EXPECTED UTILITY MAXIMISATION FOR EXPONENTIAL LEVY MODELS WITH OPTION AND INFORMATION PROCESSES

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EXPECTED UTILITY MAXIMISATION FOR EXPONENTIAL LEVY MODELS WITH OPTION

AND INFORMATION PROCESSES

Lioudmila Vostrikova

To cite this version:

Lioudmila Vostrikova. EXPECTED UTILITY MAXIMISATION FOR EXPONENTIAL LEVY

MODELS WITH OPTION AND INFORMATION PROCESSES. 2016. �hal-01388047�

EXPONENTIAL LEVY MODELS WITH OPTION AND INFORMATION PROCESSES

Lioudmila Vostrikova

LAREMA, D´epartement de Math´ematiques, Universit´e d’Angers, 2 Bd. Lavoisier - 49045, Angers Cedex 01.

Abstract. We consider expected utility maximisation problem for exponential Levy models and HARA utilities in presence of illiquid asset in portfolio . This illiquid asset is modelled by an option of European type on another risky asset which is correlated with the first one. Under some hypothesis on Levy processes, we give the expressions of information processes figured in maximum utility formula. As applications, we consider Black-Scholes models with correlated Brownian Motions, and also Black-Scholes models with jump part represented by Poisson process.

Key words and phrases : utility maximisation, exponential Levy model, f-divergence minimal martingale measure, dual approach, en- tropy, Kullback-Leibler information, information processes.

MSC 2010 subject classifications: 60G07, 60G51, 91B24

1. Introduction

Levy processes was used in Mathematical Finance since a long time.

These models contain a number of popular jump models including Gen- eral Hyperbolic models and Variance-Gamma models. The use of such processes for modelling allows an excellent fit both for daily log return and intra-day data. The class of Levy processes is also flexible enough to allow the processes with finite and infinite variation, and also with finite and infinite activity. Levy models are not only excellent to fit

11This work is supported in part by ANR-09-BLAN-0084-01 of the Department of Mathematics of Angers’s University.

1

the data but also mathematically tractable (see [8], [9] and references there).

Let X = (X

)

be a d-dimensional Levy process, d ≥ 1, with gener- ating triplet (b, c, K) where b ∈ R

is drift parameter, c is d × d matrix related with continuous martingale part of X and K is Levy measure which satisfies:

(1)

Z

R^d

( || x ||

∧ 1)K (dx) < ∞ .

As known, the law of such process is entirely determined by its one- dimensional distributions and the characteristic function of

X

=

(X

, X

, · · · , X

) at λ ∈ R

is given by:

φ

(λ) = exp { tψ(λ) } where the characteristic exponent of X

ψ(λ) = exp

i

λb − 1 2

⊤

λcλ + Z

R^d

(e

− 1 − i

λ l(x))K(dx)

with the truncation function l. In general, truncation function l : R

→ R

is a bounded function with compact support such that l(x) = x in the neighbourhood of zero. The classical choice of l is l(x) = x1

where 1

is indicator function and || · || is euclidean norm in R

(for more information on Levy processes see [2],[19]).

For given Levy process X, the modelling of risky asset can be made by the exponential process S = (S

)

with

S

=

( E (X

)

, E (X

)

, · · · , E (X

)

)

where E ( · ) is Dol´ean-Dade exponential and X

,1 ≤ i ≤ d, are the com- ponents of X. We recall that for each one-dimensional semi-martingale X

,

E (X

)

= exp

X

− 1

2 < X

>

Y

0≤s≤t

exp {− ∆X

} (1 + ∆X

) . Here < X

> is quadratic variation of continuous martingale part of X

and ∆X

are jumps of X

(see [16] for more details).

Utility maximisation of exponential Levy models with single Levy pro-

cess was considered in a number of articles (see for instance [6], [7] and

references there). However, the same questions in presence of illiquid

assets in portfolio was not completely studied.

Utility maximisation in mentioned situation was considered in a num- ber of books and papers, see for instance [3], [4], [5], [13], [17], [18].

Some explicit formulas for maximum of expected utility were obtained for Brownian motion models, where the incompleteness on the market comes from the non-traded asset (see [13], [17], [18]). The formulas for maximum of expected utility in complete markets was derived in [1].

