SOLUTIONS OF SOME ELLIPTIC EQUATIONS ASSOCIATED WITH A PIECEWISE CONTINUOUS PROCESS

ASSOCIATED WITH A PIECEWISE CONTINUOUS PROCESS

B. IFTIMIE, I. MOLNAR and C. VÂRSAN

We provide a quasimartingale representation and a stochastic rule of dierentia- tion for test functions belonging to some class of polynomials, which depend on the solution of a stochastic dynamic system dened via a piecewise continuous func- tionλ(t). This is achieved by applying the classical Itô's dierentiation rule on each continuity interval. The quasimartingale representation on each continuity interval is obtained by solving some possibly degenerate elliptic equations.

AMS 2000 Subject Classication: 35B40, 35J70, 60G44, 91B28.

Key words: stochastic rule of dierentiation, piecewise continuous process, dege- nerate elliptic equation.

1. INTRODUCTION

The evolution of a multi-dimensional piecewise continuous process {(z(t), µ(t)); t>0}is analyzed in connection with the stochastic rule of deriva- tion by using polynomial solutions for some (possibly degenerate) elliptic equa- tions. In our setting, the process{z(t)∈Rⁿ;t>0}is generated as a solution of a stochastic dierential system depending on a piecewise constant process {µ(t) ∈ S ⊆ R^d;t > 0}. The set of test functions ϕ ∈ P_p(z;µ) consists of all polynomial functions of degree p in the state variables z = (z₁, . . . , z_n), whose coecients are continuous bounded functions ofµ∈R^d. It agrees with the elliptic operators Lµdepending on the parameterµ∈R^d which determine the drift part when a pondered functional {exp(γt)ϕ(z(t); µ(t)); t > 0} is considered and the decomposition formula

exp(γt)ϕ(z(t);µ(t)) =ψt(z(·), µ(·)) + Z t

0

exp(γs)[γϕ+Lµ(ϕ)](z(s);µ(s))ds+

+M_t(z(·), µ(·)), t>0

is valid. Here, Lµ:P_p(z;µ)→ P_p(z;µ) is an elliptic operator, {M_t(z(·), µ(·));

t > 0} is a continuous martingale and {ψ_t(z(·);µ(·)); t > 0} is a piecewise constant process.

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 4, 323338

The above decomposition formula is meaningfull when describing the asymptotic behaviour of the augmented process {exp(γt)kz(t)k² | t > 0}, provided the nontrivial solution ϕγ∈ P₂(z;µ) of the elliptic equation

γϕγ(z;µ) +kyk²+Lµ(ϕγ)(z, µ) = 0, ∀z∈Rⁿ, µ∈R^d, satises the nonsingularity condition

δ(γ)kzk² 6ϕγ(z, µ)6 1 C(γ)kzk² for some positive constants δ(γ),C(γ).

2. DEFINITIONS, SETTING OF THE PROBLEM AND AUXILIARY RESULTS

LetW(t) be a standard m-dimensional Wiener process over a complete probability space {Ω,F,{F_t}_t>0,P}. On the same probability space we have piecewise constant adapted processes{µ(t);t>0}and{y(t);t>0}of dimen- sion d, respectivelyn. Set λ(t) = (y(t), µ(t)),t>0, and

λ(t, ω) =λ_k(ω) = (y_k(ω), µ_k(ω)), t∈[t_k(ω), t_k+1(ω)),

where {t_k;k >0} is an increasing sequence of positive adapted random vari- ables with t₀ = 0, t_k → ∞, a.s., as k → ∞ and (y_k, µ_k), k > 0, are mul- tidimensional F_tk-measurable random variables. We assume that the process W(t) and the sequence{(t_k, λ_k);k>1} are independent.

We are next given a nite set of vector elds gi(z;µ) ∈ Rⁿ, linear in z, dened as

(1) g_i(z;µ) =a_i(µ) +A_i(µ)z, i= 0,1, . . . , m, µ∈R^d, z ∈Rⁿ,

where the (n×n)-matrices Ai(µ) and the n-dimensional vectors ai(µ) are assumed to be continuous and bounded.

