section on short rate 2 (or 2+) factors models trip down memory lane, I am quite doubtful if anyone still use those models

1. Luc_Faucheux_2021
THE RATES WORLD – Part IV_a
Starting to look at modeling rates, a taxonomy of
models…appendix on IRMA, and mostly 2 factors
short rate models, plus my squid teacher
1

2. Luc_Faucheux_2021
Couple of notes on those slides
¨ This is expanding a little the section on the IRMA function
¨ More generally we are looking at multi factor short rate models
¨ Trip down memory lane about some short rate models that were common in the ‘80s and
‘90s
¨ Actually most of those came from Salomon Brothers, those guys had a 3 factor model up
and running in the late ‘70s, and with people moving to other firms it gradually became the
standard for a while
¨ These days I do not think that anyone still use it, as most the of the industry moved to LMM,
BGM and FMM models based on the forwards, not the short rate, as computing became
cheaper and faster (still rates is the requirements are “inordinately demanding” as Peter
Carr would say)
2

3. Luc_Faucheux_2021
IRMA
¨ If you get bored with one-factor short rate models, you can using multi-factors short rate
models.
¨ If you are a genius like Craig Fithian and worked at Salomon in 1972, you write (am using the
SIE form to be more compact) what got to be known worldwide as the 2+ IRMA model
¨ !
𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
𝑑𝑧 = −𝑘#. (𝑧 − 𝑥 − 𝑦). 𝑑𝑡 + 𝜎#. ([). 𝑑𝑊
#
¨ With: < 𝑑𝑊
!. 𝑑𝑊
! >= 𝜌. 𝑑𝑡
¨ And : 𝑟 𝑡 = 𝐼𝑅𝑀𝐴[𝑧 𝑡 + 𝜇 𝑡 ]
¨ Where 𝐼𝑅𝑀𝐴(z) is the IRMA function (Interest Rate Mapping I think) which created an
incredible stable skew, they had historical data on skew going back to the 1960
¨ IRMA was not named after the hurricane from 2017
3

4. Luc_Faucheux_2021
IRMA - II
¨ The Salomon IRMA function was NOT named after hurricane IRMA (2017)
4

5. Luc_Faucheux_2021
IRMA - III
¨ I recalled the day when working there I got my hands on the original code (which I think was
in FORTRAN) from 1972.
¨ Thinks about it, when Black Sholes came out, Salomon Brothers was running its swap and
option desk with a 3-factor short rate model with IRMA skew !!
¨ The trick was the calibration of the IRMA mapping.
¨ It was defined with 3 variables originally (we then extended to 4 to account for negative
rates, and yours truly tried to replace IRMA with SQUID, more to come on this), but the
original three variables were:
¨ <
Intercept 𝐼
Slope 𝑆
Regime Change 𝑅
5

6. Luc_Faucheux_2021
IRMA - IV
¨ The IRMA function 𝐼𝑅𝑀𝐴 𝑧 was actually defined by parametrizing the function:
¨ 𝑔 𝑟 =
$%&'( #
$#
=
$%&'(
$#
𝐼𝑅𝑀𝐴)* 𝑟 = 𝐼𝑅𝑀𝐴′(𝐼𝑅𝑀𝐴)* 𝑟 )
6
𝑟
𝑔(𝑟)
Regime Change 𝑅
Intercept 𝐼
Slope 𝑆

7. Luc_Faucheux_2021
IRMA - V
¨ Above the Regime Change 𝑅, the function IRMA is a straight line of slope 𝑆
¨ If that straight line was continued below the regime change 𝑅 it would intercept the y-axis at
the Intercept 𝐼
¨ Below the Regime Change 𝑅, the function IRMA is a quadratic function that connect with
the straight line at the point (𝑅, 𝐼 + 𝑆. 𝑅) and goes through the origin (0,0)
¨ Note, when I got there in 2002, that function was still used throughout the firm and
matched the observed skew in a very stable and remarkable manner. We dabbled into
tweaking it for negative rates by essentially adding a parameter similar to the shifted
lognormal model, so that the above sentence got changed to:
¨ Below the Regime Change 𝑅, the function IRMA is a quadratic function that connect with
the straight line at the point (𝑅, 𝐼 + 𝑆. 𝑅) and goes through a point (−𝑁, 0) left of the origin
7

8. Luc_Faucheux_2021
IRMA - VI
¨ You can see the beauty of this: 𝑟 𝑡 = 𝐼𝑅𝑀𝐴[𝑧 𝑡 + 𝜇 𝑡 ]
¨ IF (𝑆 = 0, 𝑅 = 0, 𝐼 = 𝑐𝑡𝑒) then we have: 𝑔 𝑟 = 𝐼 and so
$%&'( #
$#
= 𝐼
¨ 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 = 𝐼. 𝑧 + 𝐶 is a linear function of the Gaussian variable 𝑧 𝑡 , the model will
produce a NORMAL skew
¨ IF (𝑆 = 𝑐𝑡𝑒, 𝑅 = 0, 𝐼 = 0) then we have: 𝑔 𝑟 = 𝑆. 𝑟 and so
$%&'( #
$#
= 𝑆. 𝐼𝑅𝑀𝐴(𝑧)
¨ We then get: 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 = exp 𝑆. 𝑧 + 𝐶
¨ 𝑟 𝑡 is an exponential function of the Gaussian variable 𝑧 𝑡 , the model will produce a
LOGNORMAL skew
¨ The quadratic part under the Regime Change 𝑅 when non-zero will fold the distribution of
𝑧 𝑡 back on positive rates, so the model avoids negative rates (which for a while was
deemed to be a good thing, unless things changed)
¨ We will go back to all the beautiful ways we can parametrize IRMA to recover the market
skew and smile
8

9. Luc_Faucheux_2021
IRMA - VII
¨ The 3 parameters had been calibrated to historical data for the skew covering like 40 years
of historical market moves or so, which was in itself amazing (the fact that Salomon had a
clean database that you could use that was going back so far)
¨ The 3 parameters were surprisingly stable, and essentially produced something that was
getting Lognormal at low rates below the Regime Change 𝑅, and closer to Normal above the
Regime Change 𝑅
¨ Ask anyone who worked on the options desk there and worked with 2+IRMA, and they
might still remember by heart those parameters
¨ <
Intercept 𝐼 = 0.06
Slope 𝑆=0.2
Regime Change 𝑅 = 0.01
¨ Yours truly worked on implemented SQUID in order to recover the market skew and smile at
high strike (SQUID=Skew of Quadratic Interest rate Distribution)
9

10. Luc_Faucheux_2021
IRMA - VIII
¨ The SQUID function 𝑔(𝑟) in order to recover skew and smile. Long live baby squid Cthulhu !!
10
𝑟
𝑔(𝑟)
Intercept 𝐼
Slope 𝑆
Regime Change 𝑅! Quadratic Switch 𝑅"

11. Luc_Faucheux_2021
2+ IRMA in the literature
11

12. Luc_Faucheux_2021
IRMA - VIII
¨ The model became quite widely known and adopted in the literature and in the financial
industry.
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ 𝑑𝑧 = −𝑘#. (𝑧 − 𝑥 − 𝑦). 𝑑𝑡 + 𝜎#. ([). 𝑑𝑊
#
¨ With: < 𝑑𝑊
!. 𝑑𝑊
! >= 𝜌. 𝑑𝑡
¨ And : 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹(𝑧)
¨ It is referred to as a 3 factors OU (Ornstein-Uhlenbeck) process
¨ Remember Ornstein-Uhlenbeck is only a fancy way to say “Langevin”
12

13. Luc_Faucheux_2021
IRMA - IX
¨ At Citi, it was known as 2+
¨ The reason it was known as 2+ is actually quite funny. A two factor version had been
implemented previously and had been approved internally by Risk, modeling,…
¨ A three factor version would have been too much work to go through all the documentation
and approval processes (remember at the time things were a little more flexible), so it was
easier to pass the new 3-factor model as a modification (a +) on the existing 2-factor model
and call it 2+
¨ Every time an option trader drives on the highway and sees the sign for the HOV (High
Occupancy Lanes) with usually an indication of “2+”, meaning that you can be on the HOV if
you have 2 passengers or more in your car, he or she think with sweet longing about the 2+
IRMA model, at least I know I do
13

14. Luc_Faucheux_2021
Piterbarg 2 factor Gaussian model
14

15. Luc_Faucheux_2021
IRMA - Piterbarg - I
¨ In the literature, it is sometimes referred to (with some tweaks) as the Gaussian model.
¨ In Piterbarg (p. 494), you see the original formulation of the ”2 Factor Gaussian model”
¨ Piterbarg actually writes the equations as:
¨ 𝑑𝜀 = −𝑘+. 𝜀. 𝑑𝑡 + 𝜎+. ([). 𝑑𝑊
+
¨ 𝑑𝑟 = 𝑘,. {𝜗 𝑡 + 𝜀 𝑡 − 𝑟 𝑡 }. 𝑑𝑡 + 𝜎,. ([). 𝑑𝑊
,
¨ With: < 𝑑𝑊
+. 𝑑𝑊
, >= 𝜌. 𝑑𝑡
15

16. Luc_Faucheux_2021
IRMA - Piterbarg - II
¨ But you can break those down as:
¨ Step 1: replace 𝜀 𝑡 by 𝑥 𝑡 (that is an easy one…)
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑𝑟 = 𝑘,. {𝜗 𝑡 + 𝑥 𝑡 − 𝑟 𝑡 }. 𝑑𝑡 + 𝜎,. ([). 𝑑𝑊
,
¨ With: < 𝑑𝑊
!. 𝑑𝑊
, >= 𝜌. 𝑑𝑡
16

17. Luc_Faucheux_2021
IRMA - Piterbarg - III
¨ Step 2: forget about the second equation because this is a 2-factor model, not a 3-factor one
¨ So forget about 𝑦 being stochastic variable, it is a deterministic variable
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑦(𝑡) = 𝜗 𝑡
¨ 𝑑𝑧 = 𝑘#. {𝑦 𝑡 + 𝑥 𝑡 − 𝑧 𝑡 }. 𝑑𝑡 + 𝜎#. ([). 𝑑𝑊
#
¨ With: < 𝑑𝑊
!. 𝑑𝑊
# >= 𝜌. 𝑑𝑡
¨ And : 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹 𝑧 = 𝑧
¨ That is as simple an IRMA function as we can get, and is called Gaussian because it does not
perturb the initial Gaussian distribution of the Langevin equations. I will add the slides from
the Langevin deck to remind us that the probability distribution is a Gaussian indeed
¨ Also the linear transformation 𝐹 𝑧 = 𝑧 keeps the Gaussian untouched
17

18. Luc_Faucheux_2021
IRMA - Piterbarg - IV
¨ In that formulation:
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑦(𝑡) = 𝜗 𝑡
¨ 𝑑𝑧 = 𝑘#. {𝑦 𝑡 + 𝑥 𝑡 − 𝑧 𝑡 }. 𝑑𝑡 + 𝜎#. ([). 𝑑𝑊
#
¨ With: < 𝑑𝑊
!. 𝑑𝑊
# >= 𝜌. 𝑑𝑡
¨ And : 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹 𝑧 = 𝑧
¨ It is usually customary to think of the variable 𝑥(𝑡) as the “short-term rate” or more exactly
shocks in the front end of the curve
¨ The variable 𝑦(𝑡) = 𝜗 𝑡 is then the slope of the yield curve
¨ Bear in mind though that this is still a short rate model and not a term structure model
18

19. Luc_Faucheux_2021
IRMA - Piterbarg - V
¨ Alternatively, to make it even closer to 2+IRMA, we can write it as:
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ 𝑑𝑧 = −𝑘#. (𝑧 − 𝑥 − 𝑦). 𝑑𝑡 + 𝜎#. ([). 𝑑𝑊
#
¨ With: < 𝑑𝑊
!. 𝑑𝑊
! >= 𝜌. 𝑑𝑡
¨ And : 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹 𝑧 = 𝑧 + 𝜗 𝑡
¨ With:
¨ 𝜎" = 0
¨ 𝑘" = 0
¨ 𝑦 𝑡 = 𝑦 0 = 0
19

20. Luc_Faucheux_2021
Mercurio G2++ model
20

21. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - I
¨ In the Mercurio book it is referred to as the G2++ model, which is quite close to the original
“2+” terminology at Salomon Brothers (p.132)
21

22. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - II
¨ Mercurio writes down the model as (p.133)
¨ 𝑑𝑥 = −𝑎. 𝑥. 𝑑𝑡 + 𝜎. ([). 𝑑𝑊
!
¨ 𝑑𝑦 = −𝑏. 𝑦. 𝑑𝑡 + 𝜂. ([). 𝑑𝑊
"
¨ 𝑟 𝑡 = 𝑥 𝑡 + 𝑦 𝑡 + 𝜑(𝑡)
¨ With: < 𝑑𝑊
!. 𝑑𝑊
! >= 𝜌. 𝑑𝑡
¨ We can see again that we can recast those in the 2+IRMA format with:
¨ 𝑘! = 𝑎
¨ 𝑘" = 𝑏
¨ 𝜎! = 𝜎
¨ 𝜎! = 𝜂
22

23. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - III
23
¨ We then get:
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ 𝑟 𝑡 = 𝑥 𝑡 + 𝑦 𝑡 + 𝜑(𝑡)
¨ With: < 𝑑𝑊
!. 𝑑𝑊
! >= 𝜌. 𝑑𝑡
¨ Again because it is a 2 factor model, the third variable in this case is deterministic and set to:
¨ 𝑧 𝑡 = 𝑥 𝑡 + 𝑦 𝑡
¨ Which itself is the limit of the original:
¨ 𝑑𝑧 = −𝑘#. (𝑧 − 𝑥 − 𝑦). 𝑑𝑡 + 𝜎#. ([). 𝑑𝑊
#
¨ With 𝜎# = 0 and 𝑘# → ∞
¨ And then 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹 𝑧 = 𝑧 𝑡 + 𝜑(𝑡)

24. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - IV
¨ So the G2++ model from Mercurio is also the 2+IRMA with the following:
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ 𝑑𝑧 = −𝑘#. (𝑧 − 𝑥 − 𝑦). 𝑑𝑡 + 𝜎#. ([). 𝑑𝑊
#
¨ With: < 𝑑𝑊
!. 𝑑𝑊
! >= 𝜌. 𝑑𝑡
¨ And : 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹(𝑧)
¨ 𝜎# = 0
¨ 𝑘# → ∞ so 𝑥 + 𝑦 − 𝑧 → 0 so 𝑧 𝑡 = 𝑥 𝑡 + 𝑦(𝑡)
¨ 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹 𝑧 = 𝑧 𝑡 + 𝜑(𝑡)
¨ So again the fact that the IRMA function 𝐹 𝑧 is affine enforces the Gaussian distribution
24

25. Luc_Faucheux_2021
Why do we call those models “Gaussian” ?
25

26. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - V
¨ Why is called Gaussian by the way ? Looks pretty complicated, how do we know that
distribution is a Gaussian ?
¨ First of all, let’s look at the equations for 𝑥(𝑡) and 𝑦(𝑡)
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ They BOTH individually fit the Langevin equation (or if you are in Finance you call that OU
process, again just ask Ranjit Bhattacharjee, the king of OU models)
¨ Time for a little refresher on the results of the Langevin deck, using some of those slides
verbatim
26