But the case of correlated Levy models with jumps was not considered up to now.

To model dependent assets of Levy type, we denote by X

and X

two d-dimensional independent integrable Levy processes with gener- ating triplets (b

, c

, K

) and (b

, c

, K

) respectively where K

and K

verify the condition of type (1). For two invertible matrix ρ

and ρ

with real valued components, we introduce the process X = (X

)

as a linear map of X

and X

, namely X

= ρ

X

+ ρ

X

We suppose that our two risky assets can be modelled by the processes S

= (S

)

and S

= (S

)

with T

> T and

S

=

( E (X

)

, E (X

)

, · · · , E (X

)

) and

S

=

( E (X

)

, E (X

)

, · · · , E (X

)

).

To ensure that the components of S

and S

are positive, we assume, that for 1 ≤ i ≤ d, the jumps of X

and X

verify: ∆X

> − 1,

∆X

> − 1. Without loss of generality and up to now we assume that the interest rate r of non-risky asset is equal to zero.

In our setting, the investor, which has two assets S

and S

, can trade the first asset S

with maturity time T , but the second asset with maturity time T

> T , can not be traded because of lack of liquidity or legal restrictions. At the same time the investor has an European option g(X

) on risky asset S

, where g is some non-negative real valued Borel function on R

. In such situation the investor, who would like to sell the option, would like also to evaluate the corresponding maximal expected utility of the portfolio with option.

Let us denote by Π(F) the set of self-financing admissible strategies

with respect to the filtration F, generated by X. Then, for utility

function u and initial capital x

, the optimal expected utility U

(x

, 0)

related with the first asset S

only, verify U

(x

, 0) = sup

φ∈Π(F)

E [u(x

+ Z

0

φ

· dS

)]

and if we add the mentioned option, then the optimal expected utility will be equal to

U

(x

, g) = sup

φ∈Π(G)

E[u(x

+ Z

0

φ

· dS

+ g(X

)]

where Π( G ) is a set of self-financing admissible strategies with respect to the enlarged filtration G = ( G

)

with, for 0 ≤ t < T ,

G

= \

s>t

F

⊗ σ(X

) and G

= F

⊗ σ(X

).

This approach coincide with so called utility maximisation with distor- tion . In the case of Levy processes the distortion is δ = X

− X

, and the information contained in G

coincide with the one’s of the σ- algebra F

⊗ F

∨ σ(X

− X

), i.e. with progressive filtration at time T augmented by σ-algebra generated by distortion.

In this note we concentrate ourselves on non-complete market case modelled by correlated exponential L´evy models. We recall that very often the utility maximisation analysis is carried out for the hyperbolic absolute risk utilities (in short HARA utilities). HARA utilities can be defined trough the coefficient of absolute risk aversion:

R(x) = − u

(x) u

(x) In HARA utility case,

R(x) = 1 A + Bx

with A and B positive constants. The solutions of such differential equation for u are known, and they are logarithmic, power and expo- nential utilities given below:

u(x) = ln x, with x ∈ R

, u(x) = x

p , with x ∈ R

and p ∈ ( −∞ , 0) ∪ (0, 1), u(x) = 1 − e

, with x ∈ R and γ > 0.

The problem of utility maximisation with option, when X and X

are semimartingales, was considered in [10]. The method applied was

based on enlargement of filtration, combined with the conditioning with

respect to the variable X

and, then, with dual approach. In dual

approach we replace the problem of maximisation of expected utility by finding so-called f-divergence minimal martingale measure where f is dual to u function, namely

(2) f(x) = sup

y∈R

(u(y) − xy ), As known, if u is logarithmic, then

f(x) = − ln x − 1, if u is power, then

f (x) = 1 − p p x

, and if u is exponential,

f (x) = 1 − x

γ (1 + ln γ) + 1 γ x ln x.

In Section 2 we give, for convenience of the reader, some results about utility maximisation with option for semimartingale models. The main results of this section are the formulas for maximum of utility. These formulas contain the corresponding information quantities, like Kulback- Leibler information and Hellinger type integrals. In turn, these infor- mation quantities can be recovered from respective information pro- cesses.

