Directory UMM :Journals:Journal_of_mathematics:AUA:
ACTA UNIVERSITATIS APULENSIS
No 19/2009
DIRECTIONALLY CONVEX ORDERING IN
MULTIDIMENSIONAL JUMP DIFFUSIONS MODELS
Toufik Guendouzi
Abstract. The purpose of the present article is to use the so-called the propagation of directional convexity property and a general version of the Kolmogorov
equation to obtain ordering results in n-dimensional jump diffusion model. We give
some conditions to prove the comparison inequality for directional convex function,
and if this conditions are true for any class F of the directionally convex order then
we obtain also comparison result between two multidimensional jump diffusion in
the directionally convex order.
2000 Mathematics Subject Classification: 60E15,60G60, 60G44, 39B62, 60F10.
1. Introduction
Stochastic order and comparison inequalities have been used in some areas of
probability and statistics. Such areas include decision theory, Financial economics,
queuing theory and wireless communications, biology, see the monographs of Shaked
and Shanthikumar [10]. In statistics, a stochastic order quantifies the concept of one
random variable being smaller than another. In Finance, many criteria have been
introduced to stochastically conduct comparisons between two option prices with
respect to convex functions under different models [2]. However, in these area, the
comparison in the directional convex order is relatively less used and it is interesting
to study this tool of comparison in jump diffusions models. This paper is devoted
to formulate and study the dcx ordering in n-dimensional case, we presents an jump
diffusion model defined by the n-dimensional integro-differential equation in section
(2) and we reccal the the classical Kolmogorov equation in our diffusion. In section
(3) our main result is the theorem 3.2, we give sufficient conditions for the stochastic
comparison based on directional convexity property for what φ is dcx function by
the directional convexity assuption in the functional ν(x, t, dz). Finally we formulate
this ordering in the case of the predictable processes.
179
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
2.Preliminaries
In this section we give the framework that will be used in this article. That
is, we introduce briefly the directional convex orders notion and we present some
basic definitions on the multi-dimensional jump diffusions model, we also use the
essential property called the propagation of directional convexity property to give
the comparison result in the next section.
Let ≤ denote the componentwise partial order in IRn i.e, x ≤ y if xi ≤ yi for
i = 1, 2, . . . , n where x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ).
The function φ : IRn → IR is said to be directionally convex (dcx) if for any xi ∈ IRn ,
i = 1, . . . , 4, such that x1 ≤ x2 and x3 ≤ x4 ,
φ(x2 ) + φ(x3 ) ≤ φ(x1 ) + φ(x4 ).
If F is some class of functions from IRn to IR then for two real-valued random
vectors X and Y of the same dimension, we say that X is smaller than Y in F order
and we write X ≤F Y if E(φ(X)) ≤ E(φ(Y)), see [10] for more results in this topic.
In the following, denote F the class of dcx functions, then for two random vectors
X and Y in IRn we have
X ≤dcx Y ⇐⇒ Eφ(X) ≤ Eφ(Y),
∀φ ∈ F,
(1)
such that the integral exist for φ. The formula is similar for the convex order.
Consider the following integro-differential equation on the probability space (Ω, A, P)
associated with the n-dimensional diffusion process ξ(t), ξ(0) = ξ0 ∈ IRn defined
by the Ito differential equation
dξ(t) = σ(ξ(t), t)dWt + η(ξ(t), t)dt +
Z
|z|≤1
z µ̂(dt, dz) +
Z
zµ(dt, dz),
(2)
|z|>1
where the drift coefficient
η ∈ IRn and the diffusion coefficient σ = (σij ) is an IRn×m
X
matrix, |σ|2 =
|σij |2 . Here Wt , t ∈ IR+ is an m-dimensional standard Wiener
ij
process, µ(dt, dz) is the jump measure of ξt with the compensator ν(ξ(t− ), t, dz), and
µ̂(dt, dz) = µ(dt, dz) − ν(ξ(t− ), t, dz)dt is the corresponding martingale measure.
The infinitesimal generator of ξ is a partial differential operator defined for any
F ∈ C02 (IRn ) as
180
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
(Lt F )(x) =
+
n
n
X
1 X
∂F
∂2F
ηi (x, t)
βij (x, t)
+
2 i,j=1
∂xi ∂xj i=1
∂xi
Z "
n
X
#
∂F
(x, t)1|z|≤1 ν(x, t, dz),
zi
F (x + z, t) − F (x, t) −
∂xi
i=1
(3)
where C02 (IRn ) is the set of twice continuously differentiable functions, vanishing
at infinity, and β(x, t) = σ(x, t)σ́(x, t), (σ́ is the transpose of σ), 1|z|≤1 is the indicator function of {z ∈ IRn : |z| ≤ 1}.
We will use the following result for formulate our main idea, we can also refer to
Friedman [6] for more detail.
