆")!>"+∆"= 𝑏(𝑋 𝑡 , 𝑡)!. ∆𝑡 (diffusion term)
58

59. Luc_Faucheux_2021
Why did we go through all this trouble? - II
¨ We ALSO know that going between ITO and [𝛼]:
¨ The ITO SDE:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ Has the same solution (is the same) as the [𝛼] SDE in [𝛼] calculus:
¨ 𝑑𝑋 𝑡 = [𝑎 𝑡, 𝑋 𝑡 − 𝛼. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡 ]. 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ The [𝛼] SDE
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ Has the same solution (is the same) as the ITO SDE in ITO calculus
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
59

60. Luc_Faucheux_2021
Why did we go through all this trouble? - III
¨ And so, if we start with an [𝛼] SDE: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ It has the same solution (is the same) as the ITO SDE in ITO calculus
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ Which will then follow the ITO FORWARD Kolmogorov (FP):
¨
,@(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑃 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑃(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ With:
¨ 𝑎 𝑋 𝑡 , 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡
¨ 𝑏 𝑋 𝑡 , 𝑡 = A
𝑏 𝑡, 𝑋 𝑡
¨ 𝑎 𝑋 𝑡 , 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + 𝛼. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡
60

61. Luc_Faucheux_2021
Why did we go through all this trouble? – III - a
¨ < ∆𝑋 > = 𝐸 ∆𝑋 =< 𝑥 >"+∆" −< 𝑥 >"= 𝑎 𝑋 𝑡 , 𝑡 . ∆𝑡 (advection term)
¨ < ∆𝑋!> = 𝐸 ∆𝑋! =< (𝑥−< 𝑥 >"+∆")!>"+∆"= 𝑏(𝑋 𝑡 , 𝑡)!. ∆𝑡 (diffusion term)
¨ With:
¨ 𝑎 𝑋 𝑡 , 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡
¨ 𝑏 𝑋 𝑡 , 𝑡 = A
𝑏 𝑡, 𝑋 𝑡
¨ Remember that we started from : 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ So: < ∆𝑋 > = 𝑎 𝑋 𝑡 , 𝑡 . ∆𝑡 = {@
𝑎 𝑡, 𝑋 𝑡 + 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 }. ∆𝑡
¨ < ∆𝑋 > = @
𝑎 𝑡, 𝑋 𝑡 . ∆𝑡 + 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
¨ < ∆𝑋 > =< @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . [𝛼] . 𝑑𝑊 >
61

62. Luc_Faucheux_2021
Why did we go through all this trouble? – III - b
¨ < ∆𝑋 > = @
𝑎 𝑡, 𝑋 𝑡 . ∆𝑡 + 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
¨ < ∆𝑋 > = < @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 > + < A
𝑏 𝑡, 𝑋 𝑡 . [𝛼] . 𝑑𝑊 >
¨ And
¨ < @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 > = @
𝑎 𝑡, 𝑋 𝑡 . ∆𝑡
¨ So we have:
¨ < A
𝑏 𝑡, 𝑋 𝑡 . [𝛼] . 𝑑𝑊 > = 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
¨ In particular:
¨ ITO : < A
𝑏 𝑡, 𝑋 𝑡 . [ . 𝑑𝑊 > = 0
¨ STRATO: < A
𝑏 𝑡, 𝑋 𝑡 . ∘ . 𝑑𝑊 > = [
*
!
]. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
62

63. Luc_Faucheux_2021
Why did we go through all this trouble? – III - c
¨ ITO : < A
𝑏 𝑡, 𝑋 𝑡 . [ . 𝑑𝑊 > = 0
¨ One can also go back to the definition of the ITO integral (because remember it is never a
SDE, it is ALWAYS and SIE) , but essentially the convention [ of taking the value “before the
jump”, implies that the ITO integral is a martingale of expected value 0
¨ STRATO: < A
𝑏 𝑡, 𝑋 𝑡 . ∘ . 𝑑𝑊 > = [
*
!
]. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
¨ Again we can explicitly derive this from the integral, but the convention ∘ implies taking
the value “in the middle of the jump”, hence the STRATO integral CANNOT be a martingale
and has a non zero expected value.
¨ We did this derivation when we looking at the correspondence between the ITO and
STRATO.
¨ It has been a while so might take a few pages here to redo it.
63

64. Luc_Faucheux_2021
Why did we go through all this trouble? – III - d
¨ So essentially here is the deal:
¨ A PDE is well defined, it is just a Partial Differential Equation where the
usual/Leibniz/regular/Leibniz calculus applies
¨ A SDE is NOT well defined, unless you specify which 𝛼-calculus you use (essentially which
point within the bin you take when defining the integral as a limit of sum)
¨ Left point of the bin -> ITO , 𝛼 = 0
¨ Middle point of the bin -> STRATO, 𝛼 = 1/2
¨ Right point of the bin -> KLIMONT, 𝛼 = 1
¨ Let’s summarize again before we delve more into the reason for the controversy
64

65. Luc_Faucheux_2021
A nice little summary
65

66. Luc_Faucheux_2021
We need a nice summary to avoid any confusion
¨ 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
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨
,=
,"
= −
,
,0
𝑎𝑝 −
,
,0
%!=
!
= −
,
,0
[𝑎𝑝] +
*
!
,!
,0! [
%!=
!
]
¨
,=
,"
= −
,
,0
𝑎𝑝 − 𝑏
,%
,0
. 𝑝 −
*
!
. 𝑏! .
,
,0
𝑝
¨
,=
,"
= −
,
,-
𝑎𝑝 −
,A
,0
. 𝑝 − 𝐷.
,
,0
𝑝
¨
,=
,"
= −
,
,0
𝑎𝑝 −
,
,0
(𝐷𝑝) using the convention 𝐷 =
*
!
. 𝑏!
66

67. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - a
¨
,=
,"
= −
,
,0
𝑎𝑝 −
,
,0
(𝐷𝑝)
¨ Physicists like a diffusion equation for particles that looks like:
¨
,=
,"
= −
,
,0
𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ Because when integrating over the x-axis, and assuming some reasonable (read 0) value for
the distribution function at infinity, the integral of the density is invariant with time
(conservation of matter), or:
¨
,
,"
∫
1(
+(
𝑝 𝑥, 𝑡 . 𝑑𝑥 = 0
¨ That is a good thing.
¨ Physicists also like something called “steady-state solution”, meaning at equilibrium (think
very long time),
,=
,"
= 0
67

68. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - b
¨
,=
,"
= 0 leads to:
¨ −
,
,0
𝑎𝑝 −
,
,0
(𝐷𝑝)
¨ For simplicity of argument, in the case of no external forcing (𝑎 = 0)
¨
,
,0
−
,
,0
(𝐷𝑝) = 0
¨ So the solution of that is:
¨
,
,0
𝐷𝑝 = 𝑐𝑡𝑒
¨ Again for things to no blow up at infinity, that means:
,
,0
𝐷𝑝 = 𝑐𝑡𝑒 = 0
¨ That means that 𝐷𝑝 = 𝑐𝑡𝑒 with the normalization condition of ∫
1(
+(
𝑝 𝑥, 𝑡 . 𝑑𝑥 = 1 usually
68

69. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - c
¨ 𝐷𝑝 = 𝑐𝑡𝑒 with the normalization condition of ∫
1(
+(
𝑝 𝑥, 𝑡 . 𝑑𝑥 = 1, which allows to
calculate the value of that constant.
¨ And that is when physicists lost their wits in the 80s/90s, because the diffusion is a dynamic
effect, and should not impact the equilibrium solution.
¨ The equilibrium solution is given by the laws of thermodynamics, and the equilibrium
density should not depend on something like 𝐷
¨ Essentially the argument was that, if you give it enough time, the equilibrium distribution
will be uniform, and should not depend on how fast or slow you diffuse in some regions
¨ That is a first flavor of the controversy that arose when we started to look in earnest in non-
homogeneous diffusion processes
¨ Remember that it was not that long ago on the scale of the planet, Bachelier / Langevin /
Einstein were just over a century ago now, Ito/Doelin in the 50s, so all in all fairly new
69

70. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - II
¨ [𝛼] SDE is: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD [𝛼] Kolmogorov PDE
¨
&/(),$|+",$")
&$
= −
&
&)
{1
𝑎 𝑥, 𝑡 + 𝛼. 4
𝑏 𝑥, 𝑡 .
&
&)
4
𝑏 𝑥, 𝑡 }. 𝑝 𝑥, 𝑡 𝑋#, 𝑡# −
&
&)
[
,
-
. [4
𝑏(𝑥, 𝑡)-
. 𝑝(𝑥, 𝑡|𝑋#, 𝑡#)]
¨
,=
,"
= −
,
,0
@
𝑎𝑝 + 𝛼. A
𝑏.
,
,0
A
𝑏. 𝑝 −
,
,0
[
*
!
. [A
𝑏! . 𝑝]
¨
,=
,"
= −
,
,0
@
𝑎𝑝 + 𝛼. A
𝑏.
,
,0
A
𝑏. 𝑝 −
,
,0
B
%!=
!
= −
,
,0
@
𝑎𝑝 +
,
,0
𝛼. A
𝑏.
,
,-
A
𝑏. 𝑝 +
*
!
,!
,0! [
B
%!=
!
]
¨
,=
,"
= −
,
,0
@
𝑎𝑝 + 𝛼. A
𝑏.
,B
%
,0
. 𝑝 − A
𝑏.
,B
%
,0
. 𝑝 −
*
!
. A
𝑏!.
,
,0
𝑝
¨
,=
,"
= −
,
,0
𝑎𝑝 + 𝛼.
,C
A
,0
. 𝑝 −
,C
A
,0
. 𝑝 − b
𝐷.
,
,0
𝑝
70

71. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - III
¨ [𝛼 = 1/2] STRATO SDE is: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD STRATO Kolmogorov PDE
¨
&/(),$|+",$")
&$
= −
&
&)
{1
𝑎 𝑥, 𝑡 +
,
-
. 4
𝑏 𝑥, 𝑡 .
&
&)
4
𝑏 𝑥, 𝑡 }. 𝑝 𝑥, 𝑡 𝑋#, 𝑡# −
&
&)
[
,
-
. [4
𝑏(𝑥, 𝑡)-
. 𝑝(𝑥, 𝑡|𝑋#, 𝑡#)]
¨
,=
,"
= −
,
,0
@
𝑎𝑝 +
*
!
. A
𝑏.
,B
%
,0
. 𝑝 −
,
,0
*
!
. [A
𝑏! . 𝑝
¨
,=
,"
= −
,
,0
@
𝑎𝑝 +
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝 −
,
,0
B
%!=
!
= −
,
,0
@
𝑎𝑝 +
,
,0
*
!
. A
𝑏.
,
,-
A
𝑏. 𝑝 +
*
!
,!
,0! [
B
%!=
!
]
¨
,=
,"
= −
,
,0
@
𝑎𝑝 +
*
!
. A
𝑏.
,B
%
,0
. 𝑝 − A
𝑏.
,B
%
,0
. 𝑝 −
*
!
. A
𝑏!.
,
,0
𝑝 = −
,
,0
@
𝑎𝑝 −
*
!
A
𝑏
,B
%
,0
𝑝 −
*
!
A
𝑏! ,
,0
𝑝
¨
,=
,"
= −
,
,0
@
𝑎𝑝 −
*
!
.
,C
A
,0
. 𝑝 − b
𝐷.
,
,0
𝑝 = −
,
,0
@
𝑎𝑝 +
*
!
.
,C
A
,0
. 𝑝 −
,
,0
(b
𝐷𝑝)
71

72. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - IIIa
¨ Note that we can also write as:
¨
,=
,"
= −
,
,0
@
𝑎𝑝 +
*
!
. A
𝑏.
,B
%
,0
. 𝑝 −
,
,0
*
!
. [A
𝑏! . 𝑝
¨
,=
,"
= −
,
,0
[@
𝑎𝑝] −
,
,0
[
*
!
. A
𝑏.
,B
%
,0
. 𝑝 −
*
!
, B
%!=
,0
]
¨ And since:
*
!
, B
%!=
,0
=
*
!
,B
%.B
%=
,0
=
*
!
A
𝑏
,B
%=
,0
+
*
!
A
𝑏𝑝
,B
%
,0
¨
,=
,"
= −
,
,0
@
𝑎𝑝 +
*
!
,
,0
[A
𝑏
,(B
%=)
,0
]
¨
,=(0,"|-.,".)
,"
= −
,
,0
@
𝑎 𝑥, 𝑡 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ +
*
!
,
,0
[A
𝑏 𝑥, 𝑡
,
,0
{A
𝑏 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ }]
72

73. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - IIIb
¨ That is right there in essence the crux of the matter:
¨
,@
,"
= −
,
,0
@
𝑎𝑝 −
*
!
.
,C
A
,0
. 𝑝 − b
𝐷.
,
,0
𝑝 = −
,
,0
@
𝑎𝑝 +
*
!
.
,C
A
,0
. 𝑝 −
,
,0
(b
𝐷𝑝)
¨ Because of course:
¨
,
,0
b
𝐷𝑝 = b
𝐷.
,
,0
𝑝 + 𝑝.
,
,0
b
𝐷
¨ So you can see how that, combined with the {
*
!
.
,C
A
,0
. 𝑝} term, can easily get people confused,
because in a way you are “mixing” the drift term and the diffusion term.
¨ In some ways you should not do that, what is drift is drift, and what is diffusion is diffusion
¨ It is clear when you write the SDE/SIE, because the drift term is in 𝑑𝑡, and the diffusive term
is in 𝑑𝑊, assuming that you have specified in which calculus you operate
¨ But when writing the PDE, it looks like you can willy nilly mix the two terms.
73

74. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - IV
¨ 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
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ The PDF ALSO follows the BACKWARD ITO Kolmogorov PDE:
¨
,=(0,"|-.,".)
,".
= −𝑎 𝑋$, 𝑡$
,
,-.
𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
*
!
. 𝑏(𝑋$, 𝑡$)! ,!
,-.
! 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)
¨
,=
,".
= −𝑎
,=
,-.
−
*
!
. 𝑏! ,!=
,-.
!
¨
,=
,".
= −𝑎
,=
,-.
− 𝐷
,!=
,-.
!
¨ Where I have explicitly kept the notation 𝑡$ and 𝑋$ to indicate the fact that this is a
BACKWARD PDE (expectation of a payoff at maturity)
74

75. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - V
¨ [𝛼] SDE is: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD [𝛼] Kolmogorov PDE
¨
&/(),$|+",$")
&$
= −
&
&)
{1
𝑎 𝑥, 𝑡 + 𝛼. 4
𝑏 𝑥, 𝑡 .
&
&)
4
𝑏 𝑥, 𝑡 }. 𝑝 𝑥, 𝑡 𝑋#, 𝑡# −
&
&)
[
,
-
. [4
𝑏(𝑥, 𝑡)-
. 𝑝(𝑥, 𝑡|𝑋#, 𝑡#)]
¨ The PDF ALSO follows the BACKWARD ITO Kolmogorov PDE:
¨
!"($,&|(!,&!)
!&!
= − G
𝑎 𝑡*, 𝑋 𝑡* + 𝛼. B
𝑏 𝑡*, 𝑋 𝑡* .
!
!(!
B
𝑏 𝑡*, 𝑋 𝑡*
!
!(!
𝑝 𝑥, 𝑡 𝑋*, 𝑡* −
+
,
. 𝑏(𝑋*, 𝑡*), !"
!(!
" 𝑝(𝑥, 𝑡|𝑋*, 𝑡*)
¨
,=
,".
= − @
𝑎 + 𝛼. A
𝑏
,
,-.
A
𝑏
,
,-.
𝑝 −
*
!
. 𝑏! ,!
,-.
! 𝑝
¨
,=
,".
= − @
𝑎 + 𝛼.
,C
A
,-.
,=
,-.
− b
𝐷
,!=
,-.
!
75

76. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - VI
¨ [𝛼 = 1/2] STRATO SDE is: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD STRATO Kolmogorov PDE
¨
&/(),$|+",$")
&$
= −
&
&)
{1
𝑎 𝑥, 𝑡 +
,
-
. 4
𝑏 𝑥, 𝑡 .
&
&)
4
𝑏 𝑥, 𝑡 }. 𝑝 𝑥, 𝑡 𝑋#, 𝑡# −
&
&)
[
,
-
. [4
𝑏(𝑥, 𝑡)-
. 𝑝(𝑥, 𝑡|𝑋#, 𝑡#)]
¨ The PDF ALSO follows the BACKWARD ITO Kolmogorov PDE:
¨
!"($,&|(!,&!)
!&!
= − G
𝑎 𝑡*, 𝑋 𝑡* +
+
,
. B
𝑏 𝑡*, 𝑋 𝑡* .
!
!(!
B
𝑏 𝑡*, 𝑋 𝑡*
!
!(!
𝑝 𝑥, 𝑡 𝑋*, 𝑡* −
+
,
. 𝑏(𝑋*, 𝑡*), !"
!(!
" 𝑝(𝑥, 𝑡|𝑋*, 𝑡*)
¨
,=
,".
= − @
𝑎 +
*
!
. A
𝑏
,
,-.
A
𝑏
,
,-.
𝑝 −
*
!
. A
𝑏! ,!
,-.
! 𝑝
¨
,=
,".
= − @
𝑎 +
*
!
.
,C
A
,-.
,=
,-.
− b
𝐷
,!=
,-.
!
76

77. Luc_Faucheux_2021
We need a nice summary to avoid any confusion – VI-a
¨ Note that we can also write as:
¨
,=
,".
= − @
𝑎 +
*
!
.
,C
A
,-.
,=
,-.
− b
𝐷
,!=
,-.
!
¨
,=
,".
= − @
𝑎 +
*
!
. A
𝑏
,
,-.
A
𝑏
,
,-.
𝑝 −
*
!
. A
𝑏! ,!
,-.
! 𝑝
¨ And since:
,
,-.
A
𝑏.
,=
,-.
=
,B
%
,-.
.
,=
,-.
+ A
𝑏.
,!=
,-.
!
¨
,=
,".
= −@
𝑎.
,=
,-.
−
*
!
. A
𝑏
,
,-.
[A
𝑏.
,=
,-.
]
¨
,=(0,"|-.,".)
,"
= −@
𝑎 𝑡$, 𝑋 𝑡$ .
,=(0,"|-.,".)
,-.
−
*
!
. A
𝑏 𝑡$, 𝑋 𝑡$
,
,-.
A
𝑏 𝑡$, 𝑋 𝑡$ .
,=(0,"|-.,".)
,-.
77

78. Luc_Faucheux_2021
We need a nice summary to avoid any confusion – VII
¨ ITO FORWARD
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ ITO BACKWARD
¨
,=(0,"|-.,".)
,".
= −𝑎 𝑋$, 𝑡$
,
,-.
𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
*
!
. 𝑏(𝑋$, 𝑡$)! ,!
,-.
! 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)
¨ STRATO FORWARD
¨
,=(0,"|-.,".)
,"
= −
,
,0
@
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ +
*
!
,
,0
[A
𝑏 𝑥, 𝑡
,
,0
{A
𝑏 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ }]
¨ STRATO BACKWARD
¨
,=(0,"|-.,".)
,"
= −@
𝑎 𝑡$, 𝑋 𝑡$ .
,=(0,"|-.,".)
,-.
−
*
!
. A
𝑏 𝑡$, 𝑋 𝑡$
,
,-.
A
𝑏 𝑡$, 𝑋 𝑡$ .
,=(0,"|-.,".)
,-.
78

79. Luc_Faucheux_2021
We need a nice summary to avoid any confusion – VIII
¨ ITO FORWARD
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[𝐷(𝑥, 𝑡). 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ ITO BACKWARD
¨
,=(0,"|-.,".)
,".
= −𝑎 𝑋$, 𝑡$
,
,-.
𝑝 𝑥, 𝑡 𝑋$, 𝑡$ − 𝐷 𝑋$, 𝑡$
,!
,-.
! 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)
¨ STRATO FORWARD
¨
,=(0,"|-.,".)
,"
= −
,
,0
@
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ +
,
,0
[ b
𝐷 𝑥, 𝑡
,
,0
{ b
𝐷 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ }]
¨ STRATO BACKWARD
¨
,=(0,"|-.,".)
,"
= −@
𝑎 𝑡$, 𝑋 𝑡$ .
,=(0,"|-.,".)
,-.
− b
𝐷 𝑡$, 𝑋 𝑡$
,
,-.
b
𝐷 𝑡$, 𝑋 𝑡$ .
,=(0,"|-.,".)
,-.
79

80. Luc_Faucheux_2021
We need a nice summary to avoid any confusion – IX
¨ Note that one of the characteristic of the STRATO equations is that they are much more
symmetrical forward and backward (physicist like that because it does not break the time
symmetry as much as ITO)
¨ In Ito, there is a clear distinction: the diffusion appears INSIDE the derivative for the
FORWARD equation, and OUTSIDE for the BACKWARD
¨ In STRATO, it is split more equally
¨ STRATO FORWARD
¨
,=
,"
= −
,
,0
@
𝑎𝑝 +
,
,0
[ b
𝐷
,
,-
{ b
𝐷. 𝑝}]
¨ STRATO BACKWARD
¨
,=
,"
= −@
𝑎.
,=
,-.
− b
𝐷
,
,-.
b
𝐷.
,=
,-.
80

81. Luc_Faucheux_2021
We need a nice summary to avoid any confusion – X
¨ Note that we in doubt it is also safe to recast the STRATO as an ITO using the rules of
transformation:
¨ If we start from a STRATO SDE: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ The corresponding ITO SDE is: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ With:
¨ 𝑎 𝑋 𝑡 , 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + [
*
!
]. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡
¨ 𝑏 𝑋 𝑡 , 𝑡 = A
𝑏 𝑡, 𝑋 𝑡
¨ 𝑎 𝑋 𝑡 , 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + [
*
!
]. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡
81

82. Luc_Faucheux_2021
We need a nice summary to avoid any confusion – XI
¨ And conversely:
¨ If we start from a ITO SDE: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ The corresponding STRATO SDE is: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ With:
¨ @
𝑎 𝑡, 𝑋 𝑡 = 𝑎 𝑋 𝑡 , 𝑡 − [
*
!
]. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡
¨ A
𝑏 𝑡, 𝑋 𝑡 = 𝑏 𝑋 𝑡 , 𝑡
¨ @
𝑎 𝑡, 𝑋 𝑡 = 𝑎 𝑋 𝑡 , 𝑡 − [
*
!
]. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡
82

83. Luc_Faucheux_2021
The crux of the matter
83

84. Luc_Faucheux_2021
The crux of the matter
¨ The main reason for the confusion is that people write something that looks like an ODE and
assume that because we are using the same usual notations, a lot of the baggage we
accumulated on regular calculus (Taylor expansion, Leibniz rule, chain rule,..) will also carry
over.
¨ I think an easy rule to remember is that writing an SDE like:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊(𝑡)
¨ Makes NO sense and is not well defined.
84

85. Luc_Faucheux_2021
The crux of the matter - I
¨ A slight improvement from writing something wrong like:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊(𝑡)
¨ Would be to write something like:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ Or:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ Or:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (α). 𝑑𝑊(𝑡)
¨
85