27. Luc_Faucheux_2021
PDF for the Langevin equation - XVI
¨ Langevin equation is usually for the particle velocity:
¨ 𝑑𝑉 𝑡 = −𝑘. 𝑉 𝑡 . 𝑑𝑡 + 𝜎. ([). 𝑑𝑊
¨ With the usual Diffusion coefficient 𝐷 =
-!
.
¨ 𝑝/ 𝑣, 𝑡|𝑚* 0 , 𝑡 = 0 =
0
.12.(*)567 ).08 )
. 𝑒𝑥𝑝(−𝑘
(:);" < .567 )08 )!
.2.(*)567 ).08 )
)
¨ Remember 𝑉 𝑡 is the stochastic process
¨ 𝑣 is a regular variable
¨ 𝑃/ 𝑣, 𝑡 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑉 ≤ 𝑣, 𝑡 = ∫
"=)>
"=:
𝑝/ 𝑦, 𝑡 . 𝑑𝑦
¨ 𝑝/(𝑣, 𝑡) =
?
?:
𝑃/ 𝑣, 𝑡 sometimes just noted 𝑝 𝑣, 𝑡|𝑚* 0 , 𝑡 = 0
27

28. Luc_Faucheux_2021
PDF for the Langevin equation - XVIII
¨ After much calculation, this is the celebrated Langevin PDF:
¨ 𝑝 𝑣, 𝑡|𝑚* 0 , 𝑡 = 0 =
0
.12.(*)567 ).08 )
. 𝑒𝑥𝑝(−𝑘
(:);" < .567 )08 )!
.2.(*)567 ).08 )
)
¨ SMALL TIME LIMIT
¨ IF 𝑡 → 0
0
2.(*)567 ).08 )
=
*
.28
+ 𝕆 𝑡.
¨ 𝑝 𝑣, 𝑡|𝑚* 0 , 𝑡 = 0 →
*
@128
. 𝑒𝑥𝑝(−
(:);" < )!
@28
)
¨ At short time scales (underdamped regime), the Langevin diffuses as a regular diffusion
process
¨ 𝑑𝑉 𝑡 = −𝑘𝑉 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊 → 𝜎. 𝑑𝑊
28

29. Luc_Faucheux_2021
PDF for the Langevin equation - XIX
¨ SMALL 𝑘 limit
¨ IF 𝑘 → 0
0
2.(*)567 ).08 )
=
*
.28
+ 𝕆 𝑘.
¨ 𝑝 𝑣, 𝑡|𝑚* 0 , 𝑡 = 0 →
*
@128
. 𝑒𝑥𝑝(−
(:);" < )!
@28
)
¨ This is expected since when 𝑘 → 0 we should recover the usual diffusion:
¨ 𝑑𝑉 𝑡 = −𝑘𝑉 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊 → 𝜎. 𝑑𝑊
29

30. Luc_Faucheux_2021
PDF for the Langevin equation - XX
¨ STEADY STATE LIMIT
¨ IF 𝑡 → ∞
0
2.(*)567 ).08 )
=
0
2
+ 𝕆 𝑡)*
¨ 𝑝 𝑣, 𝑡|𝑚* 0 , 𝑡 = 0 =
0
.12.(*)567 ).08 )
. 𝑒𝑥𝑝(−𝑘
(:);" < .567 )08 )!
.2.(*)567 ).08 )
)
¨ 𝑝 𝑣, 𝑡 → ∞|𝑚* 0 , 𝑡 = 0 =
0
.12
. 𝑒𝑥𝑝(−𝑘
:!
.2
)
¨ This is referred to as the “invariant Gaussian distribution”
30

31. Luc_Faucheux_2021
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 =
*
.1;! 8
. 𝑒𝑥𝑝(−
(:);" 8 )!
.;! 8
)
¨ 𝑝 𝑣, 𝑡 = 𝑝 𝑣, 𝑡|𝑉< = 𝑚* 0 , 𝑡 = 0 =
*
.1;! <
. 𝑒𝑥𝑝(−
(:);" < )!
.;! <
)
31

32. Luc_Faucheux_2021
PDF for the Langevin equation - XXII
¨ 𝑝 𝑣, 𝑡|𝑉 𝑡A , 𝑡A =
0
.12.(*)B#!$(&#&'))
. 𝑒𝑥𝑝(−𝑘
(:);" 8' .B#$(&#&'))!
.2(*)B#!$(&#&'))
)
¨ 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 𝑡A , 𝑡A and not on the entire
history before)
¨ 𝑝 𝑣, 𝑡|{𝑉 𝑠 , 𝑠 ≤ 𝑡A} = 𝑝 𝑣, 𝑡|𝑉 𝑡A , 𝑡A
¨ The Langevin process is stationary (only depends on (𝑡 − 𝑡A))
¨ 𝑝 𝑣, 𝑡 + ℎ|𝑉 𝑡A + ℎ = 𝑉
A, 𝑡A + ℎ = 𝑝 𝑣, 𝑡|𝑉
A, 𝑡A
¨ 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
32

33. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - VI
¨ 𝑑𝑉 𝑡 = −𝑘. 𝑉 𝑡 . 𝑑𝑡 + 𝜎. ([). 𝑑𝑊
¨ 𝑝 𝑣, 𝑡|𝑉 𝑡A = 𝑣A, 𝑡A =
0
.12.(*)B#!$(&#&'))
. 𝑒𝑥𝑝(−𝑘
(:):'.B#$(&#&'))!
.2(*)B#!$(&#&'))
)
¨ 𝑑𝑋 = −𝑘!. 𝑋. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
! with 𝐷! =
-)
!
.
¨ 𝑝C 𝑥, 𝑡|𝑋 𝑡A = 𝑥A, 𝑡A =
0)
.12). *)B#!$). &#&'
. 𝑒𝑥𝑝(−𝑘!.
(!)!'.B#$).(&#&'))!
.2).(*)B#!$).(&#&'))
)
¨ 𝑑𝑌 = −𝑘". 𝑌. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
" with 𝐷" =
-+
!
.
¨ 𝑝D 𝑦, 𝑡|𝑌 𝑡A = 𝑦A, 𝑡A =
0+
.12+. *)B#!$+. &#&'
. 𝑒𝑥𝑝(−𝑘".
(")"'.B#$+.(&#&')
)!
.2+.(*)B#!$+.(&#&')
)
)
33

34. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - VII
¨ So both 𝑋 𝑡 and 𝑌 𝑡 are normally distributed (the Probability distribution function is a
Gaussian).
¨ Let’s pick one
¨ 𝑝C 𝑥, 𝑡|𝑋 𝑡A = 𝑥A, 𝑡A =
0)
.12). *)B#!$). &#&'
. 𝑒𝑥𝑝(−𝑘!.
(!)!'.B#$).(&#&'))!
.2).(*)B#!$).(&#&'))
)
¨ This has a mean (average, expected value) given by:
¨ 𝑀 𝑋 𝑡 = 𝔼 𝑋 = 𝔼8
E)
{𝑋 𝑡 𝔉 𝑡A = 𝑥A. 𝑒)0). 8)8' = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 = 𝔼 (𝑋(𝑡) − 𝑀 𝑋 𝑡 ). = 𝔼8
E)
{(𝑋(𝑡) − 𝑀 𝑋 𝑡 ). 𝔉 𝑡A
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
34

35. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - VIII
¨ Another way to see that is to start from the SDE:
¨ 𝑑𝑋 = −𝑘!. 𝑋. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
! with 𝐷! =
-)
!
.
¨ We write the SIE using the ITO integral:
¨ Let’s define j
𝑋 𝑡 = exp 𝑘!. 𝑡 . 𝑋(𝑡) and use the ITO lemma
¨ 𝑑 j
𝑋 =
? F
C
?C
. [ . 𝑑𝑋 +
*
.
.
?! F
C
?C! . 𝑑𝑋. +
? F
C
?8
. [ . 𝑑𝑡
¨
? F
C
?8
= 𝑘!. 𝑡. 𝑋 𝑡
¨
? F
C
?C
= exp(𝑘!. 𝑡)
¨
?! F
C
?C! = 0
35

36. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - IX
¨ Again, remember that we used the notation for ITO lemma for sake of ease, what you really
have is a regular function
¨ j
𝑋 = 𝑓 𝑋 Stochastic Variable
¨ l
𝑥 = 𝑓 𝑥 Regular “Newtonian” variable with well defined partial derivatives
¨ 𝛿𝑓 =
?G
?!
. 𝛿𝑋 +
*
.
.
?!G
?!! . (𝛿𝑋). +
?G
?8
. 𝑑𝑡
¨
?G
?!
=
?G
?!
|!=C 8 ,8
¨
?!G
?!! =
?!G
?!! |!=C 8 ,8
¨
?G
?8
=
?G
?8
|!=C 8 ,8
36

37. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - X
¨ We then get:
¨ 𝑑 j
𝑋 = exp 𝑘!. 𝑡 . ([). 𝑑𝑋 + 𝑘!. 𝑡. 𝑋 𝑡 . ([). 𝑑𝑡
¨ 𝑑 j
𝑋 = exp 𝑘!. 𝑡 . ([). (−𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. 𝑑𝑊
!) + 𝑘!. 𝑡. 𝑋 𝑡 . ([). 𝑑𝑡 = exp 𝑘!. 𝑡 . 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑 j
𝑋 = exp 𝑘!. 𝑡 . 𝜎!. ([). 𝑑𝑊
!
¨ Which we should really write as an SIE anyways:
¨ j
𝑋 𝑡I − j
𝑋 𝑡A = ∫
8=8A
8=8I
𝑑 j
𝑋 𝑡 = ∫
8=8A
8=8I
exp 𝑘!. 𝑡 . 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ exp 𝑘!. 𝑡I . 𝑋 𝑡I − exp 𝑘!. 𝑡A . 𝑋 𝑡A = ∫
8=8A
8=8I
exp 𝑘!. 𝑡 . 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ 𝑋 𝑡I = exp −𝑘!. 𝑡I . {exp 𝑘!. 𝑡A . 𝑋 𝑡A + ∫
8=8A
8=8I
exp 𝑘!. 𝑡 . 𝜎!. ([). 𝑑𝑊
!(𝑡)}
37

38. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XI
¨ 𝑋 𝑡I = exp −𝑘!. 𝑡I . {exp 𝑘!. 𝑡A . 𝑋 𝑡A + ∫
8=8'
8=8,
exp 𝑘!. 𝑡 . 𝜎!. ([). 𝑑𝑊
!(𝑡)}
¨ 𝑋 𝑡I = 𝑋 𝑡A . exp −𝑘!. (𝑡I − 𝑡A) + exp −𝑘!. 𝑡I . ∫
8=8'
8=8,
exp 𝑘!. 𝑡 . 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ 𝑋 𝑡I = 𝑋 𝑡A . exp −𝑘!. (𝑡I − 𝑡A) + ∫
8=8'
8=8,
exp −𝑘!. 𝑡I − 𝑡 . 𝜎!. ([). 𝑑𝑊
! 𝑡
¨ 𝑋 𝑡I = 𝑋 𝑡A . 𝑒)0).(8,)8') + ∫
8=8'
8=8,
𝑒)0). 8,)8 . 𝜎!. ([). 𝑑𝑊
! 𝑡
¨ 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0).(8)8') + ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢
¨ From there on, we know that the ITO integral is a martingale:
¨ 𝑀 𝑋 𝑡 = 𝔼 𝑋 = 𝔼8
E)
{𝑋 𝑡 𝔉 𝑡A
38

39. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XII
¨ 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0).(8)8') + ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢
¨ 𝑀 𝑋 𝑡 = 𝔼 𝑋 = 𝔼8
E)
{𝑋 𝑡 𝔉 𝑡A
¨ 𝑀 𝑋 𝑡 = 𝔼8
E)
{𝑋 𝑡A . 𝑒)0).(8)8') + ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢 𝔉 𝑡A
¨ 𝔼8
E)
{𝑋 𝑡A . 𝑒)0). 8)8' 𝔉 𝑡A = 𝑋 𝑡A . 𝑒)0). 8)8'
¨ 𝔼8
E)
{∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢 𝔉 𝑡A = 0 since the ITO integral is a martingale.
¨ This is at times a supe useful trick, to take the expected value of something that simplifies if
it contains an ITO integral
¨ Calin book page 195 has a couple of nifty applications of this trick.
¨ 𝑀 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0). 8)8'
39

40. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XIII
¨ The variance is a little more complicated but also relies on a nifty little property of the ITO
integral, the isometry property.
¨ So as a rule:
¨ Average – Mean – Expected Value: use the fact that ITO integrals are martingale
¨ Second moment – variance – standard deviation : use the fact that the ITO integral exhibits
the Isometry property
¨ 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0).(8)8') + ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢
¨ 𝑉 𝑋 𝑡 = 𝔼 (𝑋(𝑡) − 𝑀 𝑋 𝑡 ). = 𝔼8
E)
{(𝑋(𝑡) − 𝑀 𝑋 𝑡 ). 𝔉 𝑡A
¨ 𝑀 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0). 8)8'
¨ 𝑋 𝑡 − 𝑀 𝑋 𝑡 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢
40

41. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XIV
¨ 𝑋 𝑡 − 𝑀 𝑋 𝑡 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢
¨ 𝑉 𝑋 𝑡 = 𝔼 (𝑋(𝑡) − 𝑀 𝑋 𝑡 ). = 𝔼8
E)
{(𝑋(𝑡) − 𝑀 𝑋 𝑡 ). 𝔉 𝑡A
¨ 𝑉 𝑋 𝑡 = 𝔼8
E)
{(∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢 ). 𝔉 𝑡A
¨ This is where the isometry property comes handy.
¨ Let us remind us first what it is:
¨ 𝔼{ ∫
K=<
K=8
𝑓 𝑠 . 𝑑𝑊 𝑠
.
} = ∫
K=<
K=8
𝑓 𝑠 .. 𝑑𝑠
¨ 𝔼{ ∫
K=<
K=8
𝑓 𝑊 𝑠 , 𝑠 . 𝑑𝑊 𝑠
.
} = ∫
K=<
K=8
𝑓 𝑊 𝑠 , 𝑠 .. 𝑑𝑠
41

42. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XV
¨ 𝑉 𝑋 𝑡 = 𝔼8
E)
{(∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢 ). 𝔉 𝑡A
¨ 𝑉 𝑋 𝑡 = 𝔼8
E)
{∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!
.
. 𝑑𝑢 𝔉 𝑡A
¨ 𝑉 𝑋 𝑡 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!
.
. 𝑑𝑢
¨ 𝑉 𝑋 𝑡 = 𝜎!
.. ∫
J=8'
J=8
𝑒)..0). 8)J . 𝑑𝑢
¨ 𝑉 𝑋 𝑡 = 𝜎!
.. [
*
..0)
. 𝑒)..0). 8)J ]J=8'
J=8
¨ 𝑉 𝑋 𝑡 =
-)
!
..0)
. [1 − 𝑒..0). 8)8' ]
42

43. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XVI
¨ So we do recover:
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
43

44. Luc_Faucheux_2021
A sum of independent normally distributed
variables is also normally distributed
44

45. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XVII
¨ OK, so we know that both 𝑋 𝑡 and 𝑌 𝑡 are NORMALLY distributed (the Probability
Distribution function) is a Gaussian with mean and variance for 𝑋 𝑡 (and respectively same
for 𝑌 𝑡 by changing the notation)
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
¨ So that could be an indication that defining:
¨ 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹 𝑧 = 𝑧 𝑡 + 𝜑 𝑡 = 𝑥 𝑡 + 𝑦(𝑡) + 𝜑 𝑡
¨ Could lead to a Gaussian distribution for 𝑟 𝑡
¨ Really to stick to a somewhat consistent notation, we should have used capital letters for the
stochastic processes:
¨ 𝑅 𝑡 = 𝐼𝑅𝑀𝐴 𝑍 𝑡 + 𝜇 𝑡 = 𝐹 𝑍 = 𝑍 𝑡 + 𝜑 𝑡 = 𝑋 𝑡 + 𝑌(𝑡) + 𝜑 𝑡
¨ Remember that 𝜑 𝑡 is a deterministic function and not a stochastic process
45

46. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XVIII
¨ NOW, we also know by now that the sum of a bunch of independent variables that are
normally distributed is also normally distributed
¨ The mean of the sum is the sum of the means
¨ The variance of the sum is the sum of the variances
¨ Suppose that 𝑋L are a number of independent random variables normally distributed with
respective means 𝑚L and variances 𝑣L
¨ Yeah I know I have been using capital letter 𝑀L and 𝑉L for those
¨ Promised, once I get the book deal and rewrite it I will hire a couple of interns to ensure the
consistency of the notation throughout those decks
¨ Then the sum 𝑋 = ∑ 𝑋L has:
¨ Mean: 𝑀 = ∑ 𝑀L
¨ Variance: 𝑉 = 𝜎. = ∑ 𝑉L = ∑ 𝜎L
.
46

47. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XIX
¨ Another way that we knew that was in the deck on the diffusion (Stochastic calculus II if I am
not mistaken), when looking at solution of the SDE:
¨ dX(t)= a(t).dt+b(t).dW
¨ Let’s refresh our memory with a couple of slides from this deck
47

48. Luc_Faucheux_2021
Sixth simple example – VIII dX= a(t).dt+b(t).dW
¨ So we have the mapping:
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡 . 𝑑𝑡 + 𝑏(𝑡). ([). 𝑑𝑊
¨
?
?8
. 𝑝 𝑥, 𝑡 = −
?
?!
𝑎(𝑡). 𝑝(𝑥, 𝑡) −
I 8 !
.
.
?M !,8
?!
= −
?
?!
𝐽N + 𝐽2
¨ 𝐽N = 𝑎 𝑡 . 𝑝(𝑥, 𝑡) and 𝐽2 𝑥, 𝑡 = −𝐷(𝑡).
?M !,8
?!
¨ Defining:
¨ z
𝜎(𝑡).. 𝑡 = ∫
K=<
K=8
𝜎 𝑠 .. 𝑑𝑠 setting 𝜎 𝑡 = 𝑏(𝑡) and 𝐷 𝑡 =
-(8)!
.
¨ Also defining: z
𝐷 𝑡 . 𝑡 = ∫
K=<
K=8
𝐷(𝑠). 𝑑𝑠
¨ z
𝐷 𝑡 is the average diffusion coefficient over time
¨ z
𝜎 𝑡 is the average volatility coefficient over time
48

49. Luc_Faucheux_2021
Sixth simple example – IX dX= a(t).dt+b(t).dW
¨ Also defining:
¨ 𝑅 𝑡 = 𝑋 𝑡 = 𝑋< + ∫
8=8<
8
𝑎(𝑠). 𝑑𝑠
¨ 𝑉 𝑡 = z
𝑏(𝑡).. 𝑡 = ∫
K=<
K=8
𝑏 𝑠 .. 𝑑𝑠 = z
𝜎(𝑡).. 𝑡 = ∫
K=<
K=8
𝜎 𝑠 .. 𝑑𝑠 = z
2𝐷 𝑡 . 𝑡 = 2 ∫
K=<
K=8
𝐷(𝑠). 𝑑𝑠
¨ A PDF solution for the above PDE is:
¨ Subject to 𝑝 𝑥, 𝑡 = 𝑡< = 𝛿 𝑥 − 𝑋 𝑡< = 𝛿(𝑥 − 𝑋<)
¨ 𝑝 𝑥, 𝑡 =
*
.1O
- 8 ! 8)8-
. 𝑒𝑥 𝑝 −
!)& 8
!
.O
- 8 ! 8)8-
=
*
@1O
2(8) (8)8-)
. 𝑒𝑥𝑝(−
(!)& 8 )!
@O
2(8)(8)8-)
)
49

50. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XX
¨ So this makes sense if you think of the sum of independent normal variables if you index those
variables with time
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡 . 𝑑𝑡 + 𝑏(𝑡). ([). 𝑑𝑊
¨ 𝑋 𝑡 − 𝑋(𝑡A) = ∫
J=8'
J=8
𝑑𝑋(𝑢)
¨ At each time 𝑡, the little increment 𝑑𝑋 𝑡 is picked from a normal distribution with mean
𝑎 𝑡 . 𝑑𝑡 and variance 𝑏 𝑡 .. 𝑑𝑡
¨ So 𝑋 𝑡 − 𝑋(𝑡A) is picked from a normal distribution with:
¨ 𝑅 𝑡 = 𝑋 𝑡 = 𝑀[𝑋(𝑡)] = 𝑀 𝑋 𝑡 = 𝔼 𝑋 = 𝔼8
E)
{𝑋 𝑡 𝔉 𝑡A = 𝑋(𝑡A) + ∫
8=8'
8
𝑎(𝑠). 𝑑𝑠
¨ Variance: 𝑉 𝑡 = z
𝑏 𝑡 .. (𝑡 − 𝑡A) = ∫
K=8'
K=8
𝑏 𝑠 .. 𝑑𝑠 = z
𝜎 𝑡 .. (𝑡 − 𝑡A) = ∫
K=8'
K=8
𝜎 𝑠 .. 𝑑𝑠
¨ 𝑉 𝑡 = z
2𝐷 𝑡 . (𝑡 − 𝑡A) = 2 ∫
K=8'
K=8
𝐷(𝑠). 𝑑𝑠
50

51. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXI
¨ At each time 𝑡, the little increment 𝑑𝑋 𝑡 is picked from a normal distribution with mean
𝑎 𝑡 . 𝑑𝑡 and variance 𝑏 𝑡 .. 𝑑𝑡
¨ 𝑋 𝑡 − 𝑋(𝑡A) is picked from a normal distribution with:
¨ Mean: ∫
8=8'
8
𝑎(𝑠). 𝑑𝑠
¨ Variance: ∫
K=8'
K=8
𝑏 𝑠 .. 𝑑𝑠
¨ So this should not come as a surprise that a sum of normally distributed random variable is
itself normally distributed
¨ But WAIT a second you should say, the case we are looking at is NOT independent as we
have: < 𝑑𝑊
!. 𝑑𝑊
! >= 𝜌. 𝑑𝑡
51

52. Luc_Faucheux_2021
A sum of independent normally distributed
variables is also normally distributed.
How about when there is correlation?
52

53. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXII
¨ This is true indeed, so we just need to be a little more careful here.
¨ Almost there in justifying the use of the term “gaussian” for those models, so here we go:
¨ First of all, let’s see what we can say about the mean and the variance, before saying
anything about the functional form of the PDF
¨ We could also express the equations in a slightly different ways (do a Choleski
decomposition of the Covar matrix).
53

54. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXIII
¨ In any case, let’s look at the sum of two normally distributed variables that are NOT
independent (have a non-zero correlation)
¨ 𝑍 𝑡 = 𝑋 𝑡 + 𝑌 𝑡
¨ 𝑀 𝑋 𝑡 and 𝑉 𝑋 𝑡 are such that 𝑋 𝑡 ~𝑁(𝑀 𝑋 𝑡 , 𝑉 𝑋 𝑡 )
¨ 𝑀 𝑌 𝑡 and 𝑉 𝑌 𝑡 are such that 𝑌(𝑡)~𝑁(𝑀 𝑌 𝑡 , 𝑉 𝑌 𝑡 )
¨ 𝑀 𝑍 𝑡 = 𝔼 𝑋 + 𝑌 = 𝔼 𝑋} + 𝔼{𝑌 = 𝑀 𝑋 𝑡 + 𝑀 𝑌 𝑡
¨ 𝑉 𝑍 𝑡 = 𝔼 (𝑍(𝑡) − 𝑀 𝑍 𝑡 ).
¨ 𝑉 𝑍 𝑡 = 𝔼 𝑍(𝑡). + 𝑀 𝑍 𝑡 . − 2𝑀 𝑍 𝑡 . 𝑍(𝑡)
¨ 𝑉 𝑍 𝑡 = 𝔼 𝑍(𝑡).} + 𝔼{𝑀 𝑍 𝑡 .} − 𝔼{2𝑀 𝑍 𝑡 . 𝑍(𝑡)
¨ 𝑉 𝑍 𝑡 = 𝔼 𝑍(𝑡).} + 𝑀 𝑍 𝑡 . − 2𝑀 𝑍 𝑡 . 𝔼{𝑍(𝑡)
¨ 𝑉 𝑍 𝑡 = 𝔼{𝑍(𝑡).} + 𝑀 𝑍 𝑡 . − 2𝑀 𝑍 𝑡 . 𝑀 𝑍 𝑡
54

55. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXIV
¨ 𝑉 𝑍 𝑡 = 𝔼{𝑍(𝑡).} + 𝑀 𝑍 𝑡 . − 2𝑀 𝑍 𝑡 . 𝑀 𝑍 𝑡
¨ 𝑉 𝑍 𝑡 = 𝔼{𝑍(𝑡).} − 𝑀 𝑍 𝑡 .
¨ 𝑉 𝑍 𝑡 = 𝔼{(𝑋 𝑡 + 𝑌 𝑡 ).} − 𝑀 𝑍 𝑡 .
¨ 𝑉 𝑍 𝑡 = 𝔼{𝑋 𝑡 . 𝑋 𝑡 + 𝑌 𝑡 . 𝑌 𝑡 + 2𝑋 𝑡 . 𝑌(𝑡)} − 𝑀 𝑍 𝑡 .
¨ Note that I have been a little liberal on the notation for the expectation, as one needs to consider
the joint distribution. In any case:
¨ 𝑉 𝑍 𝑡 = 𝔼{𝑋 𝑡 . 𝑋 𝑡 + 𝑌 𝑡 . 𝑌 𝑡 + 2𝑋 𝑡 . 𝑌(𝑡)} − (𝑀[𝑋 𝑡 + 𝑀[𝑌 𝑡 ]).
¨ 𝑉 𝑍 𝑡 = 𝔼{𝑋 𝑡 . 𝑋 𝑡 } + 𝔼{𝑌 𝑡 . 𝑌 𝑡 } + 𝔼{2𝑋 𝑡 . 𝑌(𝑡)} − (𝑀[𝑋 𝑡 + 𝑀[𝑌 𝑡 ]).
¨ 𝑉 𝑍 𝑡 = 𝔼{𝑋 𝑡 .} + 𝔼{𝑌 𝑡 .} + 𝔼{2𝑋 𝑡 . 𝑌(𝑡)} − (𝑀[𝑋 𝑡 + 𝑀[𝑌 𝑡 ]).
¨ 𝑉 𝑍 𝑡 = 𝔼 𝑋 𝑡 .
+ 𝔼 𝑌 𝑡 .
+ 𝔼 2𝑋 𝑡 . 𝑌 𝑡 − 𝑀 𝑋 𝑡 .
− 𝑀 𝑌 𝑡 .
− 2. 𝑀 𝑋 𝑡 . 𝑀[𝑌 𝑡 ]
55

56. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXV
¨ 𝑉 𝑍 𝑡 = 𝔼 𝑋 𝑡 .
+ 𝔼 𝑌 𝑡 .
+ 𝔼 2𝑋 𝑡 . 𝑌 𝑡 − 𝑀 𝑋 𝑡 .
− 𝑀 𝑌 𝑡 .
− 2. 𝑀 𝑋 𝑡 . 𝑀[𝑌 𝑡 ]
¨ 𝑉 𝑍 𝑡 = 𝔼 𝑋 𝑡 .
− 𝑀 𝑋 𝑡 .
+ 𝔼 𝑌 𝑡 .
− 𝑀 𝑌 𝑡 .
+ 𝔼 2𝑋 𝑡 . 𝑌 𝑡 − 2. 𝑀 𝑋 𝑡 . 𝑀[𝑌 𝑡 ]
¨ 𝑉 𝑍 𝑡 = 𝑉 𝑋 𝑡 + 𝑉 𝑌 𝑡 + 𝔼 2𝑋 𝑡 . 𝑌 𝑡 − 2. 𝑀 𝑋 𝑡 . 𝑀[𝑌 𝑡 ]
¨ 𝑉 𝑍 𝑡 = 𝑉 𝑋 𝑡 + 𝑉 𝑌 𝑡 + 2. {𝔼 𝑋 𝑡 . 𝑌 𝑡 − 𝑀 𝑋 𝑡 . 𝑀 𝑌 𝑡 }
¨ Now:
¨ 𝐴 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝐴 = 𝔼 𝑋(𝑡 . 𝑌 𝑡 + 𝑀 𝑋 𝑡 . 𝑀 𝑌 𝑡 − 𝑋 𝑡 . 𝑀 𝑌 𝑡 − 𝑌 𝑡 . 𝑀[𝑋(𝑡)]}
¨ 𝐴 = 𝔼 𝑋 𝑡 . 𝑌 𝑡 + 𝑀 𝑋 𝑡 . 𝑀 𝑌 𝑡 − 𝔼{𝑋 𝑡 }. 𝑀 𝑌 𝑡 − 𝔼{𝑌 𝑡 }. 𝑀[𝑋(𝑡)]}
¨ 𝐴 = 𝔼 𝑋 𝑡 . 𝑌 𝑡 + 𝑀 𝑋 𝑡 . 𝑀 𝑌 𝑡 − 𝑀 𝑋 𝑡 . 𝑀 𝑌 𝑡 − 𝑀 𝑌 𝑡 . 𝑀[𝑋(𝑡)]}
¨ 𝐴 = 𝔼 𝑋 𝑡 . 𝑌 𝑡 − 𝑀 𝑋 𝑡 . 𝑀 𝑌 𝑡
56

57. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXVI
¨ 𝑉 𝑍 𝑡 = 𝑉 𝑋 𝑡 + 𝑉 𝑌 𝑡 + 2. {𝔼 𝑋 𝑡 . 𝑌 𝑡 − 𝑀 𝑋 𝑡 . 𝑀 𝑌 𝑡 }
¨ 𝑉 𝑍 𝑡 = 𝑉 𝑋 𝑡 + 𝑉 𝑌 𝑡 + 2. 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ That is right there the definition of the covariance
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝑉 𝑍 𝑡 = 𝑉 𝑋 𝑡 + 𝑉 𝑌 𝑡 + 2. 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡
¨ IF 𝑋 𝑡 and 𝑌 𝑡 are independent, the Covariance is 0, and we recover the fact that the
variance of the sum is the sum of the variances
¨ Note that the reverse is NOT true, you could have non-independent variable that will show a
0 covariance
57

58. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXVII
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ For example: 𝑌 𝑡 = 𝑋 𝑡 .
¨ Obviously not independent
¨ However
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑋 𝑡 . = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑋 𝑡 . − 𝑀[𝑋 𝑡 .])
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑋 𝑡 . = 𝔼 (𝑋 𝑡 P − 𝑀 𝑋 𝑡 . 𝑋 𝑡 . − 𝑀 𝑋 𝑡 . . 𝑋 𝑡 + 𝑀 𝑋 𝑡 . 𝑀[𝑋 𝑡 .])
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑋 𝑡 . = 𝔼{ 𝑋 𝑡 P − 𝑀 𝑋 𝑡 . 𝑀 𝑋 𝑡 .
¨ Suppose that 𝑋 𝑡 is normally distributed, then
¨ 𝔼{ 𝑋 𝑡 P = 0 and 𝑀 𝑋 𝑡 = 0
¨ So 𝐶𝑂𝑉 𝑋 𝑡 , 𝑋 𝑡 . = 0 even though obviously 𝑋 𝑡 and 𝑋 𝑡 . are not independent
58

59. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXVIII
¨ All right, back to:
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝑉 𝑍 𝑡 = 𝑉 𝑋 𝑡 + 𝑉 𝑌 𝑡 + 2. 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡
¨ Note that if 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 , then we also have by expanding the first equation:
¨ 𝔼{ 𝑋 𝑡 . 𝑌 𝑡 = 𝑀 𝑋 𝑡 . 𝑀 𝑌 𝑡
¨ So we have shown that the sum of two variables is such that:
¨ 𝑍 𝑡 = 𝑋 𝑡 + 𝑌 𝑡
¨ 𝑀 𝑍 𝑡 = 𝑀 𝑋 𝑡 + 𝑀 𝑌 𝑡
¨ 𝑉 𝑍 𝑡 = 𝑉 𝑋 𝑡 + 𝑉 𝑌 𝑡 + 2. 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡
¨ Note that this is NOT saying that if 𝑋 𝑡 and 𝑌 𝑡 are normally distributed, then 𝑍 𝑡 is also
normally distributed. This requires a little more work but we are almost there.
59

60. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXIX
¨ I have to admit here that I do not have a super elegant proof that the sum of correlated
Gaussians is also a Gaussian. Turns out the math get a little tricky between marginal and
jointly.
¨ The best I can sort of do on that one is following deck III of the Stochastic Calculus that we
went over a while back.
¨ So here it goes for the best I can do:
¨ You have 2 variables:
¨ 𝑑𝑋 𝑡 = 𝑎! 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏! 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
!(𝑡)
¨ 𝑑𝑌 𝑡 = 𝑎" 𝑡, 𝑌 𝑡 . 𝑑𝑡 + 𝑏" 𝑡, 𝑌 𝑡 . ([). 𝑑𝑊
"(𝑡)
¨ With : < 𝑑𝑊
!. 𝑑𝑊
" >= 𝜌. 𝑑𝑡
60

61. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXX
¨ < 𝑑𝑋 𝑡 > to denote the usual quantity 𝔼 𝑑𝑋 = 𝔼8Q$8
E)
{𝑋 𝑡 + 𝑑𝑡 − 𝑋(𝑡) 𝔉 𝑡 that we
have been using the deck II of the stochastic calculus decks
¨ 𝑑𝑋 𝑡 = 𝑎! 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏! 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
!(𝑡)
¨ 𝑑𝑌 𝑡 = 𝑎" 𝑡, 𝑌 𝑡 . 𝑑𝑡 + 𝑏" 𝑡, 𝑌 𝑡 . ([). 𝑑𝑊
"(𝑡)
¨ 𝑍 𝑡 = 𝑋 𝑡 + 𝑌(𝑡)
¨ < 𝑑𝑋 𝑡 > = 𝑎! 𝑡, 𝑋 𝑡 . 𝑑𝑡
¨ < 𝑑𝑌 𝑡 > = 𝑎" 𝑡, 𝑌 𝑡 . 𝑑𝑡
¨ < 𝑑𝑍 𝑡 > = 𝑀 𝑍 𝑡 + 𝑑𝑡 − 𝑍 𝑡 = 𝑎" 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑎" 𝑡, 𝑌 𝑡 . 𝑑𝑡
¨ < 𝑑𝑍 𝑡 > = < 𝑑𝑋 𝑡 > +< 𝑑𝑌 𝑡 >
61

62. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXI
¨ The correlation does NOT change the drift
¨ The correlation does NOT affect the expected return
¨ That is not surprising, we know that from the MPT theory (mean variance portfolio), where
the correlation between assets will change the risk (variance, volatility, standard deviation),
but NOT the expected return
¨ Conversely, the drift will NOT change the correlation
¨ We also know that from the RN (Radon-Nykodym) section with the change of measure,
being an added drift to the Brownian motion.
¨ The change of measure does NOT change the variance
¨ The change of measure does NOT change the correlation
¨ The change of measure changes the drift (expected return, mean, average, advection) but
NOT the diffusion, variance, standard deviation, correlation
62

63. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXII
¨ Let’s put here a couple of the slides from deck III on stochastic calculus.
63

64. Luc_Faucheux_2021
Couple of slides to remind us of deck III on
Stochastic Calculus
64

65. Luc_Faucheux_2021
Introducing the [𝛼] calculus - II
¨ The ITO integral is defined as:
¨ ∫
8=8A
8=8I
𝑓 𝑋 𝑡 . ([). 𝑑𝑋(𝑡) = lim
R→>
{∑0=*
0=R
𝑓(𝑋(𝑡0)). [𝑋(𝑡0Q*) − 𝑋(𝑡0)]}
¨ The Stratonovitch integral is defined as:
¨ ∫
8=8A
8=8I
𝑓 𝑋 𝑡 . (∘). 𝑑𝑋(𝑡) = lim
R→>
{∑0=*
0=R
𝑓 [𝑋(𝑡0 + 𝑋(𝑡0Q*)]/2). [𝑋(𝑡0Q*) − 𝑋(𝑡0)]}
¨ We can define the [𝛼] integral as:
¨ ∫
8=8A
8=8I
𝑓 𝑋 𝑡 . ([𝛼]). 𝑑𝑋(𝑡) = lim
R→>
{∑0=*
0=R
𝑓(𝑋(𝑡0) + 𝛼. [𝑋(𝑡0Q*) − 𝑋(𝑡0)]). [𝑋(𝑡0Q*) − 𝑋(𝑡0)]}
¨ ITO will be the case 𝛼 = 0
¨ STRATO will be the case 𝛼 = 1/2
65

66. Luc_Faucheux_2021
Introducing the [𝛼] calculus - III
¨ We had the relation on the integrals:
¨ ∫
8=8A
8=8I
𝑓 𝑊 𝑡 . (∘). 𝑑𝑊(𝑡) = ∫
8=8A
8=8I
𝑓 𝑊 𝑡 . ([). 𝑑𝑊(𝑡) +
*
.
∫
8=8A
8=8I
𝑓′ 𝑊 𝑡 . 𝑑𝑡
¨ This becomes:
¨ ∫
8=8A
8=8I
𝑓 𝑊 𝑡 . ([𝛼]). 𝑑𝑊(𝑡) = ∫
8=8A
8=8I
𝑓 𝑊 𝑡 . ([). 𝑑𝑊(𝑡) + 𝛼. ∫
8=8A
8=8I
𝑓′ 𝑊 𝑡 . 𝑑𝑡
¨ Or:
¨ ∫
8=8A
8=8I
𝑓 𝑊 𝑡 . ([𝛼]). 𝑑𝑊(𝑡) = ∫
8=8A
8=8I
𝑓 𝑊 𝑡 . ([𝛼] = 0). 𝑑𝑊(𝑡) + 𝛼. ∫
8=8A
8=8I
𝑓′ 𝑊 𝑡 . 𝑑𝑡
66

67. 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:
¨ ∫
8=8A
8=8I
𝑓 𝑋 𝑡 . ∘ . 𝑑𝑊 𝑡 = ∫
8=8A
8=8I
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) + ∫
8=8A
8=8I *
.
. 𝑏 𝑡, 𝑋 𝑡 .
?
?C
𝑓 𝑋(𝑡 . 𝑑𝑡
¨ This now becomes:
¨ ∫
8=8A
8=8I
𝑓 𝑋 𝑡 . [𝛼] . 𝑑𝑊 𝑡 = ∫
8=8A
8=8I
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) + ∫
8=8A
8=8I
𝛼. 𝑏 𝑡, 𝑋 𝑡 .
?
?C
𝑓 𝑋(𝑡 . 𝑑𝑡
67

68. 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 .
?
?C
𝑏 𝑡, 𝑋 𝑡 ]. 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ The STRATO SDE
¨ 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ Has the same solution (is the same) as the ITO SDE in ITO calculus
¨ 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 +
*
.
. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
68

69. 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 .
?
?C
𝑏 𝑡, 𝑋 𝑡 ]. 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ The [𝛼] SDE
¨ 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ Has the same solution (is the same) as the ITO SDE in ITO calculus
¨ 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
69

70. Luc_Faucheux_2021
Introducing the [𝛼] calculus - VII
¨ The ITO lemma (chain rule) reads:
¨ 𝑓 𝑋 𝑡I − 𝑓 𝑋 𝑡A = ∫
8=8A
8=8I ?G
?C
. ([). 𝑑𝑋(𝑡) +
*
.
∫
8=8A
8=8I ?!N
?!! (𝑋 𝑡 ). 𝑏 𝑡, 𝑋 𝑡
.
𝑑𝑡
¨ In the ”limit” of small time increments, this can be written formally as the Ito lemma:
¨ 𝛿𝑓 =
?G
?C
. 𝛿𝑋 +
*
.
.
?!N
?!! . 𝑏.𝛿𝑡
¨ The STRATO lemma reads:
¨ 𝑓 𝑋 𝑡I − 𝑓 𝑋 𝑡A = ∫
8=8A
8=8I ?G
?C
. (∘). 𝑑𝑋(𝑡)
¨ In the ”limit” of small time increments, this can be written formally as the Strato lemma:
¨ 𝛿𝑓 =
?G
?C
. ∘ . 𝛿𝑋
70

71. Luc_Faucheux_2021
Introducing the [𝛼] calculus - VIII
¨ In the [𝛼] calculus the [𝛼] lemma (chain rule) now reads :
¨ 𝑓 𝑋 𝑡I − 𝑓 𝑋 𝑡A = ∫
8=8A
8=8I ?G
?C
. ([𝛼]). 𝑑𝑋(𝑡) +
*
.
− 𝛼 . ∫
8=8A
8=8I ?!N
?!! 𝑋 𝑡 . 𝑏 𝑡, 𝑋 𝑡
.
. 𝑑𝑡
¨ In the ”limit” of small time increments, this can be written formally as the [𝛼] lemma:
¨ 𝛿𝑓 =
?G
?C
. 𝛿𝑋 +
*
.
− 𝛼 .
?!N
?!! . 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 𝑡I from the left point.
¨ 𝑓 𝑋 𝑡I − 𝑓 𝑋 𝑡A = lim
R→>
∑0=*
0=R
{𝑓(𝑋(𝑡0)) − 𝑓(𝑋(𝑡0)*))}
¨ 𝑓 𝑋 𝑡I − 𝑓 𝑋 𝑡A = lim
R→>
∑0=*
0=R
{
?G
?C
. ([). 𝛿𝑋 +
*
.
.
?!N
?!! . ([). (𝛿𝑋).}
71

72. 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
¨
?M(!,8|C',8')
?8
= −
?
?C
𝑎 𝑋 𝑡 , 𝑡 . 𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
?
?C
[
*
.
. [𝑏(𝑋 𝑡 , 𝑡). . 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)]
¨
?M
?8
= −
?
?C
𝑎𝑝 −
?
?C
I!M
.
= −
?
?C
[𝑎𝑝] +
*
.
?!
?C! [
I!M
.
]
¨
?M
?8
= −
?
?C
𝑎𝑝 − 𝑏
?I
?C
. 𝑝 −
*
.
. 𝑏. .
?
?C
𝑝
¨
?M
?8
= −
?
?C
𝑎𝑝 −
?2
?C
. 𝑝 − 𝐷.
?
?C
𝑝
¨
?M
?8
= −
?
?C
𝑎𝑝 −
?
?C
(𝐷𝑝)
72

73. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - II
¨ [𝛼] SDE is: 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD [𝛼] Kolmogorov PDE
¨
?M(!,8|C',8')
?8
= −
?
?C
ƒ
„
{l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 }. 𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
?
?C
[
*
.
. [j
𝑏(𝑋 𝑡 , 𝑡). . 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)]
¨
?M
?8
= −
?
?C
l
𝑎𝑝 + 𝛼. j
𝑏.
?
?C
j
𝑏. 𝑝 −
?
?C
[
*
.
. [j
𝑏. . 𝑝]
¨
?M
?8
= −
?
?C
l
𝑎𝑝 + 𝛼. j
𝑏.
?
?C
j
𝑏. 𝑝 −
?
?C
F
I!M
.
= −
?
?C
l
𝑎𝑝 +
?
?C
𝛼. j
𝑏.
?
?C
j
𝑏. 𝑝 +
*
.
?!
?C! [
F
I!M
.
]
¨
?M
?8
= −
?
?C
l
𝑎𝑝 + 𝛼. j
𝑏.
?F
I
?C
. 𝑝 − j
𝑏.
?F
I
?C
. 𝑝 −
*
.
. j
𝑏..
?
?C
𝑝
¨
?M
?8
= −
?
?C
𝑎𝑝 + 𝛼.
?U
2
?C
. 𝑝 −
?U
2
?C
. 𝑝 − …
𝐷.
?
?C
𝑝
73

74. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - III
¨ [𝛼 = 1/2] STRATO SDE is: 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD STRATO Kolmogorov PDE
¨
?V(!,8|C',8')
?8
= −
?
?C
ƒ
„
{l
𝑎 𝑡, 𝑋 𝑡 +
*
.
. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 }. 𝑃 𝑥, 𝑡 𝑋A, 𝑡A −
?
?C
[
*
.
. [j
𝑏(𝑋 𝑡 , 𝑡). . 𝑃(𝑥, 𝑡|𝑋A, 𝑡A)]
¨
?V
?8
= −
?
?C
l
𝑎𝑃 +
*
.
. j
𝑏.
?F
I
?C
. 𝑃 −
?
?C
*
.
. [j
𝑏. . 𝑃
¨
?V
?8
= −
?
?C
l
𝑎𝑃 +
*
.
. j
𝑏.
?
?C
j
𝑏. 𝑃 −
?
?C
F
I!V
.
= −
?
?C
l
𝑎𝑃 +
?
?C
*
.
. j
𝑏.
?
?C
j
𝑏. 𝑃 +
*
.
?!
?C! [
F
I!V
.
]
¨
?V
?8
= −
?
?C
l
𝑎𝑃 +
*
.
. j
𝑏.
?F
I
?C
. 𝑃 − j
𝑏.
?F
I
?C
. 𝑃 −
*
.
. j
𝑏..
?
?C
𝑃 = −
?
?C
l
𝑎𝑃 −
*
.
j
𝑏
?F
I
?C
𝑃 −
*
.
j
𝑏. ?
?C
𝑃
¨
?V
?8
= −
?
?C
l
𝑎𝑃 −
*
.
.
?U
2
?C
. 𝑃 − …
𝐷.
?
?C
𝑃 = −
?
?C
l
𝑎𝑃 +
*
.
.
?U
2
?C
. 𝑃 −
?
?C
(…
𝐷𝑃)
74

75. 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
¨
?M(!,8|C',8')
?8
= −
?
?C
𝑎 𝑋 𝑡 , 𝑡 . 𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
?
?C
[
*
.
. [𝑏(𝑋 𝑡 , 𝑡). . 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)]
¨ The PDF ALSO follows the BACKWARD ITO Kolmogorov PDE:
¨
?M(!,8|C',8')
?8'
= −𝑎 𝑋A, 𝑡A
?
?C'
𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
*
.
. 𝑏(𝑋A, 𝑡A). ?!
?C'
! 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)
¨
?M
?8'
= −𝑎
?M
?C'
−
*
.
. 𝑏. ?!M
?C'
!
¨
?M
?8'
= −𝑎
?M
?C'
− 𝐷
?!M
?C'
!
¨ Where I have explicitly kept the notation 𝑡A and 𝑋A to indicate the fact that this is a
BACKWARD PDE (expectation of a payoff at maturity)
75

76. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - V
¨ [𝛼] SDE is: 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD [𝛼] Kolmogorov PDE
¨
?M(!,8|C',8')
?8
= −
?
?C
ƒ
„
{l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 }. 𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
?
?C
[
*
.
. [j
𝑏(𝑋 𝑡 , 𝑡). . 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)]
¨ The PDF ALSO follows the BACKWARD ITO Kolmogorov PDE:
¨
?M(!,8|C',8')
?8'
= − l
𝑎 𝑡A, 𝑋 𝑡A + 𝛼. j
𝑏 𝑡A, 𝑋 𝑡A .
?
?C'
j
𝑏 𝑡A, 𝑋 𝑡A
?
?C'
𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
*
.
. 𝑏(𝑋A, 𝑡A). ?!
?C'
! 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)
¨
?M
?8'
= − l
𝑎 + 𝛼. j
𝑏
?
?C'
j
𝑏
?
?C'
𝑝 −
*
.
. 𝑏. ?!
?C'
! 𝑝
¨
?M
?8'
= − l
𝑎 + 𝛼.
?U
2
?C'
?M
?C'
− …
𝐷
?!M
?C'
!
76

77. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - VI
¨ [𝛼 = 1/2] STRATO SDE is: 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . (∘). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD STRATO Kolmogorov PDE
¨
?M(!,8|C',8')
?8
= −
?
?C
ƒ
„
{l
𝑎 𝑡, 𝑋 𝑡 +
*
.
. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 }. 𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
?
?C
[
*
.
. [j
𝑏(𝑋 𝑡 , 𝑡). . 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)]
¨ The PDF ALSO follows the BACKWARD ITO Kolmogorov PDE:
¨
?M(!,8|C',8')
?8'
= − l
𝑎 𝑡A, 𝑋 𝑡A +
*
.
. j
𝑏 𝑡A, 𝑋 𝑡A .
?
?C'
j
𝑏 𝑡A, 𝑋 𝑡A
?
?C'
𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
*
.
. 𝑏(𝑋A, 𝑡A). ?!
?C'
! 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)
¨
?M
?8'
= − l
𝑎 +
*
.
. j
𝑏
?
?C'
j
𝑏
?
?C'
𝑝 −
*
.
. j
𝑏. ?!
?C'
! 𝑝
¨
?M
?8'
= − l
𝑎 +
*
.
.
?U
2
?C'
?M
?C'
− …
𝐷
?!M
?C'
!
77

78. 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)
¨
?M(!,8|C',8')
?8
= −
?
?C
𝑎 𝑋 𝑡 , 𝑡 . 𝑝 𝑥, 𝑡 𝑋A, 𝑡A −
?
?C
[
*
.
. [𝑏(𝑋 𝑡 , 𝑡). . 𝑝(𝑥, 𝑡|𝑋A, 𝑡A)]
¨ < ∆𝑋 > = 𝐸 ∆𝑋 =< 𝑥 >8Q∆8 −< 𝑥 >8= 𝑎 𝑋 𝑡 , 𝑡 . ∆𝑡 (advection term)
¨ < ∆𝑋.> = 𝐸 ∆𝑋. =< (𝑥−< 𝑥 >8Q∆8).>8Q∆8= 𝑏(𝑋 𝑡 , 𝑡).. ∆𝑡 (diffusion term)
78

79. 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 .
?
?C
𝑏 𝑡, 𝑋 𝑡 ]. 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ The [𝛼] SDE
¨ 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ Has the same solution (is the same) as the ITO SDE in ITO calculus
¨ 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
79

80. Luc_Faucheux_2021
Why did we go through all this trouble? - III
¨ And so, if we start with an [𝛼] SDE: 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ It has the same solution (is the same) as the ITO SDE in ITO calculus
¨ 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ Which will then follow the ITO FORWARD Kolmogorov (FP):
¨
?V(!,8|C',8')
?8
= −
?
?C
𝑎 𝑋 𝑡 , 𝑡 . 𝑃 𝑥, 𝑡 𝑋A, 𝑡A −
?
?C
[
*
.
. [𝑏(𝑋 𝑡 , 𝑡). . 𝑃(𝑥, 𝑡|𝑋A, 𝑡A)]
¨ With:
¨ 𝑎 𝑋 𝑡 , 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡
¨ 𝑏 𝑋 𝑡 , 𝑡 = j
𝑏 𝑡, 𝑋 𝑡
¨ 𝑎 𝑋 𝑡 , 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. 𝑏 𝑡, 𝑋 𝑡 .
?
?C
𝑏 𝑡, 𝑋 𝑡
80

81. Luc_Faucheux_2021
Why did we go through all this trouble? – III - a
¨ < ∆𝑋 > = 𝐸 ∆𝑋 =< 𝑥 >8Q∆8 −< 𝑥 >8= 𝑎 𝑋 𝑡 , 𝑡 . ∆𝑡 (advection term)
¨ < ∆𝑋.> = 𝐸 ∆𝑋. =< (𝑥−< 𝑥 >8Q∆8).>8Q∆8= 𝑏(𝑋 𝑡 , 𝑡).. ∆𝑡 (diffusion term)
¨ With:
¨ 𝑎 𝑋 𝑡 , 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡
¨ 𝑏 𝑋 𝑡 , 𝑡 = j
𝑏 𝑡, 𝑋 𝑡
¨ Remember that we started from : 𝑑𝑋 𝑡 = l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . ([𝛼]). 𝑑𝑊
¨ So: < ∆𝑋 > = 𝑎 𝑋 𝑡 , 𝑡 . ∆𝑡 = {l
𝑎 𝑡, 𝑋 𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 }. ∆𝑡
¨ < ∆𝑋 > = l
𝑎 𝑡, 𝑋 𝑡 . ∆𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
¨ < ∆𝑋 > =< l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + j
𝑏 𝑡, 𝑋 𝑡 . [𝛼] . 𝑑𝑊 >
81

82. Luc_Faucheux_2021
Why did we go through all this trouble? – III - b
¨ < ∆𝑋 > = l
𝑎 𝑡, 𝑋 𝑡 . ∆𝑡 + 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
¨ < ∆𝑋 > = < l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 > + < j
𝑏 𝑡, 𝑋 𝑡 . [𝛼] . 𝑑𝑊 >
¨ And
¨ < l
𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 > = l
𝑎 𝑡, 𝑋 𝑡 . ∆𝑡
¨ So we have:
¨ < j
𝑏 𝑡, 𝑋 𝑡 . [𝛼] . 𝑑𝑊 > = 𝛼. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
¨ In particular:
¨ ITO : < j
𝑏 𝑡, 𝑋 𝑡 . [ . 𝑑𝑊 > = 0
¨ STRATO: < j
𝑏 𝑡, 𝑋 𝑡 . ∘ . 𝑑𝑊 > = [
*
.
]. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
𝑏 𝑡, 𝑋 𝑡 . ∆𝑡
82

83. Luc_Faucheux_2021
Why did we go through all this trouble? – III - c
¨ ITO : < j
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: < j
𝑏 𝑡, 𝑋 𝑡 . ∘ . 𝑑𝑊 > = [
*
.
]. j
𝑏 𝑡, 𝑋 𝑡 .
?
?C
j
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.
83

84. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXIII
¨ So back to the sum of two variables:
¨ 𝑑𝑋 𝑡 = 𝑎! 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏! 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
!(𝑡)
¨ 𝑑𝑌 𝑡 = 𝑎" 𝑡, 𝑌 𝑡 . 𝑑𝑡 + 𝑏" 𝑡, 𝑌 𝑡 . ([). 𝑑𝑊
"(𝑡)
¨ 𝑍 𝑡 = 𝑋 𝑡 + 𝑌(𝑡)
¨ < 𝑑𝑍 𝑡 > = < 𝑑𝑋 𝑡 > +< 𝑑𝑌 𝑡 >
¨ < 𝑑𝑋 𝑡 > = 𝑎! 𝑡, 𝑋 𝑡 . 𝑑𝑡
¨ < 𝑑𝑌 𝑡 > = 𝑎" 𝑡, 𝑌 𝑡 . 𝑑𝑡
¨ < 𝑑𝑋. > = 𝑏!(𝑋 𝑡 , 𝑡).. 𝑑𝑡
¨ < 𝑑𝑌. > = 𝑏"(𝑌 𝑡 , 𝑡).. 𝑑𝑡
84

85. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXIV
¨ < 𝑑𝑍. > =< 𝑑(𝑋 + 𝑌). >
¨ < 𝑑𝑍. > =< (𝑋 𝑡 + 𝑑𝑡 + 𝑌 𝑡 + 𝑑𝑡 − < 𝑋 𝑡 + 𝑑𝑡 > − < 𝑌(𝑡 + 𝑑𝑡) >).>
¨ < 𝑑𝑍. > =< (𝑍(𝑡 + 𝑑𝑡)− < 𝑍 𝑡 + 𝑑𝑡 >).>
¨ < 𝑑𝑍. > =< (𝑋 𝑡 + 𝑑𝑡 − < 𝑋 𝑡 + 𝑑𝑡 > +𝑌 𝑡 + 𝑑𝑡 − < 𝑌(𝑡 + 𝑑𝑡) >).>
¨ < 𝑑𝑍. > =< (𝑏! 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
!(𝑡) + 𝑏" 𝑡, 𝑌 𝑡 . ([). 𝑑𝑊
"(𝑡) ). >
¨ < 𝑑𝑍. > =< (𝑏!. 𝑑𝑊
!).+(𝑏". 𝑑𝑊
").+2. 𝑏!. 𝑑𝑊
!. 𝑏". 𝑑𝑊
" >
¨ < 𝑑𝑍. > = 𝑏!
.
. 𝑑𝑡 + 𝑏"
.
. 𝑑𝑡 + 2. 𝜌. 𝑏!. 𝑏". 𝑑𝑡
¨ < 𝑑𝑍. > =< 𝑑𝑋. > + < 𝑑𝑌. > +2. 𝜌. 𝑏!. 𝑏". 𝑑𝑡
¨ < 𝑑𝑍. > = (𝑏!
.
+ 𝑏"
.
+ 2. 𝜌. 𝑏!. 𝑏"). 𝑑𝑡
85

86. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXV
¨ 𝑑𝑋 𝑡 = 𝑎! 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏! 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
!(𝑡)
¨ 𝑑𝑌 𝑡 = 𝑎" 𝑡, 𝑌 𝑡 . 𝑑𝑡 + 𝑏" 𝑡, 𝑌 𝑡 . ([). 𝑑𝑊
"(𝑡)
¨ 𝑍 𝑡 = 𝑋 𝑡 + 𝑌(𝑡)
¨ < 𝑑𝑍 𝑡 > = < 𝑑𝑋 𝑡 > +< 𝑑𝑌 𝑡 >
¨ < 𝑑𝑍 𝑡 > = 𝑎! 𝑡, 𝑋 𝑡 + 𝑎" 𝑡, 𝑌 𝑡 . 𝑑𝑡
¨ < 𝑑𝑍. > = (𝑏!
.
+ 𝑏"
.
+ 2. 𝜌. 𝑏!. 𝑏"). 𝑑𝑡
¨ And so the process for 𝑍 𝑡 can be described (within some mathematical reasons) by the
SDE:
¨ 𝑑𝑍 𝑡 = 𝑎# 𝑡, 𝑍 𝑡 . 𝑑𝑡 + 𝑏# 𝑡, 𝑍 𝑡 . ([). 𝑑𝑊
#(𝑡)
86

87. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXVI
¨ 𝑑𝑍 𝑡 = 𝑎# 𝑡, 𝑍 𝑡 . 𝑑𝑡 + 𝑏# 𝑡, 𝑍 𝑡 . ([). 𝑑𝑊
#(𝑡)
¨ 𝑎# 𝑡, 𝑍 𝑡 = 𝑎! 𝑡, 𝑋 𝑡 + 𝑎" 𝑡, 𝑌 𝑡
¨ 𝑎# = 𝑎! + 𝑎"
¨ 𝑏#
.
= (𝑏!
.
+ 𝑏"
.
+ 2. 𝜌. 𝑏!. 𝑏")
¨ So we know that the PDF for that process will follow the FP equation
¨ This implies that the PDF follows the FORWARD ITO Kolmogorov PDE
¨
?M(#,8|X',8')
?8
= −
?
?X
𝑎 𝑍 𝑡 , 𝑡 . 𝑝 𝑧, 𝑡 𝑍A, 𝑡A −
?
?X
[
*
.
. [𝑏(𝑍 𝑡 , 𝑡). . 𝑝(𝑧, 𝑡|𝑍A, 𝑡A)]
¨ The PDF ALSO follows the BACKWARD ITO Kolmogorov PDE:
¨
?M(#,8|X',8')
?8'
= −𝑎 𝑍A, 𝑡A
?
?X'
𝑝 𝑧, 𝑡 𝑍A, 𝑡A −
*
.
. 𝑏(𝑍A, 𝑡A). ?!
?X'
! 𝑝(𝑧, 𝑡|𝑍A, 𝑡A)
87

88. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXVII
¨ 𝑑𝑍 𝑡 = 𝑎# 𝑡, 𝑍 𝑡 . 𝑑𝑡 + 𝑏# 𝑡, 𝑍 𝑡 . ([). 𝑑𝑊
#(𝑡)
¨ 𝑎# = 𝑎! + 𝑎"
¨ 𝑏#
.
= (𝑏!
.
+ 𝑏"
.
+ 2. 𝜌. 𝑏!. 𝑏")
¨
?M(#,8|X',8')
?8
= −
?
?X
𝑎 𝑍 𝑡 , 𝑡 . 𝑝 𝑧, 𝑡 𝑍A, 𝑡A −
?
?X
[
*
.
. [𝑏(𝑍 𝑡 , 𝑡). . 𝑝(𝑧, 𝑡|𝑍A, 𝑡A)]
¨ In almost all cases (and certainly in the simple cases where say the advection and diffusion
coefficients are either constant or function only of the time), the solution of that PDE will be
a Gaussian.
¨ Am sorry guys, but this is how I explain that the sum of correlated Gaussians is itself a
Gaussian
¨ There could be some weird functions 𝑎!, 𝑎", 𝑏! and 𝑏" for which that would not be the
case.
¨ Usually if you formulated a model with such behavior, switch to a simpler one
88

89. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXVIII
¨ I truly wished that I could have been more rigorous in that section, and that keeps me up at
night, so apologies for what I perceived to be a cope out, I would be happy if one of you
email me an insulting letter with a more rigorous approach to this (and also more general)
89

90. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXIX
¨ Saying it in another way, I do not have a good argument as to why:
¨ If 𝑎!, 𝑎", 𝑏! and 𝑏" are such functions so that the PDF for 𝑋(𝑡) and 𝑌(𝑡) are such that they
have for solutions a Gaussian distribution
¨ THEN the PDF for 𝑋 𝑡 + 𝑌(𝑡) which is of the form:
¨
?M(#,8|X',8')
?8
= −
?
?X
𝑎 𝑍 𝑡 , 𝑡 . 𝑝 𝑧, 𝑡 𝑍A, 𝑡A −
?
?X
[
*
.
. [𝑏(𝑍 𝑡 , 𝑡). . 𝑝(𝑧, 𝑡|𝑍A, 𝑡A)]
¨ With:
¨ 𝑎# = 𝑎! + 𝑎"
¨ 𝑏#
.
= (𝑏!
.
+ 𝑏"
.
+ 2. 𝜌. 𝑏!. 𝑏")
¨ ALSO has a Gaussian for solution of the PDF
¨ At least not obvious to me
90

91. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXX
¨ In some simple cases (like the ones we are dealing with here), the best way to look at it is if
you can find an explicit solution of the SDE:
¨ When we had:
¨ 𝑑𝑋 = −𝑘!. 𝑋. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ We had for the solution:
¨ 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0).(8)8') + ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢
¨ Which was normally distributed with moments:
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
91

92. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXXI
¨ Similarly we had for 𝑌(𝑡):
¨ 𝑑𝑌 = −𝑘". 𝑌. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ We had for the solution:
¨ 𝑌 𝑡 = 𝑌 𝑡A . 𝑒)0+.(8)8')
+ ∫
J=8'
J=8
𝑒)0+. 8)J
. 𝜎". ([). 𝑑𝑊
" 𝑢
¨ Which was normally distributed with moments:
¨ 𝑀 𝑌 𝑡 = 𝑌(𝑡A). 𝑒)0+.(8)8')
¨ 𝑉 𝑌 𝑡 =
-+
!
.0+
. 1 − 𝑒).0+. 8)8'
92