For eachx∈Rⁿ dene a piecewise continuous process {z(t, x);t>0} as the unique solution of the dynamical system

(2)















dz(t) =g₀(z(t);µ(t)) +

m

X

j=1

g_j(z(t);µ(t))dW_j(t), t∈[t_k, t_k+1), z(t_k) =z_{_}(t_k) +y(t_k), for any k>1,

z(0) =x, wherez_{_}(t_k) = lim

t↑t_kz(t).

By denition, the unique solution of system (2) is a piecewise continuous and {F_t}-adapted process{z(t, x);t>0}such that, at each jump timet_k, the jumpz(tk, x)−z_{_}(tk, x) =y(tk) occurs.

Denition 1. Denote by P_p(z;µ) the set consisting of the polynomial functions of degree p in the variables (z₁, z₂, . . . , z_n) = z, whose coecients are continuous bounded functions of µ ∈ R^d; P_p(z) ⊆ P_p(z;µ) consists of polynomials with constant coecients.

Consider the piecewise continuous scalar process (3) V_z^γ(t, x) = exp(γt)ϕ(z(t, x);µ(t)) +

Z t 0

exp(γs)f(z(s, x))ds, t>0, whereγ is a constant,f ∈ P_p(z) and ϕ∈ P_p(z;µ).

We provide a stochastic rule of dierentiation involving the piecewise con- tinuous process{V_z^γ(t, x);t>0}which contains both a continuous component and a piecewise constant process dened as

(4) D^γ_z(t, x) =





 P

0<tk6t

exp(γt_k)ψ_k(z(·, x)), t>t₁,

0, t₀6t < t₁,

whereψ_k(z(·, x)) =ϕ(z(t_k, x);µ(t_k))−ϕ(z_{_}(t_k, x);µ(tk−1)), for any k>1. Lemma 1. Let f ∈ P_p(z), ϕ ∈ P_p(z;µ) dene the piecewise continu- ous process {V_z^γ(t, x);t > 0} as in (3), corresponding to the unique solution {z(t, x);t>0} of system (2). Then Vz^γ(t, x) satises the integral equation (5) V_z^γ(t, x) =ϕ(z(0, x);µ(0)) +

Z t 0

exp(γs) [γϕ+f+L(ϕ)](z(s, x);µ(s))ds

+

m

X

j=1

Z t 0

exp(γs)h∇_zϕ(z(s, x);µ(s)), g_j(z(s, x);µ(s))idW_j(s) +D^γ_z(t, x), for t> 0 and x ∈ Rⁿ, where the piecewise constant process {D_z^γ(t, x);t >0}

is dened by (4) and the elliptic operator L:P_p(z;µ)→ P_p(z;µ) is given by (6) L(ϕ)(z;µ) =h∇_zϕ(z;µ), g0(z;µ)i+ 1

2

m

X

j=1

h∇_z∇_zϕ(z;µ)gj(z;µ), gj(z;µ)i.

Proof. The stochastic rule of dierentiation from (5) can be obtained by applying the standard rule of stochastic dierentiation associated with the continuous process{V_z^γ(t, x);t∈[tk, tk+1)}for eachk. In this respect, rewrite (7) V_z^γ(t, x) =V_z^γ(t_k, x) +U_k(t, z(t, x))−U_k(t_k, z(t_k, x))

+ Z t

tk

exp(γs)f(z(s, x))ds,

for t ∈ [tk, tk+1), where Uk(t, z) = exp(γt)ϕ(z;µ(tk)). According to the sto- chastic rule of dierentiation, we get

Uk(t, z(t, x))−Uk(tk, z(tk, x)) = Z t

tk

exp(γs) [γϕ+L(ϕ)](z(s, x);µ(tk))ds +

m

X

j=1

Z t tk

exp(γs)h∇_zϕ(z(s, x);µ(t_k)), g_j(z(s, x);µ(t_k))idW_j(s), for every t∈[tk, tk+1). Using also (7), it follows that

(8) V_z^γ(t, x) =V_z^γ(t_k, x) + Z t

tk

exp(γs) [γϕ+f+L(ϕ)](z(s, x);µ_k)ds

+

m

X

j=1

Z t tk

exp(γs)h∇_zϕ(z(s, x);µ_k), g_j(z(s, x);µ_k)idW_j(s), for t∈[t_k, t_k+1) where, by denition,µ_k=µ(t_k) and

(9) V_z^γ(tk, x) =Uk(tk, z(tk, x)) + Z tk

0

exp(γs)f(z(s, x))ds.