86. Luc_Faucheux_2021
The crux of the matter - II
¨ That way it is obvious that we are not writing the same thing:
¨ ITO
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝛼 = 0 . 𝑑𝑊(𝑡)
¨ STRATONOVITCH
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝛼 = 1/2 . 𝑑𝑊(𝑡)
86

87. Luc_Faucheux_2021
The crux of the matter - III
¨ Writing it like this makes it obvious that we are NOT writing the same SDE, we are writing
two very different processes
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ IN FACT, if we want the two different equations to represent the same process, then
obviously the coefficients cannot be the same. It turns out that we are quite lucky in the
sense that there is a simple relations between the two. We are maybe too lucky because
the fact that this relationship is simple sort of pushed people into believing that the two
representations (Ito and Strato) are very close to each other, if not similar. Maybe if would
have been easier if there had been not simple relationship between the coefficients, or a
very complicated one
87

88. Luc_Faucheux_2021
The crux of the matter – III a
¨ Say it another way
¨ Stochastic (Brownian) calculus is very different from regular (Newtonian/Leibniz) calculus
¨ ITO calculus is one possible definition of stochastic calculus
¨ STRATO calculus is one possible definition of stochastic calculus
¨ ITO calculus is very different from STRATO calculus
¨ However there is a relationship between the ITO integral and the STRATO integral
¨ ALWAYS go back to the integral form when dealing with stochastic calculus
¨ ALWAYS use an SIE if possible as opposed to an SDE
88

89. Luc_Faucheux_2021
The crux of the matter - IV
¨ So it is actually easier in order not to get confused to add the “tilde” on top of the
coefficients when dealing with Strato
¨ STRATO SDE: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ ITO SDE: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ Just to emphasize the point that we are not writing the same equation and that those are
two different processes
89

90. Luc_Faucheux_2021
The crux of the matter - V
¨ If we start from a STRATO SDE: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ The corresponding ITO SDE is: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ With:
¨ 𝑎 𝑋 𝑡 , 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + [
*
!
]. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡
¨ 𝑏 𝑡, 𝑋 𝑡 = A
𝑏 𝑡, 𝑋 𝑡
¨ 𝑎 𝑡, 𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 + [
*
!
]. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡
¨ In that case, the two different SDE will describe the same stochastic process
90

91. Luc_Faucheux_2021
The crux of the matter - VI
¨ And conversely:
¨ If we start from a ITO SDE: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ The corresponding STRATO SDE is: 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ With:
¨ @
𝑎 𝑡, 𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 − [
*
!
]. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡
¨ A
𝑏 𝑡, 𝑋 𝑡 = 𝑏 𝑡, 𝑋 𝑡
¨ @
𝑎 𝑡, 𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 − [
*
!
]. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡
¨ In that case, the two different SDE will describe the same stochastic process
91

92. Luc_Faucheux_2021
The crux of the matter - VII
¨ Some note on notation:
¨ To be even more rigorous, we should not even write something like this:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ We should say that there is an analytical function 𝑎 𝑡, 𝑥 and 𝑏 𝑡, 𝑥 , and that the process is:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑥 = 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑥 = 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑥 = 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑥 = 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ And so really we should not write:
¨ @
𝑎 𝑡, 𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 − [
*
!
]. 𝑏 𝑡, 𝑋 𝑡 .
,
,-
𝑏 𝑡, 𝑋 𝑡
¨ But:
¨ @
𝑎 𝑡, 𝑥 = 𝑎 𝑡, 𝑥 −
*
!
. 𝑏 𝑡, 𝑥 .
,
,0
𝑏 𝑡, 𝑥
92

93. Luc_Faucheux_2021
The crux of the matter - VIII
¨ In fact, whenever the notation doe not get too cumbersome, we should not even write and
SDE but and SIE
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ 𝑋 𝑡% − 𝑋 𝑡$ = ∫
"#"$
"#"%
𝑑𝑋 𝑡 = ∫
"#"$
"#"%
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡) + ∫
"#"$
"#"%
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ Knowing that all terms are well defined with the added :
¨ ∫
"#"$
"#"%
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) = lim
&→(
{∑)#*
)#&
𝑏(𝑡), 𝑋(𝑡))). [𝑊(𝑡)+*) − 𝑊(𝑡))]}
¨ Where the mesh or partition {𝑡)} is not completely pathological
¨ Also for STRATO:
¨ ∫
"#"$
"#"%
𝑎 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡) = lim
&→(
{∑)#*
)#&
𝑓 𝑡),
[-("2)+-("234)]
!
. [𝑊(𝑡)+*) − 𝑊(𝑡))]}
93

94. Luc_Faucheux_2021
The crux of the matter - IX
¨ So in the end, this is the only way to write a process that leaves no room for error:
¨ When writing something like this, we have no idea what it means
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊(𝑡)
¨ Really we should write:
¨ 𝑋 𝑡% − 𝑋 𝑡$ = ∫
"#"$
"#"%
𝑑𝑋 𝑡 = ∫
"#"$
"#"%
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡) + ∫
"#"$
"#"%
𝑏 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑊(𝑡)
¨ ∫
"#"$
"#"%
𝑏 𝑡, 𝑋 𝑡 . (𝛼). 𝑑𝑊(𝑡) = lim
&→(
{∑)#*
)#&
𝑏(𝑡), 𝑀R[𝑡), 𝑡)+*]). [𝑊(𝑡)+*) − 𝑊(𝑡))]}
¨ Where the function 𝑀R[𝑡), 𝑡)+*] indicates where in the partition [𝑡), 𝑡)+*] we should take
the value of the stochastic process 𝑋 𝑡
94

95. Luc_Faucheux_2021
The crux of the matter - X
¨ If ITO, 𝛼 = 0, and 𝑀R 𝑡), 𝑡)+* = 𝑋(𝑡))
¨ If STRATO, 𝛼 = 1/2, and 𝑀R 𝑡), 𝑡)+* =
[-("2)+-("234)]
!
¨ If anything in between, 𝑀R 𝑡), 𝑡)+* = 𝑋(𝑡)) + 𝛼. [ 𝑋(𝑡)+* − 𝑋(𝑡))]
¨ The nice thing about ITO is that the integral is a martingale, and that the integral also flows
the isometry rule.
¨ The price to pay is that the usual rules of calculus (chain rule, Leibniz, Taylor expansion,
integration by part,..) are pretty much out of the window
¨ The nice thing about STRATO is that the usual rules of calculus carry over FORMALLY in the
same manner (careful, they are not the same, it is a formal relation), so that is nice
¨ However the integral is not a martingale and does not follow the isometry rule
95

96. Luc_Faucheux_2021
A brief history of the Ito-Stratanovitch
controversy
96

97. Luc_Faucheux_2021
A brief history
¨ Because :
¨ 1) it is a nice way to apply our knowledge of stochastic calculus
¨ 2) you encounter it in textbooks
¨ 3) it is still super confusing at times, I know I am still confused
¨ It pays to explain why that came about.
¨ But first of all, let’s go one more time over the fact that actually there is NO controversy
¨ By the way, if you meet anyone who tells you that they understand Ito versus Stratanovitch
perfectly, that person is either a liar or Van Kampen
97

98. Luc_Faucheux_2021
First of all there is no controversy
98

99. Luc_Faucheux_2021
There is no controversy
¨ In order to make things less confusing that they could be, we are going to be a little literal on
the notation
¨ Essentially,
¨ PDEs are well defined, and rely on Newtonian/Leibniz/regular calculus
¨ SDEs and SIEs for non-homogeneous diffusion coefficients (non-linear), or level depenedent
volatilities, are NOT well defined, and you need to choose a value for 𝛼
¨ There is a correspondence between the different [𝛼] calculus
99