Theorem 1 Assume that the conditions of existence and uniqueness (see [6], p108)
are satisfied for (2) and if
1. There exist Dxα η(x, t) and Dxα σ(t, x) continuous for |α| ≤ 2, with
|Dxα η(x, t)| + |Dxα σ(x, t)| ≤ k0 (1 + |x|a ) ,
where k0 , a are strictly positive constants, and Dxα =
|α| ≤ 2,
∂α
, x ∈ IRn ;
∂xα
2. φ : IRn → IR is a function endowed with continuous derivatives to second
order, with
′
|Dxα φ(x)| ≤ c 1 + |x|a , |α| ≤ 2,
′
where c, a are stictly positive constants;
then, putting ν(ξ(t), t) = E (φ(ξ(x, T, t))), x ∈ IRn and t ∈ [0, T ], we have that
νt , νxi , νxi xj are continuous in (x, t) ∈ IRn × [0, T ] and ν satisfy the parabolic equation
∂ν
(x, t) + (Lt ν)(x) = 0 in IRn × [0, T ]
∂t
(4)
lim ν(x, t) = φ(x)
t↑T
For proof of this theorem we shall need the following lemma.
181
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
Lemma 1 Since ξ(t) is an diffusion in IRn with generator L, then for all
φ ∈ C02 (IRn ) the process
Mt = φ(ξ(t)) −
where ∇x φ =
Z
t
Lφ(ξ(s))ds = φ(x) +
0
∂φ
∂φ
,...,
∂x1
∂xn
tran
t
Z
tran
∇φ
0
(ξ(s))σ(ξ(s))dWs ,
(5)
.
Proof. We have by applying Itô’s formula
n
t
X
∂ν
∂ν
(ξ(s), s)ds +
(ξ(s), s)ds
ηi (ξ(s), s)
∂s
∂x
0
i
i=1 0
n Z t
n Z t
X
∂2ν
∂ν
1X
′
(ξ(s), s)dWs +
(ξ(s), s)ds
(σσ )ij (ξ(s), s)
+
σi (ξ(s), s)
∂xi
2 i,j 0
∂xi ∂xj
i=1 0
ν(ξ(t), t) = ν(ξ(0), 0) +
+
Z tZ "
Z
t
Z
n
X
∂ν
ν(ξ(s) + z, s) − ν(ξ(s), s) −
0
∂xi
i=1
n Z t
X
t
#
(ξ(s), s)zi µ(ds, dz)
∂ν
∂ν
ηi (ξ(s), s)
(ξ(s), s)ds +
(ξ(s), s)ds
∂xi
0 ∂s
i=1 0
n Z t
n Z t
X
∂ν
1X
∂2ν
′
+
σi (ξ(s), s)
(ξ(s), s)dWs +
(ξ(s), s)ds
(σσ )ij (ξ(s), s)
∂xi
2 i,j 0
∂xi ∂xj
i=1 0
= ν(ξ(0), 0) +
+
Z tZ "
0
+
Z
ν(ξ(s) + z, s) − ν(ξ(s), s) −
IR
µ(ds, dz) − 1|z|≤1 ν(ξ(s), s, dz)ds
Z tZ
0
|z|≤1
"
n
X
∂ν
i=1
ν(ξ(s) + z, s) − ν(ξ(s), s) −
∂xi
#
(ξ(s), s)zi ·
n
X
∂ν
i=1
∂xi
#
(ξ(s), s)zi ν(ξ(s), s, dz)ds.
Finally, since ν(ξ(t), t) is martingale by construction, from Lemma (2.2) we have
n
t
X
∂ν
∂ν
(ξ(s), s)ds +
(ξ(s), s)ds
ηi (ξ(s), s)
0 = ν(ξ(0), 0) +
∂xi
0 ∂s
i=1 0
n Z t
1X
∂2ν
′
+
(σσ )ij (ξ(s), s)
(ξ(s), s)ds
2 i,j 0
∂xi ∂xj
Z
+
Z tZ
0
|z|≤1
"
t
Z
ν(ξ(s) + z, s) − ν(ξ(s), s) −
n
X
∂ν
i=1
n
∂xi
#
(ξ(s), s)zi ν(ξ(s), s, dz)ds.
✷
Let now ξ ∗ be an (σ ∗ , η)-jump diffusion in IR , with the same drift coefficient as
ξ(t), defined by the equation
dξt∗ = σt∗ dWt + ηt dt +
Z
z µ̂(dt, dz) +
|z|≤1
182
Z
|z|>1
zµ(dt, dz),
(6)
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
here the compensator of the jump measure is denoted by νt (dz)dt and we write
µ̂(dt, dz) = µ(dt, dz) − νt (dz)dt. For all F ∈ C02 (IRn ), the generator of ξ ∗ is given by
(L∗t F )(x) =
+
n
n
X
1 X
∂F
∂2F
∗
ηi (x, t)
βij
(x, t)
+
2 i,j=1
∂xi ∂xj i=1
∂xi
Z "
n
X
#
∂F
F (x + z, t) − F (x, t) −
(x, t)1|z|≤1 νt (dz).
zi
∂xi
i=1
(7)
Lemma 2 ([7]) Let F be twice continuously differentiable, then F is directionally
2 F ≥ 0,
convex (dcx) if and only if ∂i,j
for all i, j ≤ n and all x ∈ IRn .
x
Finally, the main idea in this section is the following condition called the PDC
condition: We will assuming that the function ν(x, t) defined in the theorem (2.1)
is directionally convex on IRn for all t ∈ [0, T ] when the function φ is directionally
convex. we employ this idea in the next section to obtain an comparison result in
the (dcx) oredr sense.