On the other hand, we notice that

Uk(tk, z(tk, x)) =Uk−1(tk, z_{_}(tk, x)) + exp(γtk)ψk(z(·, x)) for k>1. Uk−1(t_k, z_{_}(t_k, x))is computed as

Uk−1(t_k, z_{_}(t_k, x)) =Uk−1(tk−1, z(tk−1, x)) +

Z tk

tk−1

exp(γs)[γϕ+L(ϕ)](z(s, x);µk−1)ds

+

m

X

j=1

Z _tk

tk−1

exp(γs)h∇_zϕ(z(s, x);µk−1), g_j(z(s, x);µk−1)idW_j(s).

Inserting now the last two formulas into (9), we get (10) V_z^γ(tk, x) =V_z^γ(tk−1, x) +

Z tk

tk−1

exp(γs)[γϕ+f +L(ϕ)](z(s, x);µk−1)ds

+

m

X

j=1

Z tk

tk−1

exp(γs)h∇_zϕ(z(s, x);µk−1), gj(z(s, x);µk−1)idW_j(s) + exp(γt_k)ψ_k(z(·, x)).

Now, iterating the last formula, we can rewrite equation (8) as it appears in (5).

Problem A. For a xedf∈ P_p(z), nd a constantγ 6= 0andϕγ ∈ P_p(z;µ), such that the piecewise continuous process {V_z^γ(t, x); t>0} dened by (3) is a quasi-martingale, i.e.,

(11) V_z^γ(t, x) =ϕγ(z(0, x);µ(0)) +D^γ_z(t, x) +

m

X

j=1

Z t 0

exp(γs)h∇_zϕγ(z(s, x);µ(s)), gj(z(s, x);µ(s))idW_j(s).

Problem B. For a xedf∈ P_p(z), nd a constantγ 6= 0andϕ_γ∈ P_p(z;µ) solution of the elliptic equation

(12) γϕ_γ(z;µ) +f(z) +L(ϕ_γ)(z;µ) = 0 for any z∈Rⁿ, µ∈R^d, where the linear operator L:P_p(z;µ)→ P_p(z;µ)is dened by (6).

Remark 1. Notice that each solution(γ, ϕ_γ) of Problem B is a solution of Problem A. In this respect, the scalar process {V_z^γ(t, x); t > 0} dened by (3) is a quasi-martingale expressed as in (11), provided the stochastic rule of dierentiation from Lemma 1 is used for the given solution of Problem B.

Lemma 2. Let f ∈ P₂(z) and the vector elds g_i(z;µ), i= 0,1, . . . , m, be given as in (1). Let γ <0 be such that

(13) |γ|>4

kg₀k+1 2

m

X

j=1

kg_jk²

, where kg_jk= sup

µ∈R^d

(ka_j(µ)k+kA_j(µ)k). Then there existsϕ_γ ∈ P₂(z;µ) satis- fying the elliptic equation (12).

In addition,ϕγ is the sum of the convergent series (14) ϕ_γ(z;µ) = 1

|γ|

^∞ X

k=0

L^k_|γ|(f)(z;µ)

, (z, µ)∈Rⁿ×R^d, where L_|γ|= _|γ|¹ L.

Proof. Estimates of the polynomialsL^k_|γ|(f)(z;µ)∈ P₂(z;µ),k>1, can be obtained by an induction argument. Write each ϕ∈ P₂(z;µ) as

ϕ(z;µ) =C(µ) +hh₀(µ), zi+hH(µ)z, zi,

whereH(µ) = (h₁(µ), . . . , h_n(µ) andC(µ)∈R,h_i(µ)∈Rⁿ,i= 0,1, . . . , n are continuous bounded functions with respect toµ. Denotekϕk=kCk+Pⁿ

i=0

kh_ik,

wherekCk= sup

µ∈R^d

|C(µ)|andkh_ik= sup

µ∈R^d

kh_i(µ)k, usingkh_i(µ)k= Pn j=1

|h^j_i(µ)|, i= 0,1, . . . , n. It is easily seen that

|ϕ(z;µ)|6kϕk(1 +kzk²), ∀(z, µ)∈Rⁿ×R^d, and an elementary computation shows that

(15)















(a) kL_|γ|(ϕ)k6 4

|γ|

kg₀k²+1 2

n

X

j=1

kg_jk²

kϕk,

(b) |L_|γ|(ϕ)(z;µ)|6kϕk 4

|γ|

kg₀k²+1 2

n

X

j=1

kg_jk²

(1 +kzk²) for any (z, µ).