100. Luc_Faucheux_2021
There is no controversy - II
¨ Ito is 𝛼 = 0
¨ Strato is 𝛼 = 1/2
¨ Because the factor (1/2) also shows up in Taylor expansion and Ito lemma, that is where a
lot of the confusion comes from. BTW, Ito lemma is NOT a Taylor expansion, it formally
looks like one and comes from using a Taylor expansion in calculating the integral as a limit
of sum, but it is NOT a Taylor expansion
¨ So you cannot say (as sometimes it seems to be implied in textbooks).
¨ Hey in stochastic calculus, it is the same as regular calculus, just go up one more level in the
Taylor expansion
100

101. Luc_Faucheux_2021
There is no controversy - III
¨ An ITO SDE is of the form:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊(𝑡)
¨ This implies that the PDF follows the FORWARD ITO Kolmogorov PDE
¨
,=(0,"|-.,".)
,"
= −
,
,0
𝑎 𝑥, 𝑡 . 𝑝 𝑥, 𝑡 𝑋$, 𝑡$ −
,
,0
[
*
!
. [𝑏(𝑥, 𝑡)! . 𝑝(𝑥, 𝑡|𝑋$, 𝑡$)]
¨ With simpler notation:
¨ An ITO SDE is of the form:
¨ 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝑏. ([). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD ITO Kolmogorov PDE (Fokker-Planck)
¨
,=
,"
= −
,
,0
𝑎. 𝑝 −
,
,0
[
*
!
. [𝑏! . 𝑝]
101

102. Luc_Faucheux_2021
There is no controversy - IV
¨ An STRATO SDE is of the form:
¨ 𝑑𝑋 𝑡 = @
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + A
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊(𝑡)
¨ This implies that the PDF follows the FORWARD STRATO Kolmogorov PDE
¨
!"($,&|(!,&!)
!&
= −
!
!(
{G
𝑎 𝑡, 𝑋 𝑡 +
+
,
. B
𝑏 𝑡, 𝑋 𝑡 .
!
!(
B
𝑏 𝑡, 𝑋 𝑡 }. 𝑝 𝑥, 𝑡 𝑋*, 𝑡* −
!
!(
[
+
,
. [B
𝑏(𝑋 𝑡 , 𝑡), . 𝑝(𝑥, 𝑡|𝑋*, 𝑡*)]
¨ With simpler notation:
¨ An STRATO SDE is of the form:
¨ 𝑑𝑋 = @
𝑎. 𝑑𝑡 + A
𝑏. (∘). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD STRATO Kolmogorov PDE (Fokker-Planck)
¨
,=
,"
= −
,
,0
{@
𝑎 +
*
!
. A
𝑏.
,
,0
A
𝑏}. 𝑝 −
,
,0
[
*
!
. [A
𝑏! . 𝑝]
102

103. Luc_Faucheux_2021
There is no controversy - V
¨ So and ITO SDE is:
¨ 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝑏. ([). 𝑑𝑊
¨ And corresponds to a diffusion equation for the PDF:
¨
,=
,"
= −
,
,0
𝑎. 𝑝 −
,
,0
[
*
!
. 𝑏! . 𝑝]
¨ A STRATO SDE is:
¨ 𝑑𝑋 = @
𝑎. 𝑑𝑡 + A
𝑏. (∘). 𝑑𝑊
¨ And corresponds to a diffusion equation for the PDF:
¨
,=
,"
= −
,
,0
{@
𝑎 +
*
!
. A
𝑏.
,
,0
A
𝑏}. 𝑝 −
,
,0
[
*
!
. A
𝑏! . 𝑝]
103

104. Luc_Faucheux_2021
There is no controversy - VI
¨ So if we want the same solution, meaning we want the 2 equations for the PDE to be equal:
¨
,=
,"
= −
,
,0
𝑎. 𝑝 −
,
,0
[
*
!
. 𝑏! . 𝑝]
¨
,=
,"
= −
,
,0
{@
𝑎 +
*
!
. A
𝑏.
,
,0
A
𝑏}. 𝑝 −
,
,0
[
*
!
. A
𝑏! . 𝑝]
¨ That means as we have seen before that we need to write:
¨ A
𝑏 = 𝑏
¨ 𝑎 = @
𝑎 +
*
!
. A
𝑏.
,
,0
A
𝑏
¨ That is the correspondence between the ITO and STRATO coefficients for the SDEs
¨ Pointing out the obvious one more time, when
,
,0
A
𝑏 =
,
,0
𝑏 = 0, both calculus will return
exactly the same SDE and PDE, the SDE is called linear
104

105. Luc_Faucheux_2021
Back to a brief history of the Ito-
Stratanovitch controversy
(a physicist’s point of view)
105

106. Luc_Faucheux_2021
A brief history - II
¨ So if you are a physicist, you like to write diffusion equations like:
¨
,=
,"
= −
,
,0
[𝐽/ + 𝐽A]
¨ 𝐽A is the diffusion current (hence the D notation)
¨ 𝐽/ is the drift current, coming usually from an external force (hence the F notation)
¨ In the absence of external force, physicist would expect the equilibrium solution to be
uniform (meaning 𝑝 = 𝑐𝑡𝑒), and so it makes sense for them to write the diffusion current
as:
¨ 𝐽A = −𝐷.
,=
,0
¨ If the diffusion coefficient is a function of the position (and the physical system described is
such that the diffusive process does not impact the long term equilibrium steady state
derived from thermodynamics), physicists will still expect a uniform distribution and will still
want the diffusion coefficient to be outside the derivative
¨ 𝐽A = −𝐷(𝑥).
,=
,0
106

107. Luc_Faucheux_2021
A brief history - III
¨ For sake of simplicity let’s say that there is no external forcing right now, 𝐽/ = 0
¨ So physicist in the presence of non-homogeneous diffusion would like the PDE to look like:
¨
,=
,"
= −
,
,0
𝐽A = −
,
,0
−𝐷(𝑥).
,=
,0
¨ Nothing wrong so far.
¨ The problem arose when we started using digital computers and started running Monte
Carlo simulations
¨ This is fairly recent so we should not beat ourselves too much on the fact that we tripped on
that one.
¨ Following the work of Bachelier, Bernouilli and such, we started modeling the stochastic
process as:
¨ 𝛿𝑋 𝑡 = ± 2𝐷 with the ± being a random number generator (head or tail)
107

108. Luc_Faucheux_2021
A brief history - IV
¨ 𝛿𝑋 𝑡 = ± 2𝐷
¨ Is essentially what we looked at in the firs deck, as well as the Bachelier deck
108
X
i i+1
i-1

109. Luc_Faucheux_2021
A brief history - V
¨ We know for a contant spacing (constant
diffusion coefficient), the distribution is the
usual binomial which converges to the
Gaussian, as illustrated by the Galton
machine.
¨ By the way Galton was born not even 200
years ago, so again we should go easy on
ourselves
109

110. Luc_Faucheux_2021
A brief history - VI
¨ Ok so so far so good.
¨ The issue is when we started simulating on a digital computer something like that:
¨ 𝛿𝑋 𝑡 = ± 2𝐷(𝑋 𝑡 )
¨ Because, when we write something like the above, even without knowing it, we are in ITO
calculus and we are writing really:
¨ 𝑑𝑋(𝑡) = 2𝐷(𝑋 𝑡 ). ([). 𝑑𝑊(𝑡)
¨ This will give a PDE:
¨
,=
,"
= −
,
,0
𝑎. 𝑝 −
,
,0
[
*
!
. 𝑏! . 𝑝]
¨ With:
¨ 𝑎 = 0
¨ 𝑏 = 2𝐷(𝑥)
110

111. Luc_Faucheux_2021
A brief history - VII
¨
,=
,"
= −
,
,0
−
,
,0
[
*
!
. 2𝐷(𝑥)
!
. 𝑝]
¨
,=
,"
= −
,
,0
−
,
,0
[𝐷(𝑥). 𝑝]
¨ The solution of that (subject to non-diverging boundary conditions) is:
¨ 𝑝 = 1/𝐷(𝑥)
¨ And NOT 𝑝 = 𝑐𝑡𝑒, uniform distribution as expected.
¨ So that was the first sign that something was wrong.
111