3.Main Result
In this section we are concerned with the directionally convex ordering for the
functional ν(., t). We show that under some conditions the comparison inequality
for the dcx function φ(x) can be given if the PDC hold for ξ, witch implies the
ordering result ξt∗ ≤dcx ξ(t).
We can now state a key lemma needed for deriving the main result of this section.
Lemma 3 Let ξ ∗ be a process defined as in (6) such that (L∗t ν)(ξt∗ ) ≤ (Lt ν)(ξt∗ ),
then the process ν(ξt∗ , t) is a supermartingale and satisfies the comparison inequality
∗
) At ≤ ν(ξt∗ , t),
E φ(ξT
t ∈ [0, T].
(8)
Proof. The process ν(ξt∗ , t) can be decomposed as ν(ξt∗ , t) = Mt − At such that
n Z
X
t
∂ν ∗
(ξ , s)dWs
∂xi s
0
i=1
#
Z tZ "
n
X
∂ν ∗
∗
∗
ν(ξs + z, s) − ν(ξs , s) −
+
(ξs , s)zi · µ(ds, dz) − 1|z|≤1 νs (dz)
∂xi
0 IR
i=1
ν(ξt∗ , t) = ν(ξ0∗ , 0) +
−
Z
0
t
σi (ξs∗ , s)
((Ls ν)(ξs∗ ) − (L∗s ν)(ξs∗ ))ds ,
183
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
Z
Where Mt is a martingale, At = −
0
t
((Ls ν)(ξs∗ ) − (L∗s ν)(ξs∗ ))ds is increasing and
adapted process with E(At ) < ∞ for all t ≥ 0, hence
ν(ξ t∗ , t)
supermartingale
is a
∗
∗ , t) A
by the Doop decomposition which implies that E φ(ξT ) At = E ν(ξT
t ≤
∗
ν(ξt , t).
✷
Theorem 2 (The dcx order) Let the PDC hold for ξ and assume that
(βt∗ )ij ≤ β ij (ξt∗ , t),
and for almost all t ≥ 0 we have either :
1. νt (dz) ≤dcx ν(ξt∗ , t, dz),
or
2. νt (dz) and ν(ξt∗ , t, dz) are supported by IRn+ and νt (dz) ≤idcx ν(ξt∗ , t, dz)
Then
h
i
∗
) At ≤ E φ ξ ∗ (x, T, t) ,
E φ(ξT
t ∈ [0, T],
x ∈ IRn .
Proof. Using Lemma (3.1). We have by (3) and (7),
(L∗t ν)(ξt∗ ) − (Lt ν)(ξt∗ )
n
∂2ν
1 X
(ξ ∗ , t)
(βt∗ )ij − β ij (ξt∗ , t)
=
2 i,j=1
∂xi ∂xj t
Z "
n
X
#
n
X
#
∂ν ∗
zi
ν(ξs∗ + z, s) − ν(ξs∗ , s) −
+
(ξ , s)1|z|≤1 νs (dz)
∂xi s
i=1
"
#
Z
n
X
∂ν ∗
∗
∗
−
zi
ν(ξs + z, s) − ν(ξs , s) −
(ξs , s)1|z|≤1 ν(ξs∗ , s, dz)
∂x
i
i=1
n
∂2ν
1 X
(ξ ∗ , t)
(βt∗ )ij − β ij (ξt∗ , t)
=
2 i,j=1
∂xi ∂xj t
+
=
+
Z "
∂ν ∗
+ z, s) −
−
zi
(ξs , s)1|z|≤1 νs (dz) − ν(ξs∗ , s, dz)
∂xi
i=1
n
2
X
∂ ν
1
(βt∗ )ij − β ij (ξt∗ , t)
(ξ ∗ , t)
2 i,j=1
∂xi ∂xj t
Z
ν(ξs∗
ν(ξs∗ , s)
ψs (ξs∗ , z)1|z|≤1 νs (dz) − ν(ξs∗ , s, dz) ,
184
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
where ψt (x, z) = ν(ξs∗ + z, s) − ν(ξs∗ , s) −
n
X
zi
i=1
2
∂i,j νx ≥
As ν(., t) is directionally convex,
and if β ij (ξt∗ , t) − (βt∗ )ij ≥ 0 it follows that
∂ν ∗
(ξ , s),
∂xi s
z, x ∈ IRn .
for all i, j ≤ n and all x ∈ IRn ,
0,
∗
(L∗t ν)(ξt∗ ) ≤ (L
Z t ν)(ξt )
+
ψs (ξs∗ , z)1|z|≤1 νs (dz) − ν(ξs∗ , s, dz) .
2 ψ ≥ 0,
Since ∂i,j
for all i, j ≤ n, the function ψt (x, z) for all fixed x ∈ IRn is also
z
directionally convex on IRn and by the first condition in the theorem
Z
ψs (ξs∗ , z)1|z|≤1 νs (dz) − ν(ξs∗ , s, dz) is non-positive term. Finally the function
ψt (x, z) is increasing in z ∈ IRn+ and when νt (dz), ν(ξt∗ , t, dz) are supported by IRn+
it follows that (L∗t ν)(ξt∗ ) ≤ (Lt ν)(ξt∗ ). It remains now to use the lemma (3.1) to
obtain the result.