93. Luc_Faucheux_2021
IRMA – Mercurio – G2++ - XXXXII
¨ So in that case we get an explicit solution for 𝑍 𝑡 = 𝑋 𝑡 + 𝑌(𝑡)
¨ 𝑍(𝑡) = 𝑌 𝑡! . 𝑒"#!.(&"&")
+ ∫
()&"
()&
𝑒"#!. &"(
. 𝜎*. ([). 𝑑𝑊
* 𝑢 + 𝑋 𝑡! . 𝑒"##.(&"&") + ∫
()&"
()&
𝑒"##. &"( . 𝜎+. ([). 𝑑𝑊
+ 𝑢
¨ And we know that any function of the form:
¨ ∫
J=8'
J=8
𝑓(𝑢). ([). 𝑑𝑊
" 𝑢 is normally distributed, because of the martingale and isometry
properties of the ITO integral
¨ So 𝑍(𝑡) is a mixture of normal distributions, which is normally distributed (because you can
recast the correlation using the Cholesky decomposition into two independent normal
distribution)
¨ So in that case you can say that 𝑍(𝑡) is normally distributed.
¨ Again this is in the specific case of the Langevin equation.
¨ I unfortunately do not have a solid general argument otherwise
93

94. Luc_Faucheux_2021
Re-casting the correlation
94

95. Luc_Faucheux_2021
Recasting the correlation - I
¨ The Mercurio G2++ model was written as:
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ 𝑑𝑧 = −𝑘#. (𝑧 − 𝑥 − 𝑦). 𝑑𝑡 + 𝜎#. ([). 𝑑𝑊
#
¨ With: < 𝑑𝑊
!. 𝑑𝑊
" >= 𝜌. 𝑑𝑡
¨ 𝜎# = 0
¨ 𝑘# → ∞ so 𝑥 + 𝑦 − 𝑧 → 0 so 𝑧 𝑡 = 𝑥 𝑡 + 𝑦(𝑡)
¨ 𝑟 𝑡 = 𝐼𝑅𝑀𝐴 𝑧 𝑡 + 𝜇 𝑡 = 𝐹 𝑧 = 𝑧 𝑡 + 𝜑(𝑡)
¨ Sometimes it is not that convenient to work with: < 𝑑𝑊
!. 𝑑𝑊
" >= 𝜌. 𝑑𝑡
¨ We would rather work with picking from the Normal distribution in a way where we do not
have to take the correlation into account. We can do that in the following manner:
95

96. Luc_Faucheux_2021
Recasting the correlation - II
¨ Instead of writing :
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ With: < 𝑑𝑊
!. 𝑑𝑊
" >= 𝜌. 𝑑𝑡
¨ Let us show that can write the above using 2 other independent Brownian motions:
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. 𝑑 ˆ
𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
! + 1 − 𝜌.. 𝑑 ˆ
𝑊
"}
¨ With: < 𝑑 ˆ
𝑊
!. 𝑑 ˆ
𝑊
" > = 0
96

97. Luc_Faucheux_2021
Recasting the correlation - III
¨ Alternatively, comparing the two sets of equations:
¨ 𝑑𝑊
! = 𝑑 ˆ
𝑊
!
¨ 𝑑𝑊
" = 𝜌. 𝑑 ˆ
𝑊
! + 1 − 𝜌.. 𝑑 ˆ
𝑊
"
¨ Or again:
¨ 𝑑 ˆ
𝑊
! = 𝑑𝑊
!
¨ 𝑑 ˆ
𝑊
" =
$E+
*)f!
− 𝜌.
$E)
*)f!
97

98. Luc_Faucheux_2021
Recasting the correlation - IV
¨ When we had:
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!
¨ We had for the solution:
¨ 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0).(8)8') + ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢
¨ Which was normally distributed with moments:
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
98

99. Luc_Faucheux_2021
Recasting the correlation - V
¨ Similarly we had for 𝑌(𝑡):
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"
¨ We had for the solution:
¨ 𝑌 𝑡 = 𝑌 𝑡A . 𝑒)0+.(8)8')
+ ∫
J=8'
J=8
𝑒)0+. 8)J
. 𝜎". ([). 𝑑𝑊
" 𝑢
¨ Which was normally distributed with moments:
¨ 𝑀 𝑌 𝑡 = 𝑌(𝑡A). 𝑒)0+.(8)8')
¨ 𝑉 𝑌 𝑡 =
-+
!
.0+
. 1 − 𝑒).0+. 8)8'
99

100. Luc_Faucheux_2021
Recasting the correlation - VI
¨ We are now concerning ourselves with the Covariance:
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑𝑊
! 𝑢 ). (∫
J=8'
J=8
𝑒)0+. 8)J
. 𝜎". ([). 𝑑𝑊
" 𝑢 )
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. 𝑑𝑊
! 𝑢 ). (∫
J=8'
J=8
𝑒)0+. 8)J
. 𝜎". 𝑑𝑊
" 𝑢 )
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝜎!. 𝜎". 𝔼 ∫
J=8'
J=8
𝑑𝑊
! 𝑢 ∫
J=8'
J=8
𝑑𝑊
" 𝑢 𝑒)0+. 8)J
. 𝑒)0). 8)J
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝜎!. 𝜎". 𝔼 ∫
J=8'
J=8
∫
J=8'
J=8
𝑒)0+. 8)J
. 𝑒)0). 8)J . 𝑑𝑊
! 𝑢 . 𝑑𝑊
" 𝑢
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝜎!. 𝜎". ∫
J=8'
J=8
𝑒)0+. 8)J
. 𝑒)0). 8)J . 𝜌. 𝑑𝑢
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝜎!. 𝜎". ∫
J=8'
J=8
𝑒)(0)Q0+). 8)J
. 𝜌. 𝑑𝑢
100

101. Luc_Faucheux_2021
Recasting the correlation - VII
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝜎!. 𝜎". ∫
J=8'
J=8
𝑒)(0)Q0+). 8)J
. 𝜌. 𝑑𝑢
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝜌. 𝜎!. 𝜎". [
B#($)/$+). &#0
(0)Q0+)
]J=8'
J=8
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 =
f.-).-+
(0)Q0+)
[1 − 𝑒)(0)Q0+). 8)8' ]
101

102. Luc_Faucheux_2021
Recasting the correlation - VIII
¨ All right so now let’s redo that exercise starting from:
¨ 𝑑𝑥 = −𝑘!. 𝑥. 𝑑𝑡 + 𝜎!. 𝑑 ˆ
𝑊
!
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
! + 1 − 𝜌.. 𝑑 ˆ
𝑊
"}
¨ With: < 𝑑 ˆ
𝑊
!. 𝑑 ˆ
𝑊
" > = 0
¨ The first one for 𝑋(𝑡) is formally the same as before:
¨ 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0).(8)8') + ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢
¨ Which was normally distributed with moments:
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
102

103. Luc_Faucheux_2021
Recasting the correlation - IX
¨ The second one for 𝑌 𝑡 is slightly more complicated:
¨ 𝑑𝑦 = −𝑘". 𝑦. 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
! + 1 − 𝜌.. 𝑑 ˆ
𝑊
"}
¨ Again now let’s try to be consistent in our notations (again, this is a promise, once I get that
book deal the notation will be nicely consistent throughout the book)
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ Again as before we are going to use ITO lemma on :
¨ j
𝑌 𝑡 = exp 𝑘". 𝑡 . 𝑌(𝑡)
¨ 𝑑 j
𝑌 =
? F
D
?D
. [ . 𝑑𝑌 +
*
.
.
?! F
D
?D! . 𝑑𝑌. +
? F
D
?8
. [ . 𝑑𝑡
103

104. Luc_Faucheux_2021
Recasting the correlation - XI
¨ Again, remember that we used the notation for ITO lemma for sake of ease, what you really
have is a regular function
¨ j
𝑌 = 𝑓 𝑌 Stochastic Variable
¨ l
𝑦 = 𝑓 𝑦 Regular “Newtonian” variable with well defined partial derivatives
¨ 𝛿𝑓 =
?G
?"
. 𝛿𝑌 +
*
.
.
?!G
?"! . (𝛿𝑌). +
?G
?8
. 𝑑𝑡
¨
?G
?"
=
?G
?"
|"=D 8 ,8
¨
?!G
?"! =
?!G
?"! |"=D 8 ,8
¨
?G
?8
=
?G
?8
|"=D 8 ,8
104

105. Luc_Faucheux_2021
Recasting the correlation - XII
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ j
𝑌 𝑡 = exp 𝑘". 𝑡 . 𝑌(𝑡)
¨ 𝑑 j
𝑌 =
? F
D
?D
. [ . 𝑑𝑌 +
*
.
.
?! F
D
?D! . 𝑑𝑌. +
? F
D
?8
. [ . 𝑑𝑡
¨
? F
D
?D
= exp 𝑘". 𝑡
¨
?! F
D
?D! = 0
¨
? F
D
?8
= 𝑘". exp 𝑘". 𝑡 . 𝑌 𝑡 = 𝑘".j
𝑌 𝑡
¨ 𝑑 j
𝑌 = exp 𝑘". 𝑡 . [ . 𝑑𝑌 +
*
.
. 0. 𝑑𝑌. + 𝑘".j
𝑌 𝑡 . [ . 𝑑𝑡
105

106. Luc_Faucheux_2021
Recasting the correlation - XIII
¨ 𝑑 j
𝑌 = exp 𝑘". 𝑡 . [ . 𝑑𝑌 +
*
.
. 0. 𝑑𝑌. + 𝑘".j
𝑌 𝑡 . [ . 𝑑𝑡
¨ 𝑑 j
𝑌 = exp 𝑘". 𝑡 . 𝑑𝑌 + 𝑘".j
𝑌 𝑡 . 𝑑𝑡
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ 𝑑 j
𝑌 = exp 𝑘". 𝑡 . (−𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}) + 𝑘".j
𝑌 𝑡 . 𝑑𝑡
¨ 𝑑 j
𝑌 = exp 𝑘". 𝑡 . (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)})
¨ j
𝑌 𝑡I − j
𝑌 𝑡A = ∫
8=8'
8=8,
𝑑 j
𝑌 𝑡 = ∫
8=8'
8=8,
exp 𝑘". 𝑡 . (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)})
¨ j
𝑌 𝑡 = exp 𝑘". 𝑡 . 𝑌(𝑡)
¨ j
𝑌 𝑡I = exp 𝑘". 𝑡I . 𝑌(𝑡I)
¨ j
𝑌 𝑡A = exp 𝑘". 𝑡A . 𝑌(𝑡A)
106

107. Luc_Faucheux_2021
Recasting the correlation - XIV
¨ j
𝑌 𝑡I − j
𝑌 𝑡A = ∫
8=8'
8=8,
𝑑 j
𝑌 𝑡 = ∫
8=8'
8=8,
exp 𝑘". 𝑡 . (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)})
¨ exp 𝑘". 𝑡I . 𝑌(𝑡I) − exp 𝑘". 𝑡A . 𝑌(𝑡A) = ∫
8=8'
8=8,
exp 𝑘". 𝑡 . (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)})
¨ 𝑌 𝑡I = 𝑒)0+ 8,)8' . 𝑌 𝑡A + ∫
8=8'
8=8,
𝑒)0+ 8,)8
. (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)})
¨ 𝑌 𝑡 = 𝑒)0+ 8)8' . 𝑌 𝑡A + ∫
J=8'
J=8
𝑒)0+ 8)J
. (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑢) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)})
¨ And we know apply the usual trick of:
¨ ITO integral is a martingale to compute the mean
¨ ITO integral exhibits the property of isometry to compute the variance
107

108. Luc_Faucheux_2021
Recasting the correlation - XV
¨ 𝑌 𝑡 = 𝑒)0+ 8)8' . 𝑌 𝑡A + ∫
J=8'
J=8
𝑒)0+ 8)J
. (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑢) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)})
¨ 𝑀 𝑌 𝑡 = 𝔼 𝑌 = 𝑒)0+ 8)8' . 𝑌 𝑡A as before
¨ The Variance is a little more tricky
¨ 𝑉 𝑌 𝑡 = 𝔼 (𝑌(𝑡) − 𝑀 𝑌 𝑡 ).
¨ 𝑀 𝑌 𝑡 = 𝑌 𝑡A . 𝑒)0+. 8)8'
¨ 𝑌 𝑡 − 𝑀 𝑌 𝑡 = ∫
J=8'
J=8
𝑒)0+ 8)J
. (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑢) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)})
¨ 𝑌 𝑡 − 𝑀 𝑌 𝑡 . = (∫
J=8'
J=8
𝑒)0+ 8)J
. (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑢) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)})).
¨ Where there again we are going to use the isometry property and the fact that:
¨ < 𝑑 ˆ
𝑊
!. 𝑑 ˆ
𝑊
" > = 0
108

109. Luc_Faucheux_2021
Recasting the correlation - XVI
¨ 𝑌 𝑡 − 𝑀 𝑌 𝑡 . = (∫
J=8'
J=8
𝑒)0+ 8)J
. (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑢) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)})).
¨ 𝑌 𝑡 − 𝑀 𝑌 𝑡 . = 𝜎"
.. (∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
! 𝑢 + ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)).
¨ 𝑌 𝑡 − 𝑀 𝑌 𝑡 . = 𝜎"
.. (𝐴 + 𝐵 + 𝐶)
¨ 𝐴 = ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
!(𝑢)
¨ 𝐵 = 2. ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)
¨ 𝐶 = ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢). ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)
¨ 𝑉 𝑌 𝑡 = 𝔼 (𝑌(𝑡) − 𝑀 𝑌 𝑡 ). = 𝜎"
.. 𝔼 𝐴 + 𝐵 + 𝐶
109

110. Luc_Faucheux_2021
Recasting the correlation - XVII
¨ 𝑉 𝑌 𝑡 = 𝔼 (𝑌(𝑡) − 𝑀 𝑌 𝑡 ). = 𝜎"
.. 𝔼 𝐴 + 𝐵 + 𝐶
¨ 𝐴 = ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
!(𝑢)
¨ 𝔼{𝐴} = 𝔼{∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
! 𝑢 }
¨ 𝔼 𝐴 = ∫
J=8'
J=8
𝑒).0+ 8)J
. 𝜌.. 𝑑𝑢 = 𝜌.. [
*
.0+
. 𝑒).0+ 8)J
]J=8'
J=8
¨ 𝔼 𝐴 =
f!
.0+
. [1 − 𝑒).0+ 8)8' ]
110

111. Luc_Faucheux_2021
Recasting the correlation - XVIII
¨ 𝐵 = 2. ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)
¨ 𝔼{𝐵} = 𝔼{2. ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜌. 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
" 𝑢 }
¨ Since < 𝑑 ˆ
𝑊
!. 𝑑 ˆ
𝑊
" > = 0
¨ 𝔼 𝐵 = 0
111

112. Luc_Faucheux_2021
Recasting the correlation - XIX
¨ 𝐶 = ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢). ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)
¨ 𝔼{𝐶} = 𝔼{∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢). ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)}
¨ 𝔼{𝐶} = {∫
J=8'
J=8
𝑒)..0+ 8)J
. (1 − 𝜌.). 𝑑𝑢}
¨ 𝔼 𝐶 = ∫
J=8'
J=8
𝑒).0+ 8)J
. (1 − 𝜌.). 𝑑𝑢 =. (1 − 𝜌.). [
*
.0+
. 𝑒).0+ 8)J
]J=8'
J=8
¨ 𝔼 𝐶 =
(*)f!)
.0+
. [1 − 𝑒).0+ 8)8' ]
112