In Section 3, we consider the exponential Levy models. More precisely, we verify the assumptions of Section 2 and we give the expressions for Girsanov parameters of f-divergence minimal conditional martin- gale measures. These expressions permit us to write the corresponding information processes, and, then use the results of Section 2.

In Section 4 we give the expressions of the information quantities for Black-Scholes models with correlated Brownian Motions.

In Section 5 we consider Black-Scholes models with correlated Brow- nian Motions and jumps represented by Poisson process, in order to derive the mentioned information quantities.

2. Some known results about utility maximisation with option for exponential semimartingale models.

2.1. Modelling and assumptions. We suppose that the process X =

(X

)

is given on canonical probability space (Ω, F , P ) with filtra-

tion F = ( F

)

satisfying usual properties. This process represents

stochastic logarithms of a d-dimensional liquid asset S

= (S

)

with

S

=

( E (X

)

, E (X

)

, · · · , E (X

)

) .

At the same time, we have also a d-dimensional semimartingale X

, which represents stochastic logarithms of another risky, but illiquid as- set. This illiquid asset, in turn, is represented in portfolio by European type option g(X

) where g is a positive measurable function on R

and T

> T .

To perform utility maximisation, we introduce a product space (Ω × R

, F ⊗ σ(X

), P × α)

with P ”historical” law of X and α ”historical” law of X

, endowed with enlarged filtration G = ( G

)

with

(3) G

= \

s>t

F

⊗ σ(X

) and G

= F

⊗ σ(X

).

We remark that X remains a semimartingale on product space equipped with filtration G since X and X

are independent under the proba- bility measure P × α.

Now, we denote by P the law of the couple (X, X

) and by P

the regular conditional law of X given { X

= v } , i.e. for all A ∈ F and v ∈ R

P

(A) = P(A | X

= v) .

To preserve semimartingale property of X under conditioning, we sup- pose that the following assumption holds.

Assumption 1. For each v ∈ R

the probability P

is locally absolutely continuous with respect to P , i.e

P

≪ P.

Under the Assumption 1 and according to [15] and [16], a semimartin- gale X will remain a semimartingale under each measure P

, v ∈ R

. Of course, the characteristics of a semimartingale X under P

will be changed as it was proved in [16] (cf. Theorem 3.24, p. 159).

For 0 ≤ t ≤ T , we denote by P

and P

the restrictions of the measures P

and P on the σ-algebra F

. To avoid measurability problems in semimartingale calculus depending on a parameter v (cf. [20]), we need the optional versions of likelihood processes (

t

)

with respect to

the filtration G. For that, we introduce conditional distribution of X

given F

, i.e.

α

(ω, dv) = P(X

∈ dv |F

)(ω).

We make the following assumption

Assumption 2. The conditional distributions of random variable X

given F

are absolutely continuous with respect to its law, i.e.

α

≪ α, ∀ t ∈ ]0, T ] .

According to Jacod’s lemma (see[14]), under the Assumption 2, there exists an optional version of density process (

)

.

Remark 1. It should be noticed that the Assumption 2 can be replaced by the assumption that

T

considered as a map of (ω, v) is F

⊗B (R

)- measurable. Then we can construct an optional version of density pro- cess using the results of [20].

As it was mentioned, the next step consists to solve conditional utility maximisation problem using dual approach (see, for example, [11]). Let us denote by f dual conjugate of utility function u. Let M

be the set of equivalent martingale measures on probability space (Ω, F

, P

) for exponential semimartingale S

and let

K

= n

Q

∈ M

: E

f

dQ

dP

< ∞ o .

We recall that Q

is an equivalent f-divergence minimal martingale measure if

E

f ( dQ

dP

)

= min

Q^v_T∈K^v_T

E

f( dQ

dP

) .

To use dual approach we introduce the following assumption.

Assumption 3. For each v ∈ R

, there exists f-divergence minimal equivalent martingale measure Q

, which belongs to the set K

and such that z

(v) =

T

considered as a map of (ω, v), is F

⊗ B (R

)- measurable and such that E

f (z

(v )) is integrable in v with respect to α.