✷
Assume now that the process ξt∗ , t ≥ 0 has the following representation
ξt∗ = ξ0∗ +
Z
0
t
ηs ds +
Z tD
0
E
σs∗ , dWs +
Z tD
0
E
Hs∗ , dNs∗ .
(9)
Here dN ∗ = dZ ∗ − λ∗t dt is a jump martingale and Zt∗ is a point process with compensator λ∗t , t ≥ 0, σt∗ and Ht∗ are respectively IRn×m and IRn -valued predictable
processes. Let Z(t) and Zt∗ two independent point processes in IRn with compensators respectively
m
X
λj (x, t)δH ij (x,t) and
m
X
(λ∗t )j δ(Ht∗ )ij ; i ≤ n, j ≤ m, where δ(x,t)
j=1
j=1
denotes the Dirac measure at (x, t) ∈ IRn × IR+ .
Theorem 3 (The dcx order) Assume that ξ(t) and ξt∗ are two diffusions with
same drift and have same representation as in (9), H(x, t) and Ht∗ are IRn×m -valued
integrable predictable processes. if
1. (σ ∗ σ́ ∗ )t ≤ (σσ́)(x, t),
2.
m
X
(λ∗t )j δ(Ht∗ )i,j
≤dcx
λj (x, t)δH i,j (x,t) ;
i ≤ n, j ≤ m, t ≥ 0
j=1
j=1
then
m
X
h
i
∗
E φ(ξT
) At ≤ E φ ξ ∗ (x, T, t) ,
185
t ∈ [0, T],
x ∈ IRn ,
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
for all directionally convex function φ : IRn → IR and ξ(t) satisfies the PDC
condition defined in section (2).
Proof. Using theorem (3.2), the characteristic measures νt (dz) and ν(x, t, dz)
have respectively the form
m
X
(λ∗t )j δ(Ht∗ )i,j (dz) and
j=1
m
X
λj (x, t)δH i,j (x,t) (dz); i ≤ n, j ≤
j=1
m
X
m, t ≥ 0. Thus the condition (1) of (3.2) implies that
(λ∗t )j (Ht∗ )i,j ≤
n, j ≤ m, in this case we have
(λ∗t )j ≤
λj (x, t), and the concentration inequal-
j=1
j=1
ity hold.
m
X
λj (x, t)H i,j (x, t); i ≤
j=1
j=1
m
X
m
X
✷
Theorem 4 Let H(x, t, z) and H ∗ (z) two integrable predictable IRn -valued processes
and assume that (σ ∗ σ́ ∗ )t ≤ (σσ́)(x, t) for all t ≥ 0. then the inequality
h
i
∗
E φ(ξT
) At ≤ E φ ξ ∗ (x, T, t) ,
t ∈ [0, T],
x ∈ IRn
holds for all directionally convex function φ : IRn → IR provided the PDC condition
defined in section (2) and one of the following conditions is satisfied
1.
m
X
(λ∗t )j δ(Ht∗ )i,j ≤dcx
m
X
(λ∗t )j δ(Ht∗ )i,j ≤idcx
λj (x, t)δH i,j (x,t) ,
j=1
j=1
2.
m
X
m
X
λj (x, t)δH i,j (x,t) ;
i ≤ n, j ≤ m, t ≥ 0
j=1
j=1
Proof. Using theorem (3.2), the characteristic measures νt (dz) and ν(x, t, dz)
have respectively the form 1
IRn \{0}
(z)
m
X
(λ∗t )j δ(Ht∗ )i,j ◦ (Ht∗ )−1 (dz) and
j=1
1IRn \{0} (z)
m
X
λj (x, t)δH i,j (x,t) ◦ (H(x, t))−1 (dz);
i ≤ n, j ≤ m, t ≥ 0.
j=1
νt (dz) ≤dcx ν(x, t, dz) if and only if
m
X
(λ∗t )j ≤
j=1
m
X
λj (x, t) and by the second condi-
j=1
tion of (3.2) if the characteristic measures are supported by IRn+ and H is increasing
in z, Ht∗ ≤ H(x, t), then we obtain also the concentration inequality given by the
theorem.
✷
186
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
References
[1] B. Bassan, M. Scarsini, Convex orderings for stochastic processes, Comment.
Math. Univ. Carolin., 32, (1991), 115-118.
[2] J. Bergenthum, L. Rüschendorf, Comparison of option prices in semimartingale models, Finance and Stochastics., 10, (2006), 229-249.
[3] J. Bergenthum, L. Rüschendorf, Comparison of semimartingale and Lévy
processes, Ann. Probab., 35(1), (2007).
[4] J. Bergenthum, L. Rüschendorf, Convex ordering criteria for Lévy processes,
ADAC., 1, (2007), 143-173.
[5] E. Ekström, J. Tysk, Convexity preserving jump-diffusion models for option
pricing, J. Math. Anal. Appl., 330(1), (2007), 715-728.