112. Luc_Faucheux_2021
A brief history - VIII
¨ Let’s recap.
¨ You want to model:
¨
,=
,"
= −
,
,0
𝐽A = −
,
,0
−𝐷(𝑥).
,=
,0
¨ You do a discrete numerical simulation:
¨ 𝛿𝑋 𝑡 = ± 2𝐷(𝑋 𝑡 )
¨ You get:
¨
,=
,"
= −
,
,0
−
,
,0
[𝐷(𝑥). 𝑝]
¨ Not good, you do not get what you expected
112

113. Luc_Faucheux_2021
A brief history - IX
¨ So at that point, if physicists in the 80s were as good on stochastic calculus as they should
be, they should have said.
¨ Well yeah, a discrete simulation is by definition in the world of ITO calculus
¨ If I write: 𝛿𝑋 𝑡 = ± 2𝐷(𝑋 𝑡 )
¨ I really am writing: 𝑑𝑋(𝑡) = 2𝐷(𝑋 𝑡 ). ([). 𝑑𝑊(𝑡)
¨ Which will give me:
,=
,"
= −
,
,0
−
,
,0
[𝐷(𝑥). 𝑝]
¨ But I want:
,=
,"
= −
,
,0
−𝐷(𝑥).
,=
,0
113

114. Luc_Faucheux_2021
A brief history - X
¨ Since I want:
,=
,"
= −
,
,0
−𝐷(𝑥).
,=
,0
¨ I can write:
¨
,=
,"
= −
,
,0
−
,
,0
𝐷 𝑥 . 𝑝 = −
,
,0
−
,
,0
𝐷 𝑥 . 𝑝 −
,
,0
𝑝 . 𝐷(𝑥)
¨
,=
,"
= −
,
,0
−
,
,0
𝐷 𝑥 . 𝑝 − 𝐷(𝑥).
,=
,0
¨ That is of the form:
¨
,=
,"
= −
,
,0
𝑎. 𝑝 −
,
,0
[
*
!
. 𝑏! . 𝑝]
¨ Which in ITO will give the SDE
¨ 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝑏. ([). 𝑑𝑊
¨ With: 𝑎 = −
,
,0
𝐷 𝑥 and 𝑏 = 2𝐷(𝑋 𝑡 )
114

115. Luc_Faucheux_2021
A brief history - XI
¨ Sooo… bear with me here a few more slides..
¨ If I write:
¨ 𝑑𝑋 = {
,
,0
𝐷 𝑥 }. 𝑑𝑡 + 2𝐷(𝑋 𝑡 ). ([). 𝑑𝑊
¨ Numerically I will simulate it as:
¨ 𝛿𝑋 𝑡 =
,
,0
𝐷 𝑥 ± 2𝐷(𝑋 𝑡 )
¨ That is assuming units of time equal to 1, if not you write
¨ 𝛿𝑋 𝑡 =
,
,0
𝐷 𝑥 . 𝛿𝑡 ± 2𝐷 𝑋 𝑡 . 𝛿𝑡
115

116. Luc_Faucheux_2021
A brief history - XII
¨ I simulate: 𝛿𝑋 𝑡 =
,
,0
𝐷 𝑥 . 𝛿𝑡 ± 2𝐷 𝑋 𝑡 . 𝛿𝑡
¨ That is an ITO SDE: 𝑑𝑋 = {
,
,0
𝐷 𝑥 }. 𝑑𝑡 + 2𝐷(𝑋 𝑡 ). ([). 𝑑𝑊
¨ That has a PDE solution:
,=
,"
= −
,
,0
𝑎. 𝑝 −
,
,0
[
*
!
. 𝑏! . 𝑝]
¨ With: 𝑎 =
,
,0
𝐷 𝑥 and 𝑏 = 2𝐷(𝑋 𝑡 )
¨ So you get the PDE:
,=
,"
= −
,
,0
,
,0
𝐷 𝑥 . 𝑝 −
,
,0
[𝐷. 𝑝]
¨ And:
¨
,=
,"
= −
,
,0
,
,0
𝐷 𝑥 . 𝑝 −
,
,0
𝐷. 𝑝 = −
,
,0
,A
,0
. 𝑝 −
,A
,0
. 𝑝 − 𝐷.
,=
,0
= −
,
,0
−𝐷.
,=
,0
¨ Which is what I want
116

117. Luc_Faucheux_2021
A brief history - XIII
¨ SO…
¨ If I want to simulate:
,=
,"
= −
,
,0
−𝐷(𝑥).
,=
,0
¨ Which has no drift term, and has the equilibrium long-term steady state uniform distribution
as the desired solution, I need to simulate on a discrete digital computer:
¨ 𝛿𝑋 𝑡 =
,
,0
𝐷 𝑥 . 𝛿𝑡 ± 2𝐷 𝑋 𝑡 . 𝛿𝑡
¨ Which seems weird, because the first term
,
,0
𝐷 𝑥 . 𝛿𝑡 looks like the drift term.
¨ This is when the whole confusion started
117

118. Luc_Faucheux_2021
A brief history - XIV
¨ 𝛿𝑋 𝑡 =
,
,0
𝐷 𝑥 . 𝛿𝑡 ± 2𝐷 𝑋 𝑡 . 𝛿𝑡
¨ The term
,
,0
𝐷 𝑥 . 𝛿𝑡 looks like indeed what you would simulate if you were to simulate
a drift
¨ It is NOT a physical drift
¨ Your PDE that you wanted to simulate was
,=
,"
= −
,
,0
−𝐷.
,=
,0
and had NO drift
¨ It is purely a term that you need to add to the simulation to recover your PDE
¨ This is why it was called a “spurious drift” (Ryter 1980).
¨ Again, it is NOT a drift, you are not extraction motion out of noise, you need to have it in
there because you are using Ito calculus, and plain and simple:
¨ 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝑏. ([). 𝑑𝑊 will give you
,=
,"
= −
,
,0
𝑎. 𝑝 −
,
,0
[
*
!
. 𝑏! . 𝑝]
¨ Period..end of story…
118

119. Luc_Faucheux_2021
A brief history - XV
¨ But you can see why that was confusing, and why it tripped at the time a lot of people, yours
truly included.
¨ And that was confusing for 2 main reasons:
¨ Reason 1: Maxwell demon and Thermal ratchets
¨ Reason 2: Stratanovitch
¨ They are somewhat related but distinct.
¨ Let’s go over Stratanovitch first
119

120. Luc_Faucheux_2021
A brief history – XVI
¨ To be fair, I was also a little harsh and oversimplifying.
¨ There is also another reason why the question of Ito versus Stratanovitch came about, it has
to do with the autocorrelation of the random forcing (the noise). Is it white noise with 0
memory, and so Ito would be appropriate, or is it coloured noise, with some non zero
autocorrelation function, and in this case Strato would be more appropriate.
¨ Again it really depends on the physical system, an what PDE you want
¨ Radioactive particle decay lends itself well to Ito
¨ Diffusion in a viscous medium lends itself better to Strato from a physics point of view
¨ So as always things are a little more complicated that I would make them out to be, so you
should take what I say with a grain of salt.
120

121. Luc_Faucheux_2021
A brief history – XVII
¨ In particular, if there is any kind of correlation, then an non-ITO approach is sometimes
justified.
¨ Remember that we had from the earlier part of the deck:
¨ < A
𝑏 𝑡, 𝑋 𝑡 . [𝛼] . 𝑑𝑊 > = 𝛼. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡
¨ In particular:
¨ ITO : < A
𝑏 𝑡, 𝑋 𝑡 . [ . 𝑑𝑊 > = 0
¨ STRATO: < A
𝑏 𝑡, 𝑋 𝑡 . ∘ . 𝑑𝑊 > = [
*
!
]. A
𝑏 𝑡, 𝑋 𝑡 .
,
,-
A
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡
121

