Lévy processes - a broad class of processes used in
financial modelling
Rafal M. Lochowski
Warsaw School of Economics and AIMS
Finance research group talk
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
1 / 30
Stochastic processes - mathematical tools for modelling
the evolution of phenomena with uncertain outcomes
Stochastic processes are flexible mathematical tools for modelling
the evolution of phenomena with uncertain outcomes
Examples of such phenomena are: stock prices, numbers of cases of
some disease in a given area, the level of Nile river etc.
In a most general setting, stochastic process is simple a collection of
random variables
Xt : Ω → E , t ∈ T ,
where Ω is a probability space (equipped with a probability function
P : Ω → [0, 1]), E is the space of possible values of Xt and T is some
set
For given ω ∈ Ω the random function
T 3 t 7→ Xt (ω) ∈ E
is called the trajectory of the process X
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
2 / 30
Lévy processes - definition
A Lévy process on Rd , starting from some x ∈ Rd is a collection of
random variables Xt : Ω → Rd , t ∈ [0, +∞), which has the following
properties
X0 = x almost surely (with probability 1);
the increments of X are independent, this means that for any
0 ≤ s < t < u the variables Xu − Xt and Xt − Xs are independent;
the increments of X are stationary, this means that for any
0 ≤ s < t and ∆ > 0 the variables Xt+∆ − Xt and Xs+∆ − Xs are
identically distributed (have the same probability laws). In other
words, for any (Borel) set A ⊂ Rd ,
P (Xt+∆ − Xt ∈ A) = P (Xs+∆ − Xs ∈ A) ;
last but not least, the process X is continuous in probability, which
means that for any r > 0, limt→0+ P (|Xt | > r ) = 0.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
3 / 30
Lévy processes - another (more user-friendly ;-) definition
A Lévy process on Rd , starting from some x ∈ Rd is a collection of
random variables Xt : Ω → Rd , t ∈ [0, +∞), which has the following
properties
X0 = x almost surely (with probability 1);
the increments of X are independent, this means that for any
0 ≤ s < t < u the variables Xu − Xt and Xt − Xs are independent;
the increments of X are stationary, this means that for any
0 ≤ s < t and ∆ > 0 the variables Xt+∆ − Xt and Xs+∆ − Xs are
identically distributed (have the same probability laws). In other
words, for any (Borel) set A ⊂ Rd ,
P (Xt+∆ − Xt ∈ A) = P (Xs+∆ − Xs ∈ A) ;
last but not least, the trajectories of the process X are almost surely
càdlàg = right-continuous with left limits .
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
4 / 30
Financial modeling with Lévy processes - examples
One of the first models used in financial mathematics incorporating Lévy
processes was Merton’s jump-diffusion model (1976). He modeled the
dynamics of a stock price St by an SDE of the type
dSt = St− (adt + σdBt + dLt ) for t ≥ 0 and S0 > 0.
(1)
P t
In the above B denotes a standard Brownian motion and L = N
n=1 Yn
denotes a compound Poisson process (it will be defined later). Solution to
(1) is given by
Y
Nt
σ2
(1 + Yn ) .
t + σBt
St = S0 exp
a−
2
n=1
An important generalisation of Merton’s jump-diffusion model is given by
an exponential of the form
St = S0 exp (Xt ) ,
where X denotes a Lévy process.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
5 / 30
Lévy processes - examples
The class of Lévy processes is very broad.
However, it appears that the only Lévy process on Rd with
continuous trajectories is a Brownian motion with drift, which
means that
Xt = A · Bt + M · t, t ≥ 0,
where A is a d × d real matrix, M ∈ Rd is a drift and B is a standard
Brownian motion on Rd .
Recall that for a standard Brownian motion on Rd and 0 ≤ s < t one
has that Bt − Bs has normal distribution with mean 0 and covariance
matrix (t − s)Id , where Id is the d-dimesional identity matrix.
Another fundamental example of a Lévy process is a Poisson
process. It attains values on N = {0, 1, 2, . . .} .