Using an induction argument, we can easily prove that

(16)

L^k_|γ|(f)(z;µ) 6



 4

|γ|

kg₀k²+1 2

n

X

j=1

kg_jk²





k

kfk(1 +kzk²) for any (z, µ)∈Rⁿ×R^d and k>0.

In this respect, recall thatL^k+1_|γ| :P₂(z;µ)→ P₂(z;µ) can be written as L^k+1_|γ| (f)(z, µ) =L_|γ| L^k_|γ|(f)

(z, µ), k>0, and assume that L^k_|γ|(f)∈ P₂(z;µ)and

L^k_|γ|(f) 6



 4

|γ|

kg₀k²+ 1 2

n

X

j=1

kg_jk²





k

kfk

for some xedk>1. Then, using estimates (15) for ϕ=L^k_|γ|(f), we get

L^k+1_|γ| (f) =

L|γ|(ϕ) 6 4

|γ|

kg₀k²+1 2

n

X

j=1

kg_jk²

kϕk

6



 4

|γ|

kg₀k²+1 2

n

X

j=1

kg_jk²





k+1

kfk=ρ^k+1kfk and

L^k+1_|γ| (f)(z, µ)

6ρ^k+1kfk(1 +kzk²) for any (z, µ), whereρ:= _|γ|⁴

h

kg₀k²+¹₂

n

P

j=1

kg_jk²i .

This shows that estimate (16) is satised for any k > 0 and assuming that ρ < 1 (see (13)), we easily get that the series in (14) is convergent. In addition, using (16) we obtain that

|ϕ_γ(z, µ)|6 1

|γ|

^∞ X

k=0

ρ^k

kfk(1 +kzk²), where

1

|γ|

∞

X

k=0

ρ^k= 1

|γ|

1

1−ρ = 1

|γ| −4

"

kg₀k²+1 2

n

P

j=1

kg_jk²

#>0.

By denition, ϕγ∈ P₂(z;µ)and using (14) we get that ϕγ satises the elliptic equation (12), as the following simple computation shows

L(ϕ_γ)(z;µ) =L_|γ|

"_∞ X

k=0

L^k_|γ|(f)(z, µ)

#

=

∞

X

k=1

L^k_|γ|(f)(z, µ) =−γϕ_γ(z;µ)−f(z).

The proof is complete.

Remark 2. The functional ϕ_γ ∈ P₂(z;µ) constructed in Lemma 2 is meaningful for the asymptotic analysis of the piecewise continuous process {V_z^γ(t, x);t>0} providedϕ_γ(z;µ) is a quadratic positive form satisfying

δ(γ)kzk² 6ϕ_γ(z;µ)6 1

C(γ)kzk², for any µ∈R^d. Here, δ(γ) and C(γ) are positive constants.

This can be achieved by imposing the additional conditions on the vector elds g_i(z;µ),i= 0,1, . . . , mgiven in (1)

(17)

(a) g_i(z;µ) =A_i(µ)z, i= 0,1, . . . , m,

(b) A_j(µ)A_k(µ) =A_k(µ)A_j(µ) for anyj, k∈ {0,1, . . . , m}, µ∈R^d, where recall that Ai(µ)are continuous and bounded symmetric matrices.