113. Luc_Faucheux_2021
Recasting the correlation - XX
¨ 𝑉 𝑌 𝑡 = 𝔼 (𝑌(𝑡) − 𝑀 𝑌 𝑡 ). = 𝜎"
.. 𝔼 𝐴 + 𝐵 + 𝐶
¨ 𝔼 𝐴 =
f!
.0+
. [1 − 𝑒).0+ 8)8' ]
¨ 𝔼 𝐵 = 0
¨ 𝔼 𝐶 =
(*)f!)
.0+
. [1 − 𝑒).0+ 8)8' ]
¨ 𝑉 𝑌 𝑡 = 𝔼 (𝑌(𝑡) − 𝑀 𝑌 𝑡 ). = 𝜎"
.. 𝔼 𝐴 + 𝐵 + 𝐶
¨ 𝑉 𝑌 𝑡 = 𝜎"
..
(*)f!Qf!)
.0+
. 1 − 𝑒).0+ 8)8' =
-+
!
.0+
. [1 − 𝑒).0+ 8)8' ]
113

114. Luc_Faucheux_2021
Recasting the correlation - XXI
¨ So far we have recovered the same expression for the mean and Variance for both 𝑋 𝑡 and
also 𝑌(𝑡)
¨ We are now left with the Covariance
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ All right we are almost there, and since we know that both 𝑋 𝑡 and 𝑌(𝑡) are normally
distributed, if we can prove that we also recover the Covariance then both descriptions are
identical. That will be quite an achievement (again might seem obvious if you are not too
concerned about being somewhat rigorous, or at least trying to be).
114

115. Luc_Faucheux_2021
Recasting the correlation - XXII
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝑋 𝑡 = 𝑋 𝑡A . 𝑒)0).(8)8') + ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢
¨ Which was normally distributed with moments:
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
¨ 𝑋 𝑡 − 𝑀 𝑋 𝑡 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢
115

116. Luc_Faucheux_2021
Recasting the correlation - XXIII
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝑌 𝑡 = 𝑒)0+ 8)8' . 𝑌 𝑡A + ∫
J=8'
J=8
𝑒)0+ 8)J
. (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑢) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)})
¨ 𝑀 𝑌 𝑡 = 𝔼 𝑌 = 𝑒)0+ 8)8' . 𝑌 𝑡A
¨ 𝑉 𝑌 𝑡 =
-+
!
.0+
. [1 − 𝑒).0+ 8)8' ]
¨ 𝑌 𝑡 − 𝑀 𝑌 𝑡 = ∫
J=8'
J=8
𝑒)0+ 8)J
. (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑢) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)})
¨ 𝑌 𝑡 − 𝑀 𝑌 𝑡 = ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜎". 𝜌. 𝑑 ˆ
𝑊
! 𝑢 + ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)
116

117. Luc_Faucheux_2021
Recasting the correlation - XXIV
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝑋 𝑡 − 𝑀 𝑋 𝑡 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢
¨ 𝑌 𝑡 − 𝑀 𝑌 𝑡 = ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜎". 𝜌. 𝑑 ˆ
𝑊
! 𝑢 + ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)
¨ 𝑋 𝑡 − 𝑀 𝑋 𝑡 . 𝑌 𝑡 − 𝑀 𝑌 𝑡 = 𝐴 + 𝐵
¨ 𝐴 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜎". 𝜌. 𝑑 ˆ
𝑊
! 𝑢
¨ 𝐵 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 (𝑋 𝑡 − 𝑀 𝑋 𝑡 ). (𝑌 𝑡 − 𝑀[𝑌(𝑡)])
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 𝐴 + 𝐵 = 𝔼 𝐴 + 𝔼 𝐵
117

118. Luc_Faucheux_2021
Recasting the correlation - XXV
¨ 𝐴 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜎". 𝜌. 𝑑 ˆ
𝑊
! 𝑢
¨ 𝔼{𝐴} = 𝔼{∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 𝜎". 𝜌. 𝑑 ˆ
𝑊
! 𝑢 }
¨ 𝔼{𝐴} = {∫
J=8'
J=8
𝑒)(0)Q0+). 8)J
. 𝜎!. 𝜎". 𝜌. 𝑑𝑢}
¨ 𝔼 𝐴 = 𝜌. 𝜎!. 𝜎". ∫
J=8'
J=8
𝑒)(0)Q0+). 8)J
𝑑𝑢 = 𝜌. 𝜎!. 𝜎". [
*
0)Q0+
. 𝑒)(0)Q0+). 8)J
]J=8'
J=8
¨ 𝔼 𝐴 =
f.-).-+
0)Q0+
. [1 − 𝑒)(0)Q0+). 8)8' ]
118

119. Luc_Faucheux_2021
Recasting the correlation - XXVI
¨ 𝐵 = ∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑢)
¨ 𝔼{𝐵} = 𝔼{∫
J=8'
J=8
𝑒)0). 8)J . 𝜎!. ([). 𝑑 ˆ
𝑊
! 𝑢 . ∫
J=8'
J=8
𝑒)0+ 8)J
. 1 − 𝜌.. 𝑑 ˆ
𝑊
" 𝑢 }
¨ 𝔼 𝐵 = 0
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 = 𝔼 𝐴 + 𝐵 = 𝔼 𝐴 + 𝔼 𝐵
¨ 𝔼 𝐴 =
f.-).-+
0)Q0+
. [1 − 𝑒)(0)Q0+). 8)8' ]
¨ 𝔼 𝐵 = 0
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 =
f.-).-+
0)Q0+
. [1 − 𝑒)(0)Q0+). 8)8' ]
¨ We recover indeed the same formula for the Covariance
119

120. Luc_Faucheux_2021
Recasting the correlation - XXVII
¨ This is actually seen quite often in the literature (Mercurio p.134 for example), and is usually
useful when you want to deal with independent Brownian motions, and do not want to have
to deal with the correlation when “picking” the stochastic process out of the distribution.
120

121. Luc_Faucheux_2021
Recasting the correlation - XXVIII
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"(𝑡)
¨ With: < 𝑑𝑊
!(𝑡). 𝑑𝑊
"(𝑡) >= 𝜌. 𝑑𝑡
¨ Is equivalent to:
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. 𝑑 ˆ
𝑊
!(𝑡)
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ With: < 𝑑 ˆ
𝑊
!(𝑡). 𝑑 ˆ
𝑊
"(𝑡) > = 0
121

122. Luc_Faucheux_2021
Recasting the correlation - XXIX
¨ In both formulation we have:
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
¨ 𝑀 𝑌 𝑡 = 𝑌 𝑡A . 𝑒)0+. 8)8'
¨ 𝑉 𝑌 𝑡 =
-+
!
.0+
. [1 − 𝑒).0+ 8)8' ]
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 =
f.-).-+
0)Q0+
. [1 − 𝑒)(0)Q0+). 8)8' ]
122

123. Luc_Faucheux_2021
Recasting the correlation - XXX
¨ It pays to look at the small time approximation: (𝑡 → 𝑡A)
¨ This would be also the weak mean reversion regime 𝑘! → 0 and 𝑘" → 0 (diffusive regime)
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8') → 𝑋(𝑡A)
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8' →
-)
!
.0)
. 1 − 1 + 2𝑘!. 𝑡 − 𝑡A = 𝜎!
.. 𝑡 − 𝑡A
¨ 𝑀 𝑌 𝑡 = 𝑌 𝑡A . 𝑒)0+. 8)8' → 𝑌(𝑡A)
¨ 𝑉 𝑌 𝑡 =
-+
!
.0+
. 1 − 𝑒).0+ 8)8' → 𝜎"
.. 𝑡 − 𝑡A
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 =
f.-).-+
0)Q0+
. 1 − 𝑒) 0)Q0+ . 8)8' →
f.-).-+
0)Q0+
. 1 − 1 + 𝑘! + 𝑘" . 𝑡 − 𝑡A
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 → 𝜌. 𝜎!. 𝜎". 𝑡 − 𝑡A
123

124. Luc_Faucheux_2021
Recasting the correlation - XXXI
¨ In the strong mean reversion regime, 𝑘! → ∞ and 𝑘" → ∞
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8') → 0
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8' →
-)
!
.0)
. 1 − 0 → 0
¨ 𝑀 𝑌 𝑡 = 𝑌 𝑡A . 𝑒)0+. 8)8' → 0
¨ 𝑉 𝑌 𝑡 =
-+
!
.0+
. 1 − 𝑒).0+ 8)8' → 0
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 =
f.-).-+
0)Q0+
. 1 − 𝑒) 0)Q0+ . 8)8' →
f.-).-+
0)Q0+
. 1 − 0 → 0
¨ Looks boring, but we saw that this is a nice limit of the IRMA formalism for 𝑍(𝑡)
124

125. Luc_Faucheux_2021
Recasting the correlation - XXXII
¨ And just because physicists loooove steady state (equilibrium or 𝑡 → ∞) solutions
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8') → 0
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8' →
-)
!
.0)
¨ 𝑀 𝑌 𝑡 = 𝑌 𝑡A . 𝑒)0+. 8)8' → 0
¨ 𝑉 𝑌 𝑡 =
-+
!
.0+
. 1 − 𝑒).0+ 8)8' →
-+
!
.0+
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 =
f.-).-+
0)Q0+
. 1 − 𝑒) 0)Q0+ . 8)8' →
f.-).-+
0)Q0+
¨ So the distributions are bounded, which is quite nice, which was one of the original
attraction about the Langevin equation
125

126. Luc_Faucheux_2021
Re-casting the correlation for the
instantaneous variables
126

127. Luc_Faucheux_2021
Recasting the correlation instantaneous - I
¨ Instead of looking at the exact solutions:
¨ 𝑀 𝑋 𝑡 = 𝑋(𝑡A). 𝑒)0).(8)8')
¨ 𝑉 𝑋 𝑡 =
-)
!
.0)
. 1 − 𝑒).0). 8)8'
¨ 𝑀 𝑌 𝑡 = 𝑌 𝑡A . 𝑒)0+. 8)8'
¨ 𝑉 𝑌 𝑡 =
-+
!
.0+
. [1 − 𝑒).0+ 8)8' ]
¨ 𝐶𝑂𝑉 𝑋 𝑡 , 𝑌 𝑡 =
f.-).-+
0)Q0+
. [1 − 𝑒)(0)Q0+). 8)8' ]
¨ We could have also only looked like Tuckman does p. 288 at the instantaneous quantities
𝑀 𝑑𝑋 𝑡 , 𝑉 𝑑𝑋 𝑡 , 𝑀 𝑑𝑌 𝑡 , 𝑉 𝑑𝑌 𝑡 , 𝐶𝑂𝑉 𝑑𝑋 𝑡 , 𝑑𝑌 𝑡
¨ Let’s do that to gain some intuition
127

128. Luc_Faucheux_2021
Recasting the correlation instantaneous - II
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"(𝑡)
¨ With: < 𝑑𝑊
!(𝑡). 𝑑𝑊
"(𝑡) >= 𝜌. 𝑑𝑡
¨ Is equivalent to:
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. 𝑑 ˆ
𝑊
!(𝑡)
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ With: < 𝑑 ˆ
𝑊
!(𝑡). 𝑑 ˆ
𝑊
"(𝑡) > = 0
128

129. Luc_Faucheux_2021
Recasting the correlation instantaneous - III
¨ Let’s look at :
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ Using the notation:
¨ < 𝑑𝑋 𝑡 > to denote the usual quantity 𝔼 𝑑𝑋 = 𝔼8Q$8
E)
{𝑋 𝑡 + 𝑑𝑡 − 𝑋(𝑡) 𝔉 𝑡 that we
have been using the deck II of the stochastic calculus decks
¨ < 𝑑𝑋 𝑡 > = 𝔼 𝑑𝑋 = 𝔼8Q$8
E)
{𝑋 𝑡 + 𝑑𝑡 − 𝑋(𝑡) 𝔉 𝑡
¨ < 𝑑𝑋 𝑡 > = 𝔼 𝑑𝑋 = 𝔼 −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡
¨ That is the usual drift / advection term
¨ Let’s now look at the higher moment (diffusion)
129

130. Luc_Faucheux_2021
Recasting the correlation instantaneous - IV
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ < 𝑑𝑋 𝑡 . > = 𝔼 𝑑𝑋. = 𝔼8Q$8
E)
{(𝑋 𝑡 + 𝑑𝑡 − 𝑋(𝑡)). 𝔉 𝑡
¨ < 𝑑𝑋 𝑡 . > = 𝔼 𝑑𝑋.
¨ < 𝑑𝑋 𝑡 . > = 𝔼 (−𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡) ).
¨ < 𝑑𝑋 𝑡 . > = 𝜎!
.. 𝑑𝑡
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ < 𝑑𝑋 𝑡 > = −𝑘!. 𝑋(𝑡). 𝑑𝑡
¨ < 𝑑𝑋 𝑡 . > = 𝜎!
.. 𝑑𝑡
¨ Same for the process 𝑌(𝑡)
130

131. Luc_Faucheux_2021
Recasting the correlation instantaneous - V
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌(𝑡) > = 𝔼 𝑑𝑋. 𝑑𝑌 = 𝔼8Q$8
E)
{ 𝑋 𝑡 + 𝑑𝑡 − 𝑋 𝑡 . (𝑌 𝑡 + 𝑑𝑡 − 𝑌(𝑡)) 𝔉 𝑡
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡)
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". ([). 𝑑𝑊
"(𝑡)
¨ With: < 𝑑𝑊
!(𝑡). 𝑑𝑊
"(𝑡) >= 𝜌. 𝑑𝑡
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌(𝑡) > = 𝔼 𝑑𝑋. 𝑑𝑌
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌(𝑡) > = 𝔼 (−𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡) ). (−𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. ([). 𝑑𝑊
!(𝑡) )
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌 𝑡 > = 𝔼 (𝜎!. 𝑑𝑊
!(𝑡) ). (𝜎!. 𝑑𝑊
!(𝑡) )
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌 𝑡 > = 𝜎!. 𝜎". 𝔼 𝑑𝑊
! 𝑡 . 𝑑𝑊
"(𝑡)
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌 𝑡 > = 𝜌. 𝜎!. 𝜎". 𝑑𝑡
131

132. Luc_Faucheux_2021
Recasting the correlation instantaneous - VI
¨ Starting from:
¨ 𝑑𝑋(𝑡) = −𝑘!. 𝑋(𝑡). 𝑑𝑡 + 𝜎!. 𝑑 ˆ
𝑊
!(𝑡)
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ With: < 𝑑 ˆ
𝑊
!(𝑡). 𝑑 ˆ
𝑊
"(𝑡) > = 0
¨ < 𝑑𝑋 𝑡 > = −𝑘!. 𝑋(𝑡). 𝑑𝑡
¨ < 𝑑𝑋 𝑡 . > = 𝜎!
.. 𝑑𝑡
¨ The term for the process 𝑌(𝑡) is a little more complicated, but essentially the same
derivation we had for the full explicit solution in the previous slides.
132

133. Luc_Faucheux_2021
Recasting the correlation instantaneous - VII
¨ 𝑑𝑌(𝑡) = −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ < 𝑑𝑌 𝑡 > = 𝔼 𝑑𝑌 = 𝔼8Q$8
g
E), g
E+
{𝑌 𝑡 + 𝑑𝑡 − 𝑌(𝑡) 𝔉 𝑡
¨ < 𝑑𝑌 𝑡 > = 𝔼 𝑑𝑌 = 𝔼 −𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ < 𝑑𝑌 𝑡 > = −𝑘". 𝑌(𝑡). 𝑑𝑡
¨ < 𝑑𝑌 𝑡 . > = 𝔼 𝑑𝑌.
¨ < 𝑑𝑌 𝑡 . > = 𝔼 (−𝑘". 𝑌(𝑡). 𝑑𝑡 + 𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}).
¨ < 𝑑𝑌 𝑡 . > = (𝜎". 𝜌).𝔼 (𝑑 ˆ
𝑊
!(𝑡)). + (𝜎". 1 − 𝜌.).𝔼 (𝑑 ˆ
𝑊
"(𝑡)). +
𝜎". 𝜌. 𝜎". 1 − 𝜌.. 𝔼 𝑑 ˆ
𝑊
! 𝑡 . 𝑑 ˆ
𝑊
"(𝑡)
133