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
6 / 30
Poisson process - properties
The Poisson process N with intensity λ has the following properties
it has independent and stationary increments (like any Lévy process);
N0 = 0 with probability 1;
for any 0 ≤ s < t the difference Nt − Ns has Poisson distribution with
parameter (expectation) λ(t − s). From this it follows that
P (Nt − Ns = k) = e −λ(t−s)
(λ(t − s))k
,
k!
k = 0, 1, 2, . . . ;
in particular, from the first three properties it folows that the jumps
of the process N have always size 1 and occur at the random times
τ1 , τ2 , τ3 , . . . . Setting τ0 = 0 we have that the differences τk − τk−1 ,
k = 0, 1, 2, . . . , are independent and
P (τk − τk−1 > t) = e −λt ,
k = 0, 1, 2, . . . ,
thus they have exponential distribution with parameter λ.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
7 / 30
Poisson process - construction
It is not obvious that the Poisson process exists.
A sketch of its construction is the following. Let T > 0 be a (large) fixed
number.
We take X1 , X2 , . . . i.i.d. (= independent and identically distributed)
random variables uniformly distributed on [0, T ].
Next, let N(T ) be a Poisson random variable with mean λT ,
independent from X1 , X2 , . . . .
Now, for t ∈ (0, T ] we set
N(T )
Nt :=
X
1{Xi ∈[0,t]} .
i=1
To define Nt for t ∈ (T , 2T ] we take X̃1 , X̃2 , . . . i.i.d. r. vs uniformly
distributed on [T , 2T ] and a Poisson random variable Ñ(T ) with
mean λT , independent from X1 , X2 , . . . , X̃1 , X̃2 , . . . and N(T ). Next,
PÑ(T )
for t ∈ (T , 2T ] we set Nt := N(T ) + i=1 1{X̃i ∈[T ,t]} .
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
8 / 30
Compound Poisson process
Any Poisson process has very simple structure of jumps they are always equal 1.
What happens if we allow jumps of any size ???
It appears that if N is a Poisson process, Y , Y1 , Y2 , . . . are i.i.d. real
random variables (Y1 , Y2 , . . . represent the sizes and direction of
consecutive jumps), independent also from N, then setting
Xt =
Nt
X
Yk
k=1
we obtain another Lévy process called compound Poisson process.
From the properties of N it is easy to infer that X is also a Lévy process.
Remark
Negative Yi corresponds to a negative jump at time τi .
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
9 / 30
Compound Poisson processes - building blocks of Lévy
processes
Compound Lévy processes allow for much greater flexibility than Poisson
processes.
It appears that they (together with the Brownian motion with drift) are
building blocks of all other Lévy processes.
The big variety of compound Poisson processes makes them difficult to
investigate. For example, it is quite challenging to find the distribution of
Xt for fixed t.
One of the most useful tools to deal with such processes are characteristic
functions. The characteristic function of the variable Xt is defined as
ϕt (u) = Ee iuXt .
There is a fundamental 1-1 correspondence between characteristic
functions and distributions of r. vs.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
10 / 30
Characteristic functions of Poisson and compound Poisson
processes
To warm up let us first calculate the characteristic function of a Poisson
process with inensity λ.
Ee
iuNt
=
∞
X
e
iuk
P (Nt = k) =
k=0
∞
X
e iuk e −λt
k=0
= e −λt
∞
X
k
λte iu
k!
k=0
= e −λt e λte = e λt (e
iu
Next, for the compound Poisson process Xt =
Ee iuXt =
∞
X
Ee iu
Pk
n=1
Yn
(λt)k
k!
P (Nt = k) =
PNt
k=1 Yk
∞ X
Ee iuY
iu·1 −1
).
we calculate
k
e −λt
k=0
k=0
∞
X
λtEe iuY
−λt
=e
k!
k
= e −λt e λtEe
iuY
= e λtE(e
(λt)k
k!
iuY −1
).
k=0
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
11 / 30
Characteristic functions of compound Poisson processes cont.
In the calculation of the characteristic function of the compound Poisson
process we used fundamental property of characteristic functions: if Y and
Z are independent real random variables then
ϕY +Z (u) := Ee iu(Y +Z ) = Ee iuY e iuZ
= Ee iuY Ee iuZ = ϕY (u)ϕZ (u).
From this it easily follows that Ee iu
Pk
n=1
Yn
= Ee iuY
k
.