Lemma 3. Assume that the vector eldsgi(z;µ) given by (1) satisfy as- sumptions (17). Let f(z) =kzk² and let the constant γ full

(18) γ <0, |γ|>4

kg₀k+1 2

m

X

j=1

kg_jk²

. Then there exists ϕ_γ ∈ P₂(z;µ) satisfying the equations (19)







(a) γϕγ(z;µ) +kzk²+L(ϕγ)(z;µ) = 0, ∀(z, µ)∈Rⁿ×R^d, (b) ϕ_γ(z;µ) =hR_γ(µ)z, zi, δ(γ)kzk² 6ϕ_γ(z;µ)6 1

C(γ)kzk²,

where

L(ϕγ)(z;µ) =h∇_zϕ(z, µ), A0(µ)zi+1 2

m

X

j=1

h∇_z∇_zϕ(z;µ)Aj(µ)z, Aj(µ)zi,

C(γ), δ(γ) are positive constants, Rγ(µ) =

|γ|I_n−B(µ)−1

and B(µ) = 2A₀(µ) +

m

P

j=1

A²_j(µ).

Proof. The assumptions of Lemma 2 are satised. Letϕγ∈ P₂(z;µ)be the solution of the elliptic equation (12) given by the series

ϕγ(z;µ) = 1

|γ|

"_∞ X

k=0

L^k_|γ|(f)(z, µ)

#

, (z, µ)∈Rⁿ×R^d, wheref(z) =kzk² and the linear operator L_|γ|= _|γ|¹ Lsatises L_|γ|(ϕ)(z, µ) = 1

|γ|

h∇_zϕ(z, µ), A0(µ)zi+1 2

m

X

j=1

h∇_z∇_zϕ(z;µ)Aj(µ)z, Aj(µ)zi

. By denition,

L_|γ|(f)(z, µ) = 1

|γ|

*

2A0(µ) +

m

X

j=1

A²_j(µ)

z, z +

= 1

|γ|hB(µ)z, zi.

It is obvious that the symmetric matrix B(µ) commutes with each A_i(µ), i= 0,1, . . . , m. Therefore,

L^k_|γ|(f)(z, µ) =

* B(µ)

|γ|

k

z, z +

. Here, the matrix B(µ) satises

kB(µ)k62kA₀(µ)k+

m

X

j=1

kA_j(µ)k²62kg₀k+

m

X

j=1

kg_jk² =ν, for any µ∈R^d. DenekBk= sup

µ∈R^d

kB(µ)k. Using (18), we get

(20) ρ= kBk

|γ| 6 ν

|γ| < 1 2 and the matrix series

(21) Rγ(µ) = 1

|γ|

∞

X

k=0

B(µ)

|γ| k

, µ∈R^d,

is convergent if |γ|>2ν. Hence ϕγ(z;µ) = 1

|γ|

* _∞ X

k=0

B(µ)

|γ|

k

z, z +

=hR_γ(µ)z, zi, provided |γ|>2ν. Recall that the symmetric matrixRγ(µ) fulls

(22) Rγ(µ) = 1

|γ|

In−B(µ)

|γ|

−1

.

Conclusion (19) is a direct application of the following argument. Let z = T(µ)y be an orthogonal mapping, i.e. T^∗(µ) = [T(µ)]⁻¹, such that

T⁻¹(µ)B(µ)T(µ) =D(µ) =diag(γ1(µ), . . . , γn(µ)) and write ϕ_γ(T(µ)y;µ) =hT⁻¹(µ)R_γ(µ)T(µ)y, yi as

ϕγ(T(µ)y;µ) = 1

|γ|

* _∞ X

k=0

D(µ)

|γ| k

y, y +

=hQ_γ(µ)y, yi, y∈Rⁿ, whereQγ(µ) =diag

(|γ| −γ1(µ))⁻¹, . . . ,(|γ| −γn(µ))⁻¹

. UsingB(µ)ek(µ) = γ_k(µ)e_k(µ), k= 1, . . . , n, where T(µ) =col[e₁(µ), . . . , e_n(µ)], it is easily seen that kB(µ)k > max

16k6n|γ_k(µ)|=|bγ(µ)|and |bγ(µ)|6kBk for any µ∈R^d. The hypothesis|γ|>2ν(see (18)) implies that the matrixQ_γ(µ)is positive dened, uniformly with respect to µ, and

δ(γ)kyk²6ϕγ(T(µ)y;µ)6 1

C(γ)kyk²,

with some positive constants δ(γ), C(γ). Take now y = T⁻¹(µ)z. Using kyk² =kzk², we obtain

1

C(γ)kzk²>ϕ_γ(z;µ) =hR_γz, zi>δ(γ)kzk², ∀z∈Rⁿ.