122. Luc_Faucheux_2021
Reason #2 for the confusion:
People started to hear about Stratanovitch
calculus
122

123. Luc_Faucheux_2021
Reason #2 for the confusion - Stratanovitch
¨ So first of all ITO calculus is weird because of ITO lemma, and the fact that the usual rules of
calculus (Leibniz rule, chain rule) are not formally respected
¨ Second ITO is very well adapted to DISCRETE processes in time (finance, but also simulations
on digital computers)
¨ Physicists do not like to learn new rules of calculus, they would rather stick to the usual one
¨ More seriously, in physics, it could be argued that VERY FEW processes are truly discrete
(radioactive decay for example being one of them), but most of the processes (like
diffusion), are not discrete but continuous
123

124. Luc_Faucheux_2021
Reason #2 for the confusion – Stratanovitch - II
¨ We know that using STRATO calculus:
¨ 𝑑𝑋 = @
𝑎. 𝑑𝑡 + A
𝑏. (∘). 𝑑𝑊
¨ Corresponds to a diffusion equation for the PDF:
¨
,=
,"
= −
,
,0
{@
𝑎 +
*
!
. A
𝑏.
,
,0
A
𝑏}. 𝑝 −
,
,0
[
*
!
. A
𝑏! . 𝑝]
¨ We can expand the last term into:
¨
,
,0
*
!
. A
𝑏! . 𝑝 =
,
,0
*
!
. A
𝑏. A
𝑏. 𝑝 =
*
!
.
,
,0
A
𝑏. A
𝑏. 𝑝 =
*
!
.
,
,0
A
𝑏. A
𝑏. 𝑝
¨
,
,0
*
!
. A
𝑏! . 𝑝 =
*
!
. A
𝑏. 𝑝 .
,
,0
A
𝑏 +
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝
¨
,
,0
*
!
. A
𝑏! . 𝑝 =
*
!
. A
𝑏.
,B
%
,0
. 𝑝 +
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝
124

125. Luc_Faucheux_2021
Reason #2 for the confusion – Stratanovitch - III
¨
,=
,"
= −
,
,0
{@
𝑎 +
*
!
. A
𝑏.
,
,0
A
𝑏}. 𝑝 −
,
,0
[
*
!
. A
𝑏! . 𝑝]
¨
,=
,"
= −
,
,0
@
𝑎 +
*
!
. A
𝑏.
,
,0
A
𝑏 . 𝑝 − {
*
!
. A
𝑏.
,B
%
,0
. 𝑝 +
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝 }
¨
,=
,"
= −
,
,0
@
𝑎 . 𝑝 −
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝
¨ So a STRATO SDE of the form:
¨ 𝑑𝑋 = @
𝑎. 𝑑𝑡 + A
𝑏. (∘). 𝑑𝑊
¨ Will give a PDE of the form:
¨
,=
,"
= −
,
,0
@
𝑎 . 𝑝 −
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝
125

126. Luc_Faucheux_2021
Reason #2 for the confusion – Stratanovitch - IV
¨ Again in order to illustrate let’s assume no drift term in the PDE
¨ So a STRATO SDE of the form:
¨ 𝑑𝑋 = A
𝑏. (∘). 𝑑𝑊
¨ Will give a PDE of the form:
¨
,=
,"
= −
,
,0
−
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝
¨ Again, nothing wrong there, but physicist got confused because they starting saying things
like the next slide:
126

127. Luc_Faucheux_2021
Reason #2 for the confusion – Stratanovitch - V
¨ If I write: 𝑑𝑋 = A
𝑏. (∘). 𝑑𝑊
¨ On a computer I simulate 𝛿𝑋 = ±A
𝑏
¨ And that gives me a PDE with no drift:
,=
,"
= −
,
,0
−
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝
¨ So I am happy, but now my diffusion current is no longer:
¨
,=
,"
= −
,
,0
𝐽A = −
,
,0
−𝐷(𝑥).
,=
,0
with A
𝑏 = 2𝐷
¨ But:
¨
,=
,"
= −
,
,0
−
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝
¨ So my diffusion current is really: 𝐽A = −
*
!
. A
𝑏.
,
,0
A
𝑏. 𝑝 = − 𝐷.
,
,0
𝐷. 𝑝
127

128. Luc_Faucheux_2021
Reason #2 for the confusion – Stratanovitch - VI
¨ So if I do not want to simulate a drit term (because my PDE has no drift term), then really I
need to write the diffusion current as:
¨ 𝐽A = − 𝐷.
,
,0
𝐷. 𝑝
¨ But then…wait a minute, if I plug this back into the PDE, then my equilibrium distribution is
no longer uniform (in fact it will be 𝐷. 𝑝 = 𝑐𝑡𝑒 or 𝑝 = 1/ 𝐷. )
¨ So when the diffusion coefficient is non-homogeneous it will change my equilibrium
distribution.
¨ I have “extracted” motion our of random noise
¨ This is maybe an example of the Maxwell demon
¨ I have broken the law of thermodynamics, so what is next ? Perpetual motion ?
128

129. Luc_Faucheux_2021
Reason #2 for the confusion – Stratanovitch - VII
¨ The previous 2 slides are obviously wrong but at the time that was a real debate in Physics
¨ For a constant diffusion the diffusion current is :
¨ 𝐽A = − 𝐷.
,
,0
𝐷. 𝑝 = −𝐷.
,
,0
𝑝 = −
,
,0
𝐷. 𝑝
¨ Because who cares, 𝐷 is a constant
¨ But now if we have 𝐷(𝑋(𝑡)) for a Brownian particle, which one is the real diffusion current?
¨ And then there were even more confusion about the Ryter spurious drift maybe being an
actual drift because
¨
,
,0
𝐷. 𝑝 =
,
,0
𝐷 . 𝑝 + 𝐷.
,
,0
𝑝
¨ So maybe 𝐷.
,
,0
𝑝 is the “real” diffusion current and the term
,
,0
𝐷 . 𝑝 is the Maxwell
demon term extracting directed motion out of a purely random diffusion process?
129

130. Luc_Faucheux_2021
Reason #2 for the confusion – Stratanovitch - VIII
¨ The whole controversy could have been squashed had there been one good math guy who
understood Ito and Strato well enough to explain it simply to physicist and say:
¨ Listen you idiots, all that really matters is the PDE
¨ If you want to play with a computer and burn some CPUs and start global warming by
writing something like: 𝛿𝑋 𝑡 = ± 2𝐷(𝑋 𝑡 )
¨ That is fine by me, but you are by definition using ITO calculus and writing:
¨ 𝑑𝑋(𝑡) = 2𝐷(𝑋 𝑡 ). ([). 𝑑𝑊(𝑡)
¨ Which will give you:
,=
,"
= −
,
,0
−
,
,0
[𝐷(𝑥). 𝑝]
¨ Which will not give you a uniform distribution for the steady state regime.
¨ Period, full stop, end of story.
130

131. Luc_Faucheux_2021
Reason #2 for the confusion – Stratanovitch - IX
¨ And then this math guy could have added:
¨ If you idiots want to simulate the SDE that corresponds to the PDE:
,=
,"
= −
,
,0
−𝐷(𝑥).
,=
,0
¨ And if you insist on using discrete numerical simulations, then sorry mate but you have no
choice but to write:
¨ 𝛿𝑋 𝑡 =
,
,0
𝐷 𝑥 ± 2𝐷 𝑋 𝑡
¨ So yeah, the first term looks like a drift term, but there is no physical drift in your PDE, it is
just a “spurious” drift because you now have entered the mysterious world of stochastic
calculus, and more precisely ITO calculus. And please leave at the door all the intuitions you
had built on Newtonian/Leibniz/regular calculus, and buckle up buttercup, because you not
in Kansas anymore Toto.
¨ Sometimes I wish math guys could not talk to physicists in such condescending manner.
131