[6] A. Friedman, Stochastic differential equations and applications, Vol. 1., Academic press, New York , 1975.
[7] S. Janson, J. Tysk, Preserving Of convexity of solutions to parabolic equations,
J. Diff Eq., 206, (2004), 182-226.
[8] C. Martini, Propagation of convexity by Markovian and martingalian semigroups, Potential Anal., 10(2), (1999), 133-175.
[9] Ph. Protter, Stochastic integration and differential equations. A new approach, Springer-Verlag, Berlin, 1990.
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Toufik Guendouzi
Laboratory of Mathematics
Djillali Liabes University
PO.Box 89, 22000 Sidi Bel Abbes, Algeria.
email:[email protected]
187
No 19/2009
DIRECTIONALLY CONVEX ORDERING IN
MULTIDIMENSIONAL JUMP DIFFUSIONS MODELS
Toufik Guendouzi
Abstract. The purpose of the present article is to use the so-called the propagation of directional convexity property and a general version of the Kolmogorov
equation to obtain ordering results in n-dimensional jump diffusion model. We give
some conditions to prove the comparison inequality for directional convex function,
and if this conditions are true for any class F of the directionally convex order then
we obtain also comparison result between two multidimensional jump diffusion in
the directionally convex order.
2000 Mathematics Subject Classification: 60E15,60G60, 60G44, 39B62, 60F10.
1. Introduction
Stochastic order and comparison inequalities have been used in some areas of
probability and statistics. Such areas include decision theory, Financial economics,
queuing theory and wireless communications, biology, see the monographs of Shaked
and Shanthikumar [10]. In statistics, a stochastic order quantifies the concept of one
random variable being smaller than another. In Finance, many criteria have been
introduced to stochastically conduct comparisons between two option prices with
respect to convex functions under different models [2]. However, in these area, the
comparison in the directional convex order is relatively less used and it is interesting
to study this tool of comparison in jump diffusions models. This paper is devoted
to formulate and study the dcx ordering in n-dimensional case, we presents an jump
diffusion model defined by the n-dimensional integro-differential equation in section
(2) and we reccal the the classical Kolmogorov equation in our diffusion. In section
(3) our main result is the theorem 3.2, we give sufficient conditions for the stochastic
comparison based on directional convexity property for what φ is dcx function by
the directional convexity assuption in the functional ν(x, t, dz). Finally we formulate
this ordering in the case of the predictable processes.
179
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
2.Preliminaries
In this section we give the framework that will be used in this article. That
is, we introduce briefly the directional convex orders notion and we present some
basic definitions on the multi-dimensional jump diffusions model, we also use the
essential property called the propagation of directional convexity property to give
the comparison result in the next section.
Let ≤ denote the componentwise partial order in IRn i.e, x ≤ y if xi ≤ yi for
i = 1, 2, . . . , n where x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ).
The function φ : IRn → IR is said to be directionally convex (dcx) if for any xi ∈ IRn ,
i = 1, . . . , 4, such that x1 ≤ x2 and x3 ≤ x4 ,
φ(x2 ) + φ(x3 ) ≤ φ(x1 ) + φ(x4 ).
If F is some class of functions from IRn to IR then for two real-valued random
vectors X and Y of the same dimension, we say that X is smaller than Y in F order
and we write X ≤F Y if E(φ(X)) ≤ E(φ(Y)), see [10] for more results in this topic.
In the following, denote F the class of dcx functions, then for two random vectors
X and Y in IRn we have
X ≤dcx Y ⇐⇒ Eφ(X) ≤ Eφ(Y),
∀φ ∈ F,
(1)
such that the integral exist for φ. The formula is similar for the convex order.
Consider the following integro-differential equation on the probability space (Ω, A, P)
associated with the n-dimensional diffusion process ξ(t), ξ(0) = ξ0 ∈ IRn defined
by the Ito differential equation
dξ(t) = σ(ξ(t), t)dWt + η(ξ(t), t)dt +
Z
|z|≤1
z µ̂(dt, dz) +
Z
zµ(dt, dz),
(2)
|z|>1
where the drift coefficient
η ∈ IRn and the diffusion coefficient σ = (σij ) is an IRn×m
X
matrix, |σ|2 =
|σij |2 . Here Wt , t ∈ IR+ is an m-dimensional standard Wiener
ij
process, µ(dt, dz) is the jump measure of ξt with the compensator ν(ξ(t− ), t, dz), and
µ̂(dt, dz) = µ(dt, dz) − ν(ξ(t− ), t, dz)dt is the corresponding martingale measure.
The infinitesimal generator of ξ is a partial differential operator defined for any
F ∈ C02 (IRn ) as
180
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
(Lt F )(x) =
+
n
n
X
1 X
∂F
∂2F
ηi (x, t)
βij (x, t)
+
2 i,j=1
∂xi ∂xj i=1
∂xi
Z "
n
X
#
∂F
(x, t)1|z|≤1 ν(x, t, dz),
zi
F (x + z, t) − F (x, t) −
∂xi
i=1
(3)
where C02 (IRn ) is the set of twice continuously differentiable functions, vanishing
at infinity, and β(x, t) = σ(x, t)σ́(x, t), (σ́ is the transpose of σ), 1|z|≤1 is the indicator function of {z ∈ IRn : |z| ≤ 1}.