Remark 3. The stochastic rule of dierentiation from Lemma 1 and the result stated in Lemma 3 are relevant for the piecewise continuous process {z(t, x);t>0}, dened as the unique solution of system (2). In this respect, we use the decomposition

(23) z(t, x) =z_c(t, x) +y(t), t>0,

where the continuous component {z_c(t, x);t>0}is the unique solution of the linear stochastic system

(24)







dzc(t) =A0(µ(t))zc(t)dt+

m

X

j=1

A0(µ(t))zc(t)dWj(t), t>0, zc(0) =x.

The stochastic rule of dierentiation written for zc(t, x)reads as (25) V_z^γ_c(t, x) =ϕγ(x;µ(0)) +D^γ_zc(t, x)

+ Z t

0

exp(γs)[γϕγ+f +L(ϕγ)](zc(s, x);µ(s))ds +

m

X

j=1

Z t 0

exp(γs)h∇_zϕγ(zc(s, x);µ(s)), Aj(s, x);µ(s))zc(s, x)idW_j(s), where

D_z^γ_c(t, x) = X

0<tk6t

exp(γt_k)ψ_k(zc(·, x)) and

ψk(zc(·, x)) =ϕγ(zc(tk, x);µ(tk))−ϕγ(zc(tk, x);µ(tk−1)), k>1.

Lemma 4. Let the hypotheses of Lemma 3 hold and denote U_γ(t, x) = exp(γt)ϕ_γ(z_c(t, x);µ(t)), t>0, x∈Rⁿ,

forf(z) =kzk² andϕγ(z;µ) =hR_γz, zi. Then the piecewise continuous process U_γ(t, x) satises the integral equation

(26) U_γ(t, x) =ϕγ(zc(0, x);µ(0)) +D^γ_zc(t, x) +

Z t 0

h−C(γ)U_γ(s, x) + exp(γs)β_γ(z_c(s, x);µ(s))i ds +

m

X

j=1

Z t 0

exp(γs)h∇_zϕ_γ(z_c(s, x);µ(s)), A_j(µ(s))z_c(s, x)idW_j(s), where







βγ(z;µ) =C(γ)ϕγ(z;µ)− kzk² 60, ∀(z, µ)∈Rⁿ×R^d, D^γ_zc(t, x) = X

0<t_k6t

exp(γtk)

ϕγ(zc(tk, x);µ(tk))−ϕγ(zc(tk, x);µ(tk−1)) . Proof. By hypothesis, the stochastic rule of dierentiation from Lemma 1 is fullled for the continuous process {z_c(t, x);t > 0}. If we apply it for {V_z^γ_c(t, x)}, we get equation (25). Using the elliptic equation

γϕ_γ(z;µ) +kzk²+L(ϕ_γ)(z;µ) = 0, ∀(z, µ)∈Rⁿ×R^d

and the integral equation (25), we obtain that the piecewise continuous process {U_γ(t, x);t>0} satises

(27) U_γ(t, x) =ϕγ(x;µ(0)) +D^γ_zc(t, x)− Z t

0

exp(γs)kz_c(s, x)k²ds

+

m

X

j=1

Z t 0

exp(γs)h∇_zϕγ(zc(s, x);µ(s)), Aj(µ(s))zc(s, x)idW_j(s).

Conclusion (19) of Lemma 3 shows that the equation

−kzk²=−C(γ)ϕ_γ(z;µ) +β_γ(z;µ), is satised, where β_γ(z;µ)60 for any(z, µ)∈Rⁿ×R^d.

Rewrite now (27) using the last formula. We thus get conclusion (26).

Remark 4. Assuming that the setS ⊆R^dis chosen such that Rγ(µ(t)) =Rγ(µ(0)), for anyt=t_k, k>0 (seeµ(t_k)∈S),

the piecewise constant process {D_z^γ_c(t, x); t > 0} vanishes and the integral equation (26) can be written accordingly. In addition, the same equation reduces to

(28) U_γ(t, x) =ϕγ(x;µ(0))− Z t

0

C(γ)U_γ(s, x)ds

+ 2

m

X

j=1

Z t 0

U_γ(s, x)θj(s)dWj(s) + Z t

0

exp(γs)βγ(zc(s, x);µ(s))ds, where θ_j(µ) : S → R,j = 1, . . . , m, are given such that A_j(µ) = θ_j(µ)I_n and θbj(t) =µj(µ(t)).