Using this property may also easily calculate the characteristic function of
a sum of two independent compound Poisson processes.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
12 / 30
Characteristic functions of compound Poisson processes cont.
Let
Ee
iuXt
=e
λtE(e iuY −1)
and Ee
iu X̃t
=e
λ̃tE e iu Ỹ −1
then
Ee iu(Xt +X̃t ) = e λtE(e
=e
) e λ̃tE
iuY −1
(λ+λ̃)tE
e iu Ỹ −1
λ
e iuY + λ̃ e iu Ỹ −1
λ+λ̃
λ+λ̃
.
Thus we see that the sum of two independent compound processes is again
a compound Poisson process with the intensity λ + λ̃ and the distribution
of jumps given as a mixture of the distributions of the variables Y and Ỹ .
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
13 / 30
Characteristic exponents of compound Poisson processes
This may be even better seen when we write the expectations of e iuY and
e iuỸ using the laws of the variables Y and Ỹ . For a (Borel) set A let us
define
ν (A) := P (Y ∈ A) and ν̃ (A) := P Ỹ ∈ A
then
Ee iuY − 1 =
Z
e iuy − 1 ν (dy ) and Ee iuỸ − 1 =
R
Z
iuy
e − 1 ν̃ (dy ) ,
R
and
λE e iuY − 1 + λ̃E e iuỸ − 1
)
(
Z λ̃
λ
iuy
ν (dy ) +
ν̃ (dy ) .
= λ + λ̃
e −1
λ + λ̃
λ + λ̃
R
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
14 / 30
Characteristic functions of compound Poisson processes cont.
As a result we see that any random variable whose characteristic function
is given by
Z
iuy
iuX1
Ee
= exp
e − 1 ν (dy ) ,
R
where ν is a finite and non-negative (Borel) measure on R is a
characteristic function of X1 where X is a compound Poisson process with
intensity Λ := ν (R \ {0}) such that
Ee
iuXt
Z
Z
iuy
= exp t
e − 1 ν (dy ) = exp Λt
iuy
ν (dy )
e −1
Λ
R\{0}
R
!
Moreover, the measure ν has the interpretation that ν (dy ) represents the
intensity of jumps of size y .
Unfortunately (or fortunately ;-) it does not exhaust all possible forms of
characteristic functions of Lévy processes without Brownian part.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
15 / 30
A general form of the characteristic functions of real Lévy
processes
Roughly speaking, it appears that it is possible to add small jumps, which
appear with infinite intensity, but whose signs are opposite, which results
in cancelation, or which are compensated by drift in opposite direction,
and as a result we obtain finite sum.
A general form of the characteristic function of Xt is given by the formula
1 2 2
E exp (iuXt ) = exp iaut − σ u t
2
!
Z
iuy
ν (dy )
× exp ν (R \ [−1, 1]) t
e −1
ν (R \ [−1, 1])
R\(−1,1)
!
Z
iuy
× exp t
e − 1 − iuy ν (dy ) .
[−1,1]\{0}
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
16 / 30
Interpretation of a general form of the characteristic
function of a real Lévy process
This formula makes sense under the assumption that
Z
min y 2 , 1 ν (dy ) < +∞.
R\{0}
The first factor corresponds to the Brownian motion with drift
Wt = σBt + a · t, where B is a standard Brownian motion.
the second factor corresponds to the compound Poisson process with
the intensity ν (R \ [−1, 1]) and the distribution of jums given by
ν(dy )
1R\[−1,1] ν(R\[−1,1])
.
the third factor corresponds to compensated sum of small jumps
(with possibly infinite intensity).
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
17 / 30
Interpretation of the third factor of the characteristic
function of a real Lévy process
Let us consider the sequence 1 = ε0 > ε1 > . . . > 0 such that
limn→+∞ εn = 0 and for n = 1, 2, . . . and let us consider the factor
Z
iuy
exp t
e − 1 − iuy ν (dy )
Z
An
Z
iuy
ν (dy )
− iut
y ν (dy )
,
= exp ν (An ) t
e −1
ν (An )
An
An
where we define An := [−εn−1 , −εn ) ∪ (εn , εn−1 ] . This may be viewed as a
characteristic function of a compensated compound Poisson process
(n)
(n)
Xt
:=
Nt
X
(n)
Yi
Z
−
i=1
y ν (dy ) t,
An
where N (n) is a Poisson process with the intensity ν (An ) and the
ν(dy )
distribution of jumps Y (n) is given by 1An ν(A
.