2.2. Existence of f-divergence minimal martingale measure.

We recall the results about the existence of global f-divergence min-

imal martingale measure. For that we denote by P

the restrictions of

the measure P to the σ − algebra G

and let M

be the set of equiva-

lent martingale measures for semimartingale (S

)

considered as

an application on probability space (Ω

× R

, F

⊗ B (R

), P

) with the filtration G. Let

K

= { Q

∈ M

| E

| f ( dQ

dP

) | < ∞} .

We remark that K

6 = ∅ . In fact, as Radon-Nikodym density of a measure Q

with respect to P

, we can take z

(v ) from Assumption 3 with replacement of v by X

. We recall that Q

is f-divergence minimal measure if

QT

inf

E

f ( dQ

dP

) = E

f ( dQ

dP

) .

Theorem 1. (cf. [10])Under the Assumptions 1,2,3 there exists Q

∈ K

which is f -divergence minimal martingale measure and

(4) U

(x

, g) = Z

R^d

E

[u ( − f

(λ

(v )z

(v)))] dα(v), where λ

(v) is a solution of the equation

(5) − E

[f

(λ

(v) z

(v))] = x

+ g(v) .

2.3. Conditional information quantities and maximal expected utility. Let us assume the existence of f -divergence minimal martin- gale measure Q

∈ K

and denote

z

(v) = Q

P

, p

= dP

dP

.

Now, we introduce three important quantities related with P

and Q

namely the entropy of P

with respect to Q

,

I(P

| Q

) = − E

[ln z

(v)] = − E

[p

ln z

(v)] ,

then, the entropy of Q

with respect to P

or Kulback-Leibler infor- mation

I (Q

| P

) = E

[z

(v ) ln z

(v)] = E

[p

z

(v) ln z

(v)] , and also Hellinger type integrals

H

(v) = E

[(z

(v))

] = E

[p

(z

(v))

] , where q =

with p < 1.

In the following theorem we give the expressions of the maximal ex-

pected utility involving relative entropies and Hellinger-type integrals.

Theorem 2. (cf. [10])Under the Assumptions 1, 2, 3, there exist a B (R

)-measurable versions of the information quantities. Moreover, we have the following expressions for U

(x

, g ) :

(i) If u(x) = ln x then (6) U

(x

, g) =

Z

R^d

[ ln(x

+ g(v)) + I(P

| Q

) ]dα(v) . (ii) If u(x) =

with p < 1, p 6 = 0 then

(7) U

(x

, g) = 1 p

Z

R^d

(x

+ g(v))

H

(v )

dα(v) .

(iii) If u(x) = 1 − e

with γ > 0 then (8) U

(x

, g) = 1 −

Z

R^d

exp {− [ γ(x

+ g(v)) + I(Q

| P

) ] } dα(v) .

2.4. Conditional information processes and conditional infor- mation quantities. In this subsection we recall that the conditional information quantities can be recovered from conditional information processes. To simplify the expression for information processes we sup- pose during this subsection that the process X is quasi-left continuous.

We recall that (P, F)-semimartingale X is a quasi-left continuous, if for any predictable stopping time τ , the jump ∆X

= 0 (P -a.s.) on the set { τ < ∞} . We remark that since P

≪

P , (P

, F) semi-martingale X will be also quasi-left continuous.

Let us denote by β

and Y

two (P

, F)-predictable processes known as Girsanov parameters for the change of measure P

into Q

such that: ∀ t ≥ 0 and P

-a.s.

Z

Z

R^d

|| l(x) || | (Y

(x) − 1) | ν

(ds, dx) < ∞ , and

Z

0

|| c

β

) || ds < ∞ , Z

0

⊤

β

c

β

ds < ∞ .

Here ν

stands for the compensator of the jump measure of X with respect to (P

, F), l is the truncation function and c is the density of the predictable variation of continuous martingale part of X, w.r.t.

Lebesgue measure.

In the case of logarithmic utility we consider the entropy I(P

| Q

) and also the corresponding information process I

(v) = ( I

(v))

with

(9) I

(u) = 1 2

Z

⊤

β

c

β

ds

− Z

0

Z

R^d

[ln(Y

(x)) − Y

(x) + 1] ν

(ds, dx).

