DIRECTIONALLY CONVEX ORDERING IN MULTIDIMENSIONAL JUMP DIFFUSIONS MODELS
Toufik Guendouzi
Abstract. The purpose of the present article is to use the so-called the prop-agation of directional convexity property and a general version of the Kolmogorov equation to obtain ordering results inn-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 classF of the directionally convex order then we obtain also comparison result between two multidimensional jump diffusion in the directionally convex order.
2000Mathematics Subject Classification: 60E15,60G60, 60G44, 39B62, 60F10. 1. Introduction
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, . . . , nwherex= (x1,x2, . . . ,xn) and y= (y1,y2, . . . ,yn).
The functionφ:IRn→IRis said to be directionally convex (dcx) if for anyxi∈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 toIR then for two real-valued random vectorsXandYof the same dimension, we say thatXis smaller thanYinF order and we write X≤F YifE(φ(X))≤E(φ(Y)), see [10] for more results in this topic. In the following, denoteF the class ofdcxfunctions, then for two random vectors X and Y inIRn 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
zµˆ(dt, dz) +
Z
|z|>1
zµ(dt, dz), (2) where the drift coefficientη∈IRnand the diffusion coefficientσ= (σij) is anIRn×m matrix, |σ|2 = X
ij
|σij|2. Here Wt, t ∈ IR+ is an m-dimensional standard Wiener
process,µ(dt, dz) is the jump measure ofξtwith the compensatorν(ξ(t−), t, dz), and
ˆ
µ(dt, dz) =µ(dt, dz)−ν(ξ(t−), t, dz)dt is the corresponding martingale measure. The infinitesimal generator ofξ is a partial differential operator defined for any
(LtF)(x) =
1 2
n
X
i,j=1
βij(x, t) ∂2F ∂xi∂xj
+
n
X
i=1
ηi(x, t) ∂F ∂xi
+
Z "
F(x+z, t)−F(x, t)−
n
X
i=1 zi
∂F ∂xi
(x, t)1|z|≤1
#
ν(x, t, dz),
(3)
whereC02(IRn) is the set of twice continuously differentiable functions, vanishing at infinity, and β(x, t) = σ(x, t)´σ(x, t), (´σis the transpose of σ), 1|z|≤1 is the
indi-cator 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 Dα
xη(x, t) and Dαxσ(t, x) continuous for |α| ≤2, with
|Dxαη(x, t)|+|Dxασ(x, t)| ≤k0(1 +|x|a), |α| ≤2,
where k0, a are strictly positive constants, and Dα x =
∂α
∂xα, x∈IR n
;
2. φ : IRn → IR is a function endowed with continuous derivatives to second order, with
|Dαxφ(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, νxixj are continuous in(x, t)∈IRn×[0, T]and ν satisfy the parabolic
equa-tion
∂ν
∂t(x, t) + (Ltν)(x) = 0 inIR
n×[0, T]
lim
t↑T ν(x, t) = φ(x)
(4)
Lemma 1 Sinceξ(t) is an diffusion inIRn with generator L, then for all
φ∈C02(IRn) the process
Mt=φ(ξ(t))−
Z t
0
Lφ(ξ(s))ds
=φ(x) +
Z t
0
∇φtran(ξ(s))σ(ξ(s))dWs
, (5)
where ∇xφ=
∂φ ∂x1 , . . . , ∂φ ∂xn tran .
Proof. We have by applying Itˆo’s formula
ν(ξ(t), t) = ν(ξ(0),0) +
Z t
0 ∂ν
∂s(ξ(s), s)ds+ n
X
i=1
Z t
0
ηi(ξ(s), s) ∂ν ∂xi
(ξ(s), s)ds
+ n X i=1 Z t 0
σi(ξ(s), s) ∂ν ∂xi
(ξ(s), s)dWs+
1 2 n X i,j Z t 0 (
σσ′)ij(ξ(s), s) ∂2ν ∂xi∂xj
(ξ(s), s)ds
+
Z t
0
Z "
ν(ξ(s) +z, s)−ν(ξ(s), s)−
n
X
i=1 ∂ν ∂xi
(ξ(s), s)zi
#
µ(ds, dz) = ν(ξ(0),0) +
Z t
0 ∂ν
∂s(ξ(s), s)ds+ n
X
i=1
Z t
0
ηi(ξ(s), s) ∂ν ∂xi
(ξ(s), s)ds
+ n X i=1 Z t 0
σi(ξ(s), s)∂ν
∂xi
(ξ(s), s)dWs+1 2 n X i,j Z t 0
(σσ′)ij(ξ(s), s) ∂2ν ∂xi∂xj
(ξ(s), s)ds
+ Z t 0 Z IR "
ν(ξ(s) +z, s)−ν(ξ(s), s)−
n
X
i=1 ∂ν ∂xi
(ξ(s), s)zi
#
·
µ(ds, dz)−1|z|≤1ν(ξ(s), s, dz)ds
+
Z t
0
Z
|z|≤1
"
ν(ξ(s) +z, s)−ν(ξ(s), s)−
n
X
i=1 ∂ν
∂xi(ξ(s), s)zi
#
ν(ξ(s), s, dz)ds.