We will use the following result for formulate our main idea, we can also refer to
Friedman [6] for more detail.
Theorem 1 Assume that the conditions of existence and uniqueness (see [6], p108)
are satisfied for (2) and if
1. There exist Dxα η(x, t) and Dxα σ(t, x) continuous for |α| ≤ 2, with
|Dxα η(x, t)| + |Dxα σ(x, t)| ≤ k0 (1 + |x|a ) ,
where k0 , a are strictly positive constants, and Dxα =
|α| ≤ 2,
∂α
, x ∈ IRn ;
∂xα
2. φ : IRn → IR is a function endowed with continuous derivatives to second
order, with
′
|Dxα φ(x)| ≤ c 1 + |x|a , |α| ≤ 2,
′
where c, a are stictly positive constants;
then, putting ν(ξ(t), t) = E (φ(ξ(x, T, t))), x ∈ IRn and t ∈ [0, T ], we have that
νt , νxi , νxi xj are continuous in (x, t) ∈ IRn × [0, T ] and ν satisfy the parabolic equation
∂ν
(x, t) + (Lt ν)(x) = 0 in IRn × [0, T ]
∂t
(4)
lim ν(x, t) = φ(x)
t↑T
For proof of this theorem we shall need the following lemma.
181
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
Lemma 1 Since ξ(t) is an diffusion in IRn with generator L, then for all
φ ∈ C02 (IRn ) the process
Mt = φ(ξ(t)) −
where ∇x φ =
Z
t
Lφ(ξ(s))ds = φ(x) +
0
∂φ
∂φ
,...,
∂x1
∂xn
tran
t
Z
tran
∇φ
0
(ξ(s))σ(ξ(s))dWs ,
(5)
.
Proof. We have by applying Itô’s formula
n
t
X
∂ν
∂ν
(ξ(s), s)ds +
(ξ(s), s)ds
ηi (ξ(s), s)
∂s
∂x
0
i
i=1 0
n Z t
n Z t
X
∂2ν
∂ν
1X
′
(ξ(s), s)dWs +
(ξ(s), s)ds
(σσ )ij (ξ(s), s)
+
σi (ξ(s), s)
∂xi
2 i,j 0
∂xi ∂xj
i=1 0
ν(ξ(t), t) = ν(ξ(0), 0) +
+
Z tZ "
Z
t
Z
n
X
∂ν
ν(ξ(s) + z, s) − ν(ξ(s), s) −
0
∂xi
i=1
n Z t
X
t
#
(ξ(s), s)zi µ(ds, dz)
∂ν
∂ν
ηi (ξ(s), s)
(ξ(s), s)ds +
(ξ(s), s)ds
∂xi
0 ∂s
i=1 0
n Z t
n Z t
X
∂ν
1X
∂2ν
′
+
σi (ξ(s), s)
(ξ(s), s)dWs +
(ξ(s), s)ds
(σσ )ij (ξ(s), s)
∂xi
2 i,j 0
∂xi ∂xj
i=1 0
= ν(ξ(0), 0) +
+
Z tZ "
0
+
Z
ν(ξ(s) + z, s) − ν(ξ(s), s) −
IR
µ(ds, dz) − 1|z|≤1 ν(ξ(s), s, dz)ds
Z tZ
0
|z|≤1
"
n
X
∂ν
i=1
ν(ξ(s) + z, s) − ν(ξ(s), s) −
∂xi
#
(ξ(s), s)zi ·
n
X
∂ν
i=1
∂xi
#
(ξ(s), s)zi ν(ξ(s), s, dz)ds.
Finally, since ν(ξ(t), t) is martingale by construction, from Lemma (2.2) we have
n
t
X
∂ν
∂ν
(ξ(s), s)ds +
(ξ(s), s)ds
ηi (ξ(s), s)
0 = ν(ξ(0), 0) +
∂xi
0 ∂s
i=1 0
n Z t
1X
∂2ν
′
+
(σσ )ij (ξ(s), s)
(ξ(s), s)ds
2 i,j 0
∂xi ∂xj
Z
+
Z tZ
0
|z|≤1
"
t
Z
ν(ξ(s) + z, s) − ν(ξ(s), s) −
n
X
∂ν
i=1
n
∂xi
#
(ξ(s), s)zi ν(ξ(s), s, dz)ds.
✷
Let now ξ ∗ be an (σ ∗ , η)-jump diffusion in IR , with the same drift coefficient as
ξ(t), defined by the equation
dξt∗ = σt∗ dWt + ηt dt +
Z
z µ̂(dt, dz) +
|z|≤1
182
Z
|z|>1
zµ(dt, dz),
(6)
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
here the compensator of the jump measure is denoted by νt (dz)dt and we write
µ̂(dt, dz) = µ(dt, dz) − νt (dz)dt. For all F ∈ C02 (IRn ), the generator of ξ ∗ is given by
(L∗t F )(x) =
+
n
n
X
1 X
∂F
∂2F
∗
ηi (x, t)
βij
(x, t)
+
2 i,j=1
∂xi ∂xj i=1
∂xi
Z "
n
X
#
∂F
F (x + z, t) − F (x, t) −
(x, t)1|z|≤1 νt (dz).
zi
∂xi
i=1
(7)
Lemma 2 ([7]) Let F be twice continuously differentiable, then F is directionally
2 F ≥ 0,
convex (dcx) if and only if ∂i,j
for all i, j ≤ n and all x ∈ IRn .
x
Finally, the main idea in this section is the following condition called the PDC
condition: We will assuming that the function ν(x, t) defined in the theorem (2.1)
is directionally convex on IRn for all t ∈ [0, T ] when the function φ is directionally
convex. we employ this idea in the next section to obtain an comparison result in
the (dcx) oredr sense.