Proposition 1. Let Q

∈ K

. Then the corresponding relative en- tropy is well-defined and

(10) I(P

| Q

) = E

I

(v).

In the case of exponential utility we consider Kullback-Leiber informa- tion I( Q

| P

) and we introduce the corresponding Kullback-Leiber process I

(v ) = (I

(v))

with

(11) I

(v) = 1 2

Z

⊤

β

c

β

ds

+ Z

0

Z

R^d

[Y

(x) ln(Y

(x)) − Y

(x) + 1] ν

(ds, dx).

Proposition 2. Let Q

∈ K

. Then, the corresponding Kullback- Leibler information is well defined and

(12) I( Q

| P

) = E

Z

0

z

(v) dI

(v)

= E

( I

(v) ) . For the case of power utility we consider Hellinger types integrals

H

(v) = E

[(z

(v ))

] , where q =

< 1.

We introduce the corresponding predictable process called Hellinger type process h

(v) = (h

(v))

(13) h

(v) = 1

2 q(1 − q) Z

0

⊤

β

c

β

ds

− Z

0

Z

R^d

[(Y

(x))

− q(Y

(x) − 1) − 1] ν

(ds, dx) .

Proposition 3. Let Q

∈ K

. Then respective Hellinger type integral H

(v) is well defined and

(14) H

(v) = 1 − E

Z

0

(z

(v))

dh

(v)

or, in the terms of the stochastic exponential,

(15) H

(v) = E

E − h

(v)

T

where R

is some locally absolutely continuous w.r.t. P

measure.

3. Utility maximisation with option for exponential L´ evy models

We begin with some basic notations for the exponential L´evy models involved in the utility maximisation calculus.

3.1. Description of the model. Let X

= (X

)

and X

= (X

)

be two independent d-dimensional Levy processes starting from zero with generating triplets (b

, c

, K

) and (b

, c

, K

) respec- tively. Each process is given on its own filtered canonical probabil- ity space (Ω

, F

, F

, P

) and (Ω

, F

, F

, P

) respectively where F

= ( F

)

and F

= ( F

)

are the corresponding filtrations verifying usual properties. Let X = (X

)

be the linear map of the processes X

and X

, namely, for t ∈ [0, T ]

(16) X

= ρ

X

+ ρ

X

involving non-random invertible matrices ρ

and ρ

. As it was men- tioned, the process X is considered on canonical probability space (Ω, F , F, P ) with filtration F = ( F

)

which satisfy usual prop- erties.

We introduce also the enlarged space (Ω × R

, F ⊗ σ(X

), G), corre- sponding to the couple (X, X

) with enlarged filtration G = ( G

)

where for 0 ≤ t < T G

= \

s>t

F

⊗ σ(X

) and G

= F

⊗ σ(X

).

We remark that on the space (Ω, F , P ) the process X, is, evidently,

a Levy process. Now, if we equip (Ω × R

, F ⊗ B ( R

), G ) with the

probability P × α, where α is the law of X

, then the process X will

remain a Levy process with the same triplet. We recall that, as before,

we use the notation P for joint law of (X, X

) and P

for conditional law of X given X

= v.

3.2. Assumptions 1 and 2. In this subsection we show that the As- sumptions 1 and 2 of Section 2 will be verified under the following hypothesis on L´evy processes.

Hypothesis H1: The processes X and X

are integrable on [0, T ] and [0, T

] respectively.

Hypothesis H2: The process (X, X

) has a transition density w.r.t.

a product η = η

× η

of two σ-finite measures η

and η

on R

, and the marginal transition densities f and f

of X and X

w.r.t. η and η

respectively, are strictly positive.

Remark 2. It should be noticed that in the case when η

and η

are Lebesgue measures, the Hypothesis H2 is equivalent to the existence of marginal strictly positive transition densities f

and f

of the processes of X

and X

. This fact follows from the independence of X

and X

.