Finally, sinceν(ξ(t), t) is martingale by construction, fromLemma (2.2)we have 0 = ν(ξ(0),0) +
Z t
0 ∂ν
∂s(ξ(s), s)ds+ n
X
i=1
Z t
0
ηi(ξ(s), s) ∂ν ∂xi
(ξ(s), s)ds
+ 1 2 n X i,j Z t 0 (
σσ′)ij(ξ(s), s) ∂2ν
∂xi∂xj(ξ(s), s)ds
+
Z t
0
Z
|z|≤1
"
ν(ξ(s) +z, s)−ν(ξ(s), s)−
n
X
i=1 ∂ν ∂xi
(ξ(s), s)zi
#
ν(ξ(s), s, dz)ds. ✷ Let nowξ∗ be an (σ∗, η)-jump diffusion inIRn, with the same drift coefficient as
ξ(t), defined by the equation
dξt∗=σ∗tdWt+ηtdt+
Z
|z|≤1
zµˆ(dt, dz) +
Z
|z|>1
here the compensator of the jump measure is denoted by νt(dz)dt and we write ˆ
µ(dt, dz) =µ(dt, dz)−νt(dz)dt. For allF ∈C02(IRn), the generator ofξ∗ is given by
(L∗tF)(x) = 1 2
n
X
i,j=1
βij∗(x, t) ∂
2F
∂xi∂xj
+
n
X
i=1
ηi(x, t) ∂F ∂xi
+
Z "
F(x+z, t)−F(x, t)−
n
X
i=1 zi
∂F
∂xi(x, t)1|z|≤1
#
νt(dz).
(7)
Lemma 2 ([7]) Let F be twice continuously differentiable, then F is directionally convex (dcx) if and only if ∂2i,jFx ≥0, for all i, j≤n and all x∈IRn.
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∗≤dcxξ(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∗tν)(ξt∗) ≤ (Ltν)(ξt∗),
then the process ν(ξ∗t, t)is a supermartingale and satisfies the comparison inequality Eφ(ξT∗)
At
≤ν(ξt∗,t), t∈[0,T]. (8)
Proof. The processν(ξt∗, t) can be decomposed as ν(ξt∗, t) =Mt−At such that
ν(ξ∗t, t) = ν(ξ0∗,0) +
n
X
i=1
Z t
0
σi(ξs∗, s) ∂ν ∂xi
(ξs∗, s)dWs
+
Z t
0
Z
IR
"
ν(ξs∗+z, s)−ν(ξs∗, s)−
n
X
i=1 ∂ν ∂xi
(ξs∗, s)zi
#
·µ(ds, dz)−1|z|≤1νs(dz)
−
Z t
0((
Lsν)(ξs∗)−(L∗sν)(ξs∗))ds
Where Mt is a martingale,At=−
Z t
0 ((
Lsν)(ξ∗s)−(L∗sν)(ξs∗))ds
is increasing and adapted process with E(At) <∞ for allt ≥0, henceν(ξt∗, t) is a supermartingale
by the Doop decomposition which implies that Eφ(ξT∗)
At
= Eν(ξ∗T,t) At
≤
ν(ξt∗,t). ✷
Theorem 2 (The dcx order) Let the PDC hold for ξ and assume that (βt∗)ij ≤βij(ξ∗
t, t),
and for almost all t≥0 we have either : 1. νt(dz)≤dcxν(ξ∗t, t, dz),
or
2. νt(dz) and ν(ξ∗t, t, dz) are supported by IRn+ and νt(dz)≤idcxν(ξt∗, t, dz)
Then
Eφ(ξ∗T)
At
≤Ehφξ∗(x,T,t)i, t∈[0,T], x∈IRn. Proof. Using Lemma (3.1). We have by (3) and (7),
(L∗tν)(ξ∗t) − (Ltν)(ξt∗)
= 1
2
n
X
i,j=1
(βt∗)ij −βij(ξt∗, t) ∂
2ν
∂xi∂xj
(ξt∗, t)
+
Z "
ν(ξs∗+z, s)−ν(ξ∗s, s)−
n
X
i=1 zi
∂ν ∂xi
(ξs∗, s)1|z|≤1
#
νs(dz)
−
Z "
ν(ξs∗+z, s)−ν(ξ∗s, s)−
n
X
i=1 zi∂ν
∂xi
(ξs∗, s)1|z|≤1
#
ν(ξs∗, s, dz)
= 1
2
n
X
i,j=1
(βt∗)ij −βij(ξ∗
t, t)
∂2ν
∂xi∂xj
(ξt∗, t)
+
Z "
ν(ξs∗+z, s)−ν(ξ∗s, s)−
n
X
i=1 zi∂ν
∂xi
(ξs∗, s)1|z|≤1
#
νs(dz)−ν(ξs∗, s, dz)
= 1
2
n
X
i,j=1
(βt∗)ij −βij(ξ∗
t, t)
∂2ν
∂xi∂xj
(ξt∗, t) +
Z
ψs(ξs∗, z)1|z|≤1
νs(dz)−ν(ξs∗,s,dz)
where ψt(x, z) =ν(ξs∗+z, s)−ν(ξs∗, s)− n
X
i=1 zi
∂ν ∂xi
(ξs∗, s), z, x∈IRn.