n)
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
18 / 30
X (n) is a square integrable martingale
Having the characteristic function of a variable X it is possible to calculate
its moments:
∂ k E exp (iuX )
EX k = (−i)k
|u=0
∂u k
(kth moment exist iff the above derivative exists). This follows from the
expansion E exp (iuX ) = 1 + iuEX +
(easily ???) calculate
(n)
EXt
= 0,
E
(iu)2
2
2! EX
(n) 2
Xt
+ . . . . Having this we
Z
=t
y 2 ν (dy ) .
An
From these equalities and the independence of increments of X (n) we infer
that X (n) is a square integrable martingale.
Moreover, taking sequence of independent square integrable martingales
(n)
X
1, 2, . . . (on the same probability space) we get that the series
Pm , n =
(n) converges (as m → ∞, in an appropriate topology) to a square
X
n=1
integrable martingale.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
19 / 30
Characteristic function of the sum
P∞
n=1 X
(n)
Since
∞ Z
X
n=1
e iuy − 1 − iuy ν (dy ) =
An
Z
e iuy − 1 − iuy ν (dy )
[−1,1]\{0}
we get that the characteristic function of
∞
Y
Z
exp t
e
is
iuy
− 1 − iuy ν (dy )
iuy
An
n=1
= exp t
(n)
n=1 Xt
P∞
∞ Z
X
n=1
!
e
− 1 − iuy ν (dy )
An
!
iuy
e − 1 − iuy ν (dy ) .
Z
= exp t
[−1,1]\{0}
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
20 / 30
α-stable processes (α ∈ (0, 2)) - processes with infinite
intensity of small jumps
An α-stable process with the stability parameter α ∈ (0, 2) is a Lévy
process with the following characteristic function
Z
E exp (iuXt ) = exp iaut + t
e
iuy
!
− 1 − iuy 1[−1,1] (y ) ν (dy ) ,
R\{0}
where
ν (dy ) =
1 C1 1(−∞,0) (y ) + C2 1(0,∞) (y ) dy
α+1
|y |
for some non-negative C1 and C2 such that C1 + C2 > 0.
Since ν ([−1, 1]) = +∞ we infer that these processes have infinite
intensity of small jumps.
An α-stable process with the stability parameter α = 2 is continuous - this
is a Brownian motion with drift.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
21 / 30
Strictly α-stable processes
If α ∈ (0, 1) then the characteristic function of an α-stable process Xt
may be represented as
!
Z
iuy
E exp (iuXt ) = exp ibut + t
e − 1 ν (dy ) ,
R\{0}
if α ∈ (1, 2) then the characteristic function of an α-stable process Xt
may be represented as
!
Z
iuy
E exp (iuXt ) = exp ibut + t
e − 1 − iuy ν (dy ) ,
R\{0}
the characteristic function of a 1-stable process Xt may be
represented as
2
E exp (iuXt ) = exp ibut + ictβu ln |u| + ct|u| ,
π
where β ∈ [−1, 1].
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
22 / 30
Strictly α-stable processes (α ∈ (0, 2))- self-similar
processes with infinite intensity of small jumps
If α ∈ (0, 1) ∪ (1, 0) and b = 0 (or α = 1 and β = 0) then the process is
self-similar. More precisely, for any A > 0 one has (verify this using the
characteristic function!)
A−1/α XA·s
=law (Xs )s≥0 .
s≥0
This may be easily verified using the characteristic functions (change of
variable u).
The variable Xt is called (strictly) α-stable variable and it has the
(1)
(n)
following property: if Xt , . . . , Xt are independent copies of Xt then
(1)
Xt
(n)
+ . . . + Xt
=law Xt .
n1/α
This follows from the following easy calculation:
(1)
Xt
(2)
+ Xt
=law
(n)
+ . . . + Xt
(1)
(1)
(1)
(1)
(1)
(1)
+ . . . + Xnt − X(n−1)t = Xnt =law n1/α Xt .