Proposition 4. Under hypotheses (H1) and (H2), the Assumptions 1 and 2 are satisfied and there exists a function F

: [T

− T, T

] × R

→ R

depending on a parameter v ∈ R

, such that

(17) dα

dα (v) = F

(T

− t, X

) F

(T

, 0) Moreover,

dα

dα = E (M )

with M = (M

)

which is a (P, F)- martingale such that M

=

Z

⊤

β

X

+ Z

0

Z

R^d

l(x)(Y

(x) − 1)dK(x)ds

where (β

, Y

) are the Girsanov parameters for the change the mea- sure P into P

, and K is Levy measure of X.

If F

∈ C

([T

− T, T

] × R

) and c is invertible, then the mentioned Girsanov parameters (β

, Y

) can be calculated by the following for- mulas:

⊤

β

=

∂ ln F

∂x

(T

− s, X

), · · · , ∂ ln F

∂x

(T

− s, X

)

and for x ∈ R

\ { 0 }

Y

(x) = F

(T − s, X

+ x) F

(T

− s, X

) .

Proof: Conditionally to X

= v , the process X is distributed as ρ

X

+ ρ

V

where V

is a Levy bridge of X

starting at (0, 0) and ending at (v, T

). Under the hypothesis (H2) and according to [12], the law of (V

)

is absolutely continuous w.r.t. the law of (X

)

and

(18) dP

dP

(T, v) = f

(T

− T, v − X

) f

(T

, v ) .

Since the process X

is independent from X

and also from V

, the conditional distributions of X given X

and the conditional distribu- tions of X given V

coincide as maps, under the measure P . Let us denote this map by q(A, x), A ∈ F

, x ∈ R

. Then,

P (A) = Z

R^d

P (ρ

X

+ρ

X

∈ A | X

= x)dP

(x) = Z

R^d

q(A, x)dP

(x) and

P

(A) = P (ρ

X

+ ρ

V

∈ A)

= Z

R^d

P (ρ

X

+ ρ

V

∈ A | V

= x) dP

(x)

= Z

R^d

q(A, x) dP

dP

(T, v) dP

(x) . Finally, if P (A) = 0 then q(A, x) = 0 (P

-a.s.) and, hence P

(A) = 0.

Hence, the Assumption 1 is verified.

The Assumption 1 and Bayes formula for conditional densities gives us:

P (X

∈ dv | X

= y) = P (X

∈ dv, X

∈ dy) P (X

∈ dy) = P (X

∈ dy | X

= v ) P (X

∈ dv)

P (X

∈ dy) .

This means that the Assumption 2 is verified. Using Markov property we write:

α

(dv) = P (X

∈ dv | F

) = P (X

∈ dv | X

) =

P (X

− X

+ X

∈ dv | X

) = P ( ˜ X

+ X

∈ dv | X

)

where ˜ X

is a process, which is independent from X

and X

, and is distributed as X

. Then, we see that α

(dv) is a function of T

− t, X

and the parameter v, denoted F

(T

− t, X

). At the same time α

(dv) = α(dv) = F

(T

, 0) since F

= {∅ , Ω } . It gives us (17).

Now, we use Ito formula to obtain that

F

(T

− t, X

) = F

(T

, 0)

− Z

0

∂F

∂s (T

− s, X

) ds +

d

X

i=1

Z

∂F

∂x

(T

− s, X

) dX

+ 1

2

d

X

i=1 d

X

j=1

Z

∂

F

∂x

∂x

(T

− s, X

) d < X

, X

>

+ X

0<s≤t

F

(T

− s, X

) − F

(T

− s, X

) −

d

X

i=1

∂F

∂x

(T

− s, X

) ∆X

. Under the conditions P

<< P

and α

<< α for t ∈ ]0, T ], we know from Jacod’s lemma (cf.[14]) that

is a (P, F) martingale. Let us put for t ∈ [0, T ], p

= dα

dα (v). Then, we divide the above expression for F

(T

− t, X

) by F

(T

, 0) and we identify its continuous martingale part. We get that

p

= 1 F

(T

, 0)

d

X

i=1

Z

∂F

dx

(T

− s, X

) dX

and, hence,

< p

, X

>

= 1 F

(T

, 0)

d

X

i=1

Z

c

∂F

dx

(T

− s, X

) ds . In addition, according to Girsanov theorem,

< p

, X

>

=

d

X

i=1

Z

c

(β

)

p

ds.