As ν(., t) is directionally convex, ∂i,j2 νx ≥ 0, for all i, j ≤n and all x ∈ IRn,
and if βij(ξt∗, t)−(βt∗)ij ≥0 it follows that
(L∗tν)(ξt∗) ≤ (Ltν)(ξt∗)
+
Z
ψs(ξs∗, z)1|z|≤1
νs(dz)−ν(ξ∗s,s,dz).
Since ∂i,j2 ψz ≥0, for all i, j≤n, the functionψt(x, z) for all fixedx∈IRn is also
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 inz∈IRn+ and whenνt(dz),ν(ξt∗, t, dz) are supported byIRn+
it follows that (L∗tν)(ξt∗) ≤ (Ltν)(ξt∗). It remains now to use the lemma (3.1) to
obtain the result.
✷
Assume now that the processξt∗, t≥0 has the following representation
ξt∗ =ξ0∗+
Z t
0
ηsds+
Z t
0
D
σ∗s, dWsE+
Z t
0
D
Hs∗, dNs∗E. (9) Here dN∗ =dZ∗−λ∗tdt is a jump martingale and Zt∗ is a point process with com-pensator λ∗t, t ≥ 0, σ∗t and Ht∗ are respectively IRn×m and IRn-valued predictable processes. Let Z(t) and Zt∗ two independent point processes in IRn with compen-sators respectively
m
X
j=1
λj(x, t)δHij(x,t)and m
X
j=1
(λ∗t)jδ (H∗
t)ij; i≤n, j ≤m, where δ(x,t)
denotes the Dirac measure at (x, t)∈IRn×IR+.
Theorem 3 (The dcx order) Assume that ξ(t) and ξ∗t are two diffusions with same drift and have same representation as in (9), H(x, t)andHt∗ areIRn×m-valued integrable predictable processes. if
1. (σ∗σ´∗)t≤(σ´σ)(x, t),
2.
m
X
j=1
(λ∗t)jδ(H∗
t)i,j ≤dcx m
X
j=1
λj(x, t)δHi,j(x,t); i≤n, j ≤m, t≥0
then
Eφ(ξ∗T)
At
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
j=1
(λ∗t)jδ(H∗
t)i,j(dz) and m
X
j=1
λj(x, t)δHi,j(x,t)(dz);i≤n, j ≤
m, t≥0. Thus the condition (1) of (3.2) implies that
m
X
j=1
(λ∗t)j(Ht∗)i,j ≤
m
X
j=1
λj(x, t)Hi,j(x, t);i≤
n, j ≤m, in this case we have
m
X
j=1
(λ∗t)j ≤ m
X
j=1
λj(x, t), and the concentration
inequal-ity hold. ✷
Theorem 4 LetH(x, t, z)andH∗(z)two integrable predictableIRn-valued processes and assume that (σ∗σ´∗)t≤(σσ´)(x, t) for allt≥0. then the inequality
Eφ(ξT∗)
At
≤Ehφξ∗(x,T,t)i, t∈[0,T], x∈IRn
holds for all directionally convex function φ:IRn→IRprovided thePDCcondition defined in section (2) and one of the following conditions is satisfied
1.
m
X
j=1
(λ∗t)jδ(H∗
t)i,j ≤dcx m
X
j=1
λj(x, t)δHi,j(x,t),
2.
m
X
j=1
(λ∗t)jδ(H∗
t)i,j ≤idcx m
X
j=1
λj(x, t)δHi,j(x,t); i≤n, j≤m, t≥0
Proof. Using theorem (3.2), the characteristic measures νt(dz) and ν(x, t, dz)
have respectively the form 1IRn\{0}(z) m
X
j=1
(λ∗t)jδ(H∗
t)i,j◦(H
∗
t)−1(dz) and
1IRn\{0}(z) m
X
j=1
λj(x, t)δHi,j(x,t)◦(H(x, t))−1(dz); i≤n, j≤m, t≥0.
νt(dz)≤dcxν(x, t, dz) if and only if m
X
j=1
(λ∗t)j ≤
m
X
j=1
λj(x, t) and by the second condi-tion of (3.2) if the characteristic measures are supported byIRn
+andH is increasing
in z, Ht∗ ≤ H(x, t), then we obtain also the concentration inequality given by the
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Toufik Guendouzi
Laboratory of Mathematics Djillali Liabes University