3.Main Result
In this section we are concerned with the directionally convex ordering for the
functional ν(., t). We show that under some conditions the comparison inequality
for the dcx function φ(x) can be given if the PDC hold for ξ, witch implies the
ordering result ξt∗ ≤dcx ξ(t).
We can now state a key lemma needed for deriving the main result of this section.
Lemma 3 Let ξ ∗ be a process defined as in (6) such that (L∗t ν)(ξt∗ ) ≤ (Lt ν)(ξt∗ ),
then the process ν(ξt∗ , t) is a supermartingale and satisfies the comparison inequality
∗
) At ≤ ν(ξt∗ , t),
E φ(ξT
t ∈ [0, T].
(8)
Proof. The process ν(ξt∗ , t) can be decomposed as ν(ξt∗ , t) = Mt − At such that
n Z
X
t
∂ν ∗
(ξ , s)dWs
∂xi s
0
i=1
#
Z tZ "
n
X
∂ν ∗
∗
∗
ν(ξs + z, s) − ν(ξs , s) −
+
(ξs , s)zi · µ(ds, dz) − 1|z|≤1 νs (dz)
∂xi
0 IR
i=1
ν(ξt∗ , t) = ν(ξ0∗ , 0) +
−
Z
0
t
σi (ξs∗ , s)
((Ls ν)(ξs∗ ) − (L∗s ν)(ξs∗ ))ds ,
183
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
Z
Where Mt is a martingale, At = −
0
t
((Ls ν)(ξs∗ ) − (L∗s ν)(ξs∗ ))ds is increasing and
adapted process with E(At ) < ∞ for all t ≥ 0, hence
ν(ξ t∗ , t)
supermartingale
is a
∗
∗ , t) A
by the Doop decomposition which implies that E φ(ξT ) At = E ν(ξT
t ≤
∗
ν(ξt , t).
✷
Theorem 2 (The dcx order) Let the PDC hold for ξ and assume that
(βt∗ )ij ≤ β ij (ξt∗ , t),
and for almost all t ≥ 0 we have either :
1. νt (dz) ≤dcx ν(ξt∗ , t, dz),
or
2. νt (dz) and ν(ξt∗ , t, dz) are supported by IRn+ and νt (dz) ≤idcx ν(ξt∗ , t, dz)
Then
h
i
∗
) At ≤ E φ ξ ∗ (x, T, t) ,
E φ(ξT
t ∈ [0, T],
x ∈ IRn .
Proof. Using Lemma (3.1). We have by (3) and (7),
(L∗t ν)(ξt∗ ) − (Lt ν)(ξt∗ )
n
∂2ν
1 X
(ξ ∗ , t)
(βt∗ )ij − β ij (ξt∗ , t)
=
2 i,j=1
∂xi ∂xj t
Z "
n
X
#
n
X
#
∂ν ∗
zi
ν(ξs∗ + z, s) − ν(ξs∗ , s) −
+
(ξ , s)1|z|≤1 νs (dz)
∂xi s
i=1
"
#
Z
n
X
∂ν ∗
∗
∗
−
zi
ν(ξs + z, s) − ν(ξs , s) −
(ξs , s)1|z|≤1 ν(ξs∗ , s, dz)
∂x
i
i=1
n
∂2ν
1 X
(ξ ∗ , t)
(βt∗ )ij − β ij (ξt∗ , t)
=
2 i,j=1
∂xi ∂xj t
+
=
+
Z "
∂ν ∗
+ z, s) −
−
zi
(ξs , s)1|z|≤1 νs (dz) − ν(ξs∗ , s, dz)
∂xi
i=1
n
2
X
∂ ν
1
(βt∗ )ij − β ij (ξt∗ , t)
(ξ ∗ , t)
2 i,j=1
∂xi ∂xj t
Z
ν(ξs∗
ν(ξs∗ , s)
ψs (ξs∗ , z)1|z|≤1 νs (dz) − ν(ξs∗ , s, dz) ,
184
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
where ψt (x, z) = ν(ξs∗ + z, s) − ν(ξs∗ , s) −
n
X
zi
i=1
2
∂i,j νx ≥
As ν(., t) is directionally convex,
and if β ij (ξt∗ , t) − (βt∗ )ij ≥ 0 it follows that
∂ν ∗
(ξ , s),
∂xi s
z, x ∈ IRn .
for all i, j ≤ n and all x ∈ IRn ,
0,
∗
(L∗t ν)(ξt∗ ) ≤ (L
Z t ν)(ξt )
+
ψs (ξs∗ , z)1|z|≤1 νs (dz) − ν(ξs∗ , s, dz) .