Since c is invertible, this implies the formula for β

. Now, we compute the jumps of p

:

∆p

= F

(T

− t, X

) − F

(T

− t, X

)

F

(T

, 0)

and ∆p

p

= F

(T

− t, X

+ ∆X

) F

(T

− t, X

) − 1 .

Then, according to the Theorem 3.24,p.159, Chapter 3 in [16]

Y

= F

(T

− t, X

+ x)) F

(T

− t, X

) and the proposition is proved. 2

3.3. Conditional locally equivalent martingale measures. The main difficulty related with the application of the results of Section 2 is the verification of the Assumption 3. The first step for this verifi- cation, consists in the complete description of the set of conditional locally equivalent martingale measures. This step can be done by use of semimartingale calculus.

We recall that the process X is defined by (16). As before we denote by (β

, Y

) the Girsanov parameters for the change of the measure P into P

. We denote also by M (P

) the set of locally equivalent to P

martingale measures Q

. We denote by (β

, Y

) the Girsanov parameters for the change of the measure P

into Q

. We notice that X is (P

, F)- semimartingale, and hence, (Q

, F) semimartingale. In the following proposition we give the triplet of predictable characteristics of X w.r.t. Q

.

Proposition 5. The triplet of predictable characteristics (B

, C

, ν

) of X with respect to (Q

, F ) is given by the expressions:

B

= (ρ

b

+ ρ

b

)t + ρ

c

R

0

β

ds +ρ

R

R

R^d

→

l

(x) (Y

(ρ

x) − 1)(K

◦ ρ

)(dx)ds + R

0

R

R^d

→

l(x) (Y

(x) − 1)K

(dx)ds + (ρ

c

ρ

+ ρ

c

ρ

) R

β

ds, C

= (ρ

c

ρ

+ ρ

c

ρ

)t,

dν

(x, s) = Y

K

(dx)ds,

where K

(dx) = (K

◦ ρ

)(dx) + Y

(ρ

x) (K

◦ ρ

)(dx). More- over, an equivalent to P

martingale measure Q

satisfy : for s ∈ [0, T ] (19) ρ

b

+ρ

b

+ρ

c

β

+ρ

Z

R^d

→

l

(x) (Y

(ρ

x) − 1)(K

◦ ρ

)(dx)

+ Z

R^d

→

l(x) (Y

(x) − 1)K

(dx) + (ρ

c

ρ

+ ρ

c

ρ

)β

= 0.

Proof: We use Girsanov theorem for successive change of the mea- sures: P → P

→ Q

. For that we write first a semimartingale decom- position of X:

X

= B

+X

+ Z

0

Z

R^d

→

l(x) (µ

(dx, ds) − ν

(dx, ds))+ X

s≤t

(∆X

− l(∆X

)) Here B is the drift part of semilmartingale decomposition, X

is its continuous martingale part, µ

and ν

are the measure of jumps and its compensator, and l is the truncation function, l(x) = x1

, x ∈ R

. It should be noticed that the integral on R

in previous expression is taken in component by component way, namely for each x ∈ R

and l(x) =

(l

(x), · · · , l

(x)) the integral

Z

Z

R^d

→

l(x) (µ

(dx, ds) − ν

(dx, ds)) is a vector with components

Z

Z

R^d

l

(x)(µ

(dx, ds) − ν

(dx, ds)) where 1 ≤ i ≤ d. We use the notation

→

l(x) to underline this particular integration.

At the same time we write a semi-martingale decompositions of the processes X

and X

:

X

= b

t + X

+ Z

0

Z

R^d

→

l

(x) (µ

(dx, ds) − K

(dx)ds)

+ X

s≤t

(∆X

− l

(∆X

)) X

= b

t + X

+

Z

Z

R^d

→

l

(x) (µ

(dx, ds) − K

(dx)ds)

+ X

s≤t

(∆X

− l

(∆X

)) with truncation functions l

(x) = x1

and l

(x) = x1

respectively.

We compare now the linear combination of the canonical decomposi-

tions of the processes X

and X

given above with canonical de-

composition of X. We can easily identify a drift part of X, which is