Xt + X2t − Xt
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
23 / 30
The jumps of a Lévy process - Poisson random measures
Let now X be a Lévy process. Let t ∈ [0, +∞) and A be a Borel subset of
R \ {0} . We consider the following random measure N on [0, +∞) × R :
X
N ([0, t] × A) :=
1A (∆Xs ) .
0<s≤t
The measure N ([0, t] × dy ) counts how many jumps of size y occured till
the moment t.
Note that
Z
EN (t, A) =
N (t, A) dP (ω)
is a Borel measure on R \ {0} . We will write µ(·) = EN (1, ·) .
Definition
We say that the Borel set A ⊂ R \ {0} is bounded from below if 0 ∈
/ Ā.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
24 / 30
Poisson random measure related to a Lévy process properties
The introduced random measures have the following properties.
For each t > 0 and ω ∈ Ω, N(t, ·)(ω) is a counting measure on
R \ {0} ;
For each A bounded from below, N(·, A) is a Poisson process with the
intensity µ(A);
The compensated measure
Ñ(t, ·)(ω) := N(t, ·)(ω) − tµ(·)
is a martingale-valued measure, i.e. for any A bounded from below,
Ñ(·, A) is a martingale.
Moreover, if A ∩ B = ∅ then N(·, A) and N(·, B) are independent.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
25 / 30
Poisson integrals
Let now f : R → R be a Borel-measurable function and A ⊂ R be a Borel
set bounded from below. Now for t ≥ 0 we define
Z
X
f (x)N(t, dx)(ω) =
f (x)N(t, {x})(ω)
A
=
x∈A
X
f (∆Xs (ω))1A (∆Xs (ω)).
0<s≤t
We also define for f ∈ L1 (A, µ) the compensated process
Z
Z
Z
f (x)Ñ(t, dx)(ω) =
f (x)N(t, dx)(ω) − t
f (x)µ(dx)
A
A
A
which is a martingale.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
26 / 30
Poisson integrals for f (x) ≡ x
For f (x) ≡ x and A(0) = R \ [−1, 1] we get that
Z
X
xN(t, dx)(ω) =
∆Xs (ω)1A(0) (∆Xs (ω))
A(0)
0<s≤t
is a compound Poisson process with the jumps > 1 occuring at exactly the
same moments at the jumps of X . Moreover, the jumps of this process
have exactly the same size as the jumps of of X . The difference
Z
X−
xN(t, dx)
A(0)
is a Lévy process with jumps smaller or equal 1. It can be shown that such
a process has finite moments of any order (it takes some time for the
process |X | to grow and we control the rate of this growth).
Let us now consider the Lévy process defined as
Z
Z
X̃t := Xt −
xN(t, dx) − tE X1 −
xN(1, dx)
A(0)
A(0)
This is a square-integrable, zero-mean martingale.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
27 / 30
Proceeding to the limit
Next, for n = 1, 2, . . . we consider A(n) = (1/(n + 1), 1/n] and the
sequence of independent, zero-mean, square integrable martingales
Z
(n)
M =
x Ñ(t, dx).
A(n)
For fixed (large) T > 0 and N ∈ N we have that X̃T −
PN
(n)
n=1 MT are independent, thus
N
X
E
(n) 2
MT
= E X̃T
n=1
2
−
N
X
X̃T −
n=1
N
X
!2
(n)
MT
(n)
n=1 MT
PN
and
2
≤ E X̃T .
n=1
P
(n) converges (at least on [0, T ]) to some
From this we infer that N
n=1 M
square integrable martingale M. The difference X̃ − M is a Lévy process
without jumps (it is continuous), thus it is a Brownian motion.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
28 / 30
Some references
A. Kyprianou, (2014) Introductory Lectures on Fluctuations of Lévy
Processes with Applications, 2nd ed. Springer, Berlin.
D. Applebaum, (2004) Lévy Processes and Stochastic Calculus.
Cambridge University Press, Cambridge.
K. Sato, (1999) Lévy Processes and Infinitely Divisible Distributions.
Cambridge University Press, Cambridge.
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
29 / 30
Thank you for your attention!
R. Lochowski (WSE and AIMS)
Lévy processes
Muizenberg, April 2017
30 / 30