132. Luc_Faucheux_2021
Reason #1 for the confusion:
Maxwell demon, Brownian motors, Thermal
ratchets
132

133. Luc_Faucheux_2021
Maxwell demon - I
¨ The other reason why there was some confusion is that at the same time a lot of physicists
were looking at issues like Maxwell demon, Brownian engines, Brownian ratchets, Thermal
ratchets…
¨ The Maxwell demon is a possible (but very improbable) entity that drive a system towards a
state that is possible but very improbable
¨ To quote the adventures of Mr. Tompkins, it is possible (but very unlikely) the ice cube that
you melted in your coke will spontaneously re-form as an ice cube and that the surrounding
liquid will warm back up again.
¨ That is possible, there is nothing in Physics that says that it is impossible, it is just very
unlikely
¨ Winning the lottery is also possible, it is just also highly unlikely
133

134. Luc_Faucheux_2021
Maxwell demon - II
¨ The Maxwell demon is intriguing because the 2nd Law of Thermodynamics says that usually
things tend to decay, and that a measure of disorder that you can define (entropy) usually
always increase at the end.
¨ That law makes sense, we all see it in our lives everyday, things always tend to go to sh..,
never (or highly unlikely) the other way around
¨ What is true for our personal lives is true for the Universe
¨ Nothing shocking there, since we are part of the universe, so the laws should be somewhat
universal
134

135. Luc_Faucheux_2021
Maxwell demon - III
¨ The Maxwell demon putting the moves on Mr. Tompkins fiancée and trying to impress her
with his tennis skills by reconstructing an ice cube without violating the 2nd law of
Thermodynamics
135

136. Luc_Faucheux_2021
Maxwell demon - IV
¨ A more common representation of the demon
136

137. Luc_Faucheux_2021
Brownian Engines
¨ So it would make sense for physicists to ask if it was possible to build a Maxwell demon.
¨ Feynman in his lecture has a great chapter on this.
¨ He even offered a prize to the first person / team who could build one.
137

138. Luc_Faucheux_2021
Brownian engines - II
¨ The canonical example of a thermal ratchet from the Feynman’s lectures in Physics
138

139. Luc_Faucheux_2021
Brownian engines - III
¨ There was also another reason why physicists were excited by all this, (none withstanding
the fact that if you could receive a check from Richard Feynman, that would be the coolest
thing ever, and you would have so much street cred), was that we were starting to look in
real-time in-vivo inside the cells and human body, and whether we like it or not (and again
nothing really to do with intelligent design theory), there were some really cool things
happening there.
¨ Vesicles are transported within the cell, to the place where they should be, in a very highly
“noisy” (read bombarded by random fluctuations) environment that should make it
impossible. Yet it works.
¨ Spermatozoides are able to “swim” to where they were supposed to go, again in a very
random and viscous environment.
¨ Trying to build a typical inertial engine and computing how much energy that required lead
to the conclusion that it was impossible, so clearly there was something akin to a Thermal
ratchet, or Brownian engine at work there. So somehow Nature found a way to extract
directed motion out of random noise, or at least found a way to still be able to build things
in a highly noisy, random and viscous regime without “forcing” its way through the way an
inertial engine (think of a car or a rocket) would do at the scale we are used to
139

140. Luc_Faucheux_2021
Brownian engines - IV
¨ I was lucky enough to work at the time in the lab of Albert Libchaber, who was
fascinated with those problems.
¨ I have never met in my life someone who was so pure in his search of the truth and who
was so keen on lifting the curtains, and who had the intellectual power to cut through
the math, or the obstacles between him and the hidden simple principles at work
¨ I count myself as being fortunate to have been able to witness him at work, and in his
life.
140

141. Luc_Faucheux_2021
Brownian engines - V
¨ I could write a book about how much fun we had in his lab (and maybe I should), but suffice
to say that we were all involved in the subject of Brownian engines, whether on the biology
side or on the more Physics side. If you guys are interested you have to read this one:
141

142. Luc_Faucheux_2021
Brownian engines - VI
¨ So anyways long story short, I spent a lot of time then on those issues, had the pleasure of
working with some of the finest minds of the time (too numerous to name all, and am not
even going to try, but of course Albert Libchaber, Mike Shelley, Dave Muraki, Marcello
Magnasco, Erez Braun, Elisha Moses, Drew Belmonte, Deborah Fygenson, Albrecht Ott,
Andreas Tilgner, Gustavo Stolovitsky, Rolf Landauer and is famous shout “complex is complex
is complex!”, and again way too many I apologize for not including in the short list above)
¨ But anyways we built the first optical thermal ratchet !
142

143. Luc_Faucheux_2021
Brownian engines - VII
¨ It turns out that there is a cool connection to Finance.
¨ If you replace the position by the accumulated PL, the thermal ratchet becomes what is
known as the Parrondo Paradox.
¨ If we just had thought about replacing X by $, that could have been known as the Libchaber
paradox
¨ Another example of the work of an illustrious Frenchman that is easily applicable to finance
but somehow someone else takes it through the goal line…
¨ Essentially, just like the thermal ratchet, you can “extract” some direction our of random
fluctuations (no worries about the 2nd law, it is still intact, because you need to dump energy
into the ratchet to make it work)
¨ So in Finance, you could have 2 losing strategies (2 loser PM in a fund), but if you switch
randomly the allocation of $ between those 2 strategies, you can find a regime where the
resulting strategy will make $ on average
143

144. Luc_Faucheux_2021
Brownian engines - VIII
¨ There are actually applications in Finance of the Maxell demon, known as the Parrondo
paradox.
¨ Two trading strategies (PM at a hedge fund) on average lose money (B and C, blue and green
line)
¨ However you can alternate between the two strategies to create one (A-red line) that on
average will be profitable, like the thermal ratchet who is extracting work out of thermal
noise, this Parrondo construct extract positive return out of random switches between two
losing strategies
144

145. Luc_Faucheux_2021
Another draft of retelling the story of the Ito-
Stratanovitch controversy
145

146. Luc_Faucheux_2021
Another way to tell the story – I
¨ This was a first draft of the story, it was a little more complicated than the one I went
through, but thought I would keep it here for sake of completeness.
¨ No need to read it
¨ It has nice pictures though
¨ Please enjoy
146

147. Luc_Faucheux_2021
Another way to tell the story - II
¨ All right, this is where all the confusion came about the Ito-Stratanovitch controversy (and
also that physicists tend to glance over stochastic calculus and be quite liberal and not that
mathematically rigorous, I have to admit)
¨ Suppose that we start with a PDE describing the following process:
¨
,=(0,")
,"
= −
,
,-
𝑎 𝑋 𝑡 , 𝑡 . 𝑝(𝑥, 𝑡) −
,
,-
[
*
!
. [𝑏(𝑋 𝑡 , 𝑡)! . 𝑝(𝑥, 𝑡)]
¨
,
,"
. 𝑝 𝑥, 𝑡 = −
,
,0
[𝐽/+𝐽A]
¨ 𝐽/ 𝑥, 𝑡 = 𝑎(𝑡). 𝑝 𝑥, 𝑡 and 𝐽A 𝑥, 𝑡 = −
,
,-
[
*
!
. [𝑏(𝑋 𝑡 , 𝑡)! . 𝑝(𝑥, 𝑡)]
¨ 𝐽A 𝑥, 𝑡 = −
,
,-
[
*
!
. [𝜎(𝑋 𝑡 , 𝑡)! . 𝑝(𝑥, 𝑡)]
¨ 𝐽A 𝑥, 𝑡 = −
,
,-
[𝐷(𝑥, 𝑡). 𝑝(𝑥, 𝑡)] with 𝐷 𝑥, 𝑡 =
*
!
. [𝑏(𝑋 𝑡 , 𝑡)! ]
147