2 ψ ≥ 0,
Since ∂i,j
for all i, j ≤ n, the function ψt (x, z) for all fixed x ∈ IRn is also
z
directionally convex on IRn and by the first condition in the theorem
Z
ψs (ξs∗ , z)1|z|≤1 νs (dz) − ν(ξs∗ , s, dz) is non-positive term. Finally the function
ψt (x, z) is increasing in z ∈ IRn+ and when νt (dz), ν(ξt∗ , t, dz) are supported by IRn+
it follows that (L∗t ν)(ξt∗ ) ≤ (Lt ν)(ξt∗ ). It remains now to use the lemma (3.1) to
obtain the result.
✷
Assume now that the process ξt∗ , t ≥ 0 has the following representation
ξt∗ = ξ0∗ +
Z
0
t
ηs ds +
Z tD
0
E
σs∗ , dWs +
Z tD
0
E
Hs∗ , dNs∗ .
(9)
Here dN ∗ = dZ ∗ − λ∗t dt is a jump martingale and Zt∗ is a point process with compensator λ∗t , t ≥ 0, σt∗ and Ht∗ are respectively IRn×m and IRn -valued predictable
processes. Let Z(t) and Zt∗ two independent point processes in IRn with compensators respectively
m
X
λj (x, t)δH ij (x,t) and
m
X
(λ∗t )j δ(Ht∗ )ij ; i ≤ n, j ≤ m, where δ(x,t)
j=1
j=1
denotes the Dirac measure at (x, t) ∈ IRn × IR+ .
Theorem 3 (The dcx order) Assume that ξ(t) and ξt∗ are two diffusions with
same drift and have same representation as in (9), H(x, t) and Ht∗ are IRn×m -valued
integrable predictable processes. if
1. (σ ∗ σ́ ∗ )t ≤ (σσ́)(x, t),
2.
m
X
(λ∗t )j δ(Ht∗ )i,j
≤dcx
λj (x, t)δH i,j (x,t) ;
i ≤ n, j ≤ m, t ≥ 0
j=1
j=1
then
m
X
h
i
∗
E φ(ξT
) At ≤ E φ ξ ∗ (x, T, t) ,
185
t ∈ [0, T],
x ∈ IRn ,
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
for all directionally convex function φ : IRn → IR and ξ(t) satisfies the PDC
condition defined in section (2).
Proof. Using theorem (3.2), the characteristic measures νt (dz) and ν(x, t, dz)
have respectively the form
m
X
(λ∗t )j δ(Ht∗ )i,j (dz) and
j=1
m
X
λj (x, t)δH i,j (x,t) (dz); i ≤ n, j ≤
j=1
m
X
m, t ≥ 0. Thus the condition (1) of (3.2) implies that
(λ∗t )j (Ht∗ )i,j ≤
n, j ≤ m, in this case we have
(λ∗t )j ≤
λj (x, t), and the concentration inequal-
j=1
j=1
ity hold.
m
X
λj (x, t)H i,j (x, t); i ≤
j=1
j=1
m
X
m
X
✷
Theorem 4 Let H(x, t, z) and H ∗ (z) two integrable predictable IRn -valued processes
and assume that (σ ∗ σ́ ∗ )t ≤ (σσ́)(x, t) for all t ≥ 0. then the inequality
h
i
∗
E φ(ξT
) At ≤ E φ ξ ∗ (x, T, t) ,
t ∈ [0, T],
x ∈ IRn
holds for all directionally convex function φ : IRn → IR provided the PDC condition
defined in section (2) and one of the following conditions is satisfied
1.
m
X
(λ∗t )j δ(Ht∗ )i,j ≤dcx
m
X
(λ∗t )j δ(Ht∗ )i,j ≤idcx
λj (x, t)δH i,j (x,t) ,
j=1
j=1
2.
m
X
m
X
λj (x, t)δH i,j (x,t) ;
i ≤ n, j ≤ m, t ≥ 0
j=1
j=1
Proof. Using theorem (3.2), the characteristic measures νt (dz) and ν(x, t, dz)
have respectively the form 1
IRn \{0}
(z)
m
X
(λ∗t )j δ(Ht∗ )i,j ◦ (Ht∗ )−1 (dz) and
j=1
1IRn \{0} (z)
m
X
λj (x, t)δH i,j (x,t) ◦ (H(x, t))−1 (dz);
i ≤ n, j ≤ m, t ≥ 0.
j=1
νt (dz) ≤dcx ν(x, t, dz) if and only if
m
X
(λ∗t )j ≤
j=1
m
X
λj (x, t) and by the second condi-
j=1
tion of (3.2) if the characteristic measures are supported by IRn+ and H is increasing
in z, Ht∗ ≤ H(x, t), then we obtain also the concentration inequality given by the
theorem.
✷
186
T. Guendouzi - Directionally convex ordering in multidimensional jump ...
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Toufik Guendouzi
Laboratory of Mathematics
Djillali Liabes University
PO.Box 89, 22000 Sidi Bel Abbes, Algeria.
email:[email protected]
187