134. Luc_Faucheux_2021
Recasting the correlation instantaneous - VIII
¨ < 𝑑𝑌 𝑡 .
> = (𝜎1. 𝜌).
𝔼 (𝑑 6
𝑊
2(𝑡)).
+ (𝜎1. 1 − 𝜌.).
𝔼 (𝑑 6
𝑊
1(𝑡)).
+ 𝜎1. 𝜌. 𝜎1. 1 − 𝜌.. 𝔼 𝑑 6
𝑊
2 𝑡 . 𝑑 6
𝑊
1(𝑡)
¨ < 𝑑 ˆ
𝑊
!(𝑡). 𝑑 ˆ
𝑊
"(𝑡) > = 0
¨ < 𝑑𝑌 𝑡 . > = (𝜎". 𝜌).𝔼 (𝑑 ˆ
𝑊
!(𝑡)). + (𝜎". 1 − 𝜌.).𝔼 (𝑑 ˆ
𝑊
"(𝑡)).
¨ < 𝑑𝑌 𝑡 . > = (𝜎". 𝜌).. 𝑑𝑡 + (𝜎". 1 − 𝜌.).. 𝑑𝑡
¨ < 𝑑𝑌 𝑡 . > = 𝜎"
.. 𝑑𝑡. {𝜌. + 1 − 𝜌.
.
}
¨ < 𝑑𝑌 𝑡 . > = 𝜎"
.. 𝑑𝑡. {𝜌. + 1 − 𝜌.}
¨ < 𝑑𝑌 𝑡 . > = 𝜎"
.. 𝑑𝑡
134

135. Luc_Faucheux_2021
Recasting the correlation instantaneous - IX
¨ And now to finish with the cross term:
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌(𝑡) > = 𝔼 𝑑𝑋. 𝑑𝑌 = 𝔼8Q$8
g
E), g
E+
{ 𝑋 𝑡 + 𝑑𝑡 − 𝑋 𝑡 . (𝑌 𝑡 + 𝑑𝑡 − 𝑌(𝑡)) 𝔉 𝑡
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌(𝑡) > = 𝔼 𝑑𝑋. 𝑑𝑌
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌(𝑡) > = 𝔼 (−𝑘2. 𝑋(𝑡). 𝑑𝑡 + 𝜎2. 𝑑 6
𝑊
2(𝑡)). (−𝑘1. 𝑌(𝑡). 𝑑𝑡 + 𝜎1. {𝜌. 𝑑 6
𝑊
2(𝑡) + 1 − 𝜌.. 𝑑 6
𝑊
1(𝑡)})
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌 𝑡 > = 𝔼{𝜎!. 𝑑 ˆ
𝑊
! 𝑡 . (𝜎". {𝜌. 𝑑 ˆ
𝑊
!(𝑡) + 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)})}
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌 𝑡 > = 𝜌. 𝜎!. 𝜎". 𝔼 𝑑 ˆ
𝑊
! 𝑡 . 𝑑 ˆ
𝑊
! 𝑡 + 𝔼{𝜎!. 𝑑 ˆ
𝑊
! 𝑡 . 𝜎". 1 − 𝜌.. 𝑑 ˆ
𝑊
"(𝑡)}
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌 𝑡 > = 𝜌. 𝜎!. 𝜎". 𝑑𝑡 + 1 − 𝜌.. 𝜎!. 𝜎". 𝔼{𝑑 ˆ
𝑊
! 𝑡 . 𝑑 ˆ
𝑊
"(𝑡)}
¨ < 𝑑 ˆ
𝑊
!(𝑡). 𝑑 ˆ
𝑊
"(𝑡) > = 0
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌 𝑡 > = 𝜌. 𝜎!. 𝜎". 𝑑𝑡
135

136. Luc_Faucheux_2021
Recasting the correlation instantaneous - X
¨ So we do recover the same expressions in both sets of equations for the quantities:
¨ < 𝑑𝑋 𝑡 . 𝑑𝑌(𝑡) >
¨ < 𝑑𝑋 𝑡 . 𝑑𝑋(𝑡) >
¨ < 𝑑𝑌 𝑡 . 𝑑𝑌(𝑡) >
¨ < 𝑑𝑋 𝑡 >
¨ < 𝑑𝑌(𝑡) >
¨ Following the logic of the deck on stochastic calculus, we will also recover the same PDE
(Partial Differential equations) for the PDF (Probability Distribution Function) of the
processes that we described through the SDE (stochastic Differential Equations).
¨ Hence the 2 descriptions are equivalent
136

137. Luc_Faucheux_2021
Tuckman “Gauss+” model
137

138. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ I
138

139. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ II
¨ Bruce Tuckman worked at Salomon on the ”2+ IRMA” model with other intellectual giants
like Craig Fithian, Francis Longstaff and other too numerous to name here.
¨ SO he knows what he is talking about.
¨ His book is awesome to read.
¨ He expresses the model on page 288 as:
¨ 𝑑𝑚(𝑡) = −𝛼;. (𝑚(𝑡) − 𝑙(𝑡)). 𝑑𝑡 + 𝜎;. (𝜌. 𝑑𝑊*(𝑡) + 1 − 𝜌.. 𝑑𝑊.(𝑡))
¨ 𝑑𝑙(𝑡) = −𝛼h. (𝑙(𝑡) − 𝜃). 𝑑𝑡 + 𝜎h. 𝑑𝑊*(𝑡)
¨ 𝑑𝑟(𝑡) = −𝛼,. (𝑟(𝑡) − 𝑚(𝑡)). 𝑑𝑡
¨ With: < 𝑑𝑊*(𝑡). 𝑑𝑊.(𝑡) > = 0
¨ Takes a little work to show that this can be casted into the 2+ IRMA formalism, but let’s do it.
¨ It’s worth it, plus I got let go of my job at Natixis, so I need to get my bearings back….J
139

140. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ III
¨ 𝑑𝑚(𝑡) = −𝛼;. (𝑚(𝑡) − 𝑙(𝑡)). 𝑑𝑡 + 𝜎;. (𝜌. 𝑑𝑊*(𝑡) + 1 − 𝜌.. 𝑑𝑊.(𝑡))
¨ 𝑑𝑙(𝑡) = −𝛼h. (𝑙(𝑡) − 𝜃). 𝑑𝑡 + 𝜎h. 𝑑𝑊*(𝑡)
¨ 𝑑𝑟(𝑡) = −𝛼,. (𝑟(𝑡) − 𝑚(𝑡)). 𝑑𝑡
¨ With: < 𝑑𝑊*(𝑡). 𝑑𝑊.(𝑡) > = 0
¨ So first of all we can recast the correlation as:
¨ 𝑑𝑚(𝑡) = −𝛼;. (𝑚(𝑡) − 𝑙(𝑡)). 𝑑𝑡 + 𝜎;. 𝑑 …
𝑊.(𝑡)
¨ 𝑑𝑙(𝑡) = −𝛼h. (𝑙(𝑡) − 𝜃). 𝑑𝑡 + 𝜎h. 𝑑 …
𝑊*(𝑡)
¨ 𝑑𝑟(𝑡) = −𝛼,. (𝑟(𝑡) − 𝑚(𝑡)). 𝑑𝑡
¨ With: < 𝑑 …
𝑊* 𝑡 . 𝑑 …
𝑊. 𝑡 > = 𝜌. 𝑑𝑡
140

141. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ IV
¨ 𝑑𝑚(𝑡) = −𝛼;. (𝑚(𝑡) − 𝑙(𝑡)). 𝑑𝑡 + 𝜎;. 𝑑 …
𝑊.(𝑡)
¨ 𝑑𝑙(𝑡) = −𝛼h. (𝑙(𝑡) − 𝜃). 𝑑𝑡 + 𝜎h. 𝑑 …
𝑊*(𝑡)
¨ 𝑑𝑟(𝑡) = −𝛼,. (𝑟(𝑡) − 𝑚(𝑡)). 𝑑𝑡
¨ With: < 𝑑 …
𝑊* 𝑡 . 𝑑 …
𝑊. 𝑡 > = 𝜌. 𝑑𝑡
¨ Let’s rename with 𝑥 and 𝑦:
¨ 𝑑𝑦(𝑡) = −𝛼". (𝑦(𝑡) − 𝑥(𝑡)). 𝑑𝑡 + 𝜎". 𝑑 …
𝑊
"(𝑡)
¨ 𝑑𝑥(𝑡) = −𝛼!. (𝑥(𝑡) − 𝜃). 𝑑𝑡 + 𝜎!. 𝑑 …
𝑊
!(𝑡)
¨ 𝑑𝑟(𝑡) = −𝛼,. (𝑟(𝑡) − 𝑦(𝑡)). 𝑑𝑡
¨ With: < 𝑑 …
𝑊
! 𝑡 . 𝑑 …
𝑊
" 𝑡 > = 𝜌. 𝑑𝑡
141

142. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ V
¨ 𝑑𝑦(𝑡) = −𝛼". (𝑦(𝑡) − 𝑥(𝑡)). 𝑑𝑡 + 𝜎". 𝑑 …
𝑊
"(𝑡)
¨ 𝑑𝑥(𝑡) = −𝛼!. (𝑥(𝑡) − 𝜃). 𝑑𝑡 + 𝜎!. 𝑑 …
𝑊
!(𝑡)
¨ 𝑑𝑟(𝑡) = −𝛼,. (𝑟(𝑡) − 𝑦(𝑡)). 𝑑𝑡
¨ With: < 𝑑 …
𝑊
! 𝑡 . 𝑑 …
𝑊
" 𝑡 > = 𝜌. 𝑑𝑡
¨ Let’s “cascade” (this is a Bruce Tuckman term) l
𝑥 𝑡 = 𝑥 𝑡 − 𝜃 (remember that is a
constant)
¨ 𝑑𝑦(𝑡) = −𝛼". (𝑦 𝑡 − l
𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑 …
𝑊
"(𝑡)
¨ 𝑑 l
𝑥(𝑡) = −𝛼!. l
𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑 …
𝑊
!(𝑡)
¨ 𝑑𝑟(𝑡) = −𝛼,. (𝑟(𝑡) − 𝑦(𝑡)). 𝑑𝑡
¨ With: < 𝑑 …
𝑊i
! 𝑡 . 𝑑 …
𝑊
" 𝑡 > = 𝜌. 𝑑𝑡
142

143. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ VI
¨ 𝑑𝑦(𝑡) = −𝛼". (𝑦 𝑡 − l
𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑 …
𝑊
"(𝑡)
¨ 𝑑 l
𝑥(𝑡) = −𝛼!. l
𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑 …
𝑊
!(𝑡)
¨ 𝑑𝑟(𝑡) = −𝛼,. (𝑟(𝑡) − 𝑦(𝑡)). 𝑑𝑡
¨ With: < 𝑑 …
𝑊i
! 𝑡 . 𝑑 …
𝑊
" 𝑡 > = 𝜌. 𝑑𝑡
¨ Replacing l
𝑥 by 𝑥, the 𝛼 with 𝑘 and 𝑟 with 𝑧
¨ 𝑑𝑥(𝑡) = −𝑘!. 𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑𝑊
!(𝑡)
¨ 𝑑𝑦(𝑡) = −𝑘". (𝑦 𝑡 − 𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑𝑊
"(𝑡)
¨ 𝑑𝑧 𝑡 = −𝑘#. 𝑧 𝑡 − 𝑦 𝑡 . 𝑑𝑡 + 𝜎#. 𝑑𝑊
#(𝑡) with 𝜎# = 0
¨ With: < 𝑑𝑊
! 𝑡 . 𝑑𝑊
" 𝑡 > = 𝜌. 𝑑𝑡
¨ 𝑟(𝑡) = 𝑧(𝑡)
143

144. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ VII
¨ 𝑑𝑥(𝑡) = −𝑘!. 𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑𝑊
!(𝑡)
¨ 𝑑𝑦(𝑡) = −𝑘". (𝑦 𝑡 − 𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑𝑊
"(𝑡)
¨ 𝑑𝑧 𝑡 = −𝑘#. 𝑧 𝑡 − 𝑦 𝑡 . 𝑑𝑡 + 𝜎#. 𝑑𝑊
#(𝑡)
¨ With: < 𝑑𝑊
! 𝑡 . 𝑑𝑊
" 𝑡 > = 𝜌. 𝑑𝑡
¨ 𝑟(𝑡) = 𝑧(𝑡)
¨ This one has the “cascade” form because first you have 𝑥, then (𝑦 − 𝑥) then (𝑧 − 𝑦)
¨ We can use matrices like Tuckman does in the appendix or just “cascade” the change of
variables
¨ l
𝑥 = 𝑥
¨ l
𝑦 = 𝑘!. 𝑦 − 𝑘". 𝑥
144

145. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ VIII
¨ 𝑑𝑥(𝑡) = −𝑘!. 𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑𝑊
!(𝑡)
¨ 𝑑𝑦(𝑡) = −𝑘". (𝑦 𝑡 − 𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑𝑊
"(𝑡)
¨ l
𝑥 = 𝛼!,!. 𝑥 + 𝛼!,". 𝑦
¨ l
𝑦 = 𝛼",!. 𝑥 + 𝛼",". 𝑦
¨ And then we will solve for the variables 𝛼!,! to reduce the mean reversion to be diagonal
¨ 𝑑 l
𝑥 = 𝛼!,!. 𝑑𝑥 + 𝛼!,". 𝑑𝑦
¨ 𝑑 l
𝑦 = 𝛼",!. 𝑑𝑥 + 𝛼",". 𝑑𝑦
¨ 𝑑 l
𝑥 = 𝛼!,!. (−𝑘!. 𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑𝑊
!(𝑡)) + 𝛼!,". (−𝑘". (𝑦 𝑡 − 𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑𝑊
"(𝑡))
¨ 𝑑 l
𝑦 = 𝛼",!. (−𝑘!. 𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑𝑊
!(𝑡)) + 𝛼",". (−𝑘". (𝑦 𝑡 − 𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑𝑊
"(𝑡))
145

146. Luc_Faucheux_2021
IRMA – Tuckman – Gauss+ IX
¨ 𝑑 l
𝑥 = 𝛼!,!. (−𝑘!. 𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑𝑊
!(𝑡)) + 𝛼!,". (−𝑘". (𝑦 𝑡 − 𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑𝑊
"(𝑡))
¨ 𝑑 l
𝑦 = 𝛼",!. (−𝑘!. 𝑥(𝑡). 𝑑𝑡 + 𝜎!. 𝑑𝑊
!(𝑡)) + 𝛼",". (−𝑘". (𝑦 𝑡 − 𝑥 𝑡 + 𝜃). 𝑑𝑡 + 𝜎". 𝑑𝑊
"(𝑡))
¨ And then we replace 𝑥 and 𝑦 by l
𝑥 and l
𝑦, which is a matrix inversion.
¨ l
𝑥 = 𝛼!,!. 𝑥 + 𝛼!,". 𝑦
¨ l
𝑦 = 𝛼",!. 𝑥 + 𝛼",". 𝑦
¨ 𝛼",! . l
𝑥 = 𝛼",!. 𝛼!,!. 𝑥 + 𝛼",!. 𝛼!,". 𝑦
¨ 𝛼!,!. l
𝑦 = 𝛼!,!. 𝛼",!. 𝑥 + 𝛼!,!. 𝛼",". 𝑦
¨ 𝛼",! . l
𝑥 − 𝛼!,!. l
𝑦 = 𝛼!,!. 𝛼",!. 𝑥 − 𝛼!,!. 𝛼",!. 𝑥 + (𝛼",!. 𝛼!," − 𝛼!,!. 𝛼","). 𝑦
¨ 𝛼",! . l
𝑥 − 𝛼!,!. l
𝑦 = 𝛼",!. 𝛼!," − 𝛼!,!. 𝛼","). 𝑦
146