If it is the case, then dene an exponential martingale (29) ξ(t) = exp



2

m

X

j=1

Z t 0

θbj(s)dWj(s)− Z t

0

(θbj(s))²ds



, t>0, which satises the equation

(30)







dξ(t) = 2ξ(t)

m

X

j=1

bθj(t)dWj(t), t>0;

ξ(0) = 1.

Lemma 5. Let hypotheses (17), (18) and (19) hold. In addition, assume that A_j(µ) = θ_j(µ)I_n, j = 1, . . . , m, and that the matrix B(µ) = 2A₀(µ) +

m

P

j=1

A²_j(µ) is constant. Denote hγ(t, x) = exp(γt)kz_c(t, x)k², t>0. Then (31) 06U_γ(t, x)6ϕγ(x;µ(0)) exp(−C(γ)t)ξ(t),

(32) 06hγ(t, x)6 ϕγ(x;µ(0))

δ(γ) exp(−C(γ)t)ξ(t), for any t>0, x∈Rⁿ.

Proof. By hypothesis, the conclusions of Lemma 4 hold. Using Remark 2 we rewrite the integral equation (26) as in (28). Set Φ_γ(t) = exp(−C(γ)t)ξ(t), t>0. Represent the solution U_γ(t, x)of equation (26) as

(33) U_γ(t, x) = Φ_γ(t)

ϕ_γ(x;µ(0))+

Z t 0

Φ⁻¹_γ (s) exp(γs)β_γ(z_c(s, x);µ(s))ds

. Recall that βγ(z;µ) 6 0 for any (z, µ) ∈ Rⁿ×S. Thus, (31) follows. Notice that h_γ(t, x)6 _δ(γ)¹ U_γ(t, x). We easily get (32).

Remark 5. A relaxation of the assumptions made consists of admitting a common eigen-subspace generated by the constant vectors {q₁, . . . , q_d} ⊆Rⁿ, d6n. In this respect, real symmetric(d×d)-matrices Bi(µ)such that

Q^∗A_i(µ)z=B_i(µ)y, i= 0,1, . . . , m, µ∈S, z∈Rⁿ,

will exist, where Q = [q₁, . . . , q_d], y = col(q^∗₁z, . . . , q_d^∗z). The original vector elds gi(z, µ) =ai(µ) +Ai(µ)z will be replaced by

h_i(y, µ) =b_i(µ) +B_i(µ)y, b_i(µ) =Q^∗a_i(µ),

wherebi(µ)∈R^dand the(d×d)-matricesBi(µ) are continuous and bounded.

For each x ∈ Rⁿ, dene a piecewise continuous process {z(t, x);b t > 0}

as the unique solution of the dynamic system (34)







dbz(t) =h₀(z(t);b µ(t))dt+

m

X

j=1

h_j(z(t);b µ(t))dW_j(t), t∈[t_k, t_k+1), bz(t_k) =zb_{_}(t_k) +Q^∗µ(t_k) for anyk>1, bz(0) =Q^∗[x+µ(0)].

Remark 6. The analysis contained in Lemmas 15 has an adequate ver- sion for the piecewise continuous process {bz(t, x); t > 0} and, in particular, the decomposition

z(t, x) =b bz_c(t, x) +Q^∗µ(t)

can be used. Here, the continuous argument bzc(t, x) stands for the unique solution of the linear stochastic system

(35)







dzb_c(t) =h₀(zb_c(t);µ(t))dt+

m

X

j=1

h_j(bz_c(t);µ(t))dW_j(t), t>0, zb_c(0) =Q^∗x.

Problem C. Prove that under suitable conditions imposed on the matrices B_i(µ), we get the asymptotic behaviour of the continuous process {˜z(t, x); t>0}, satisfying

(a)exp(γt)Ek˜z(t, x)k²→0ast→ ∞for eachx∈Rⁿprovidedγ <0and

|γ|>kB₀k+¹₂

m

P

j=1

kB_jk², wherehi(y;µ) =Bi(µ)y and kB_ik= sup

µ∈S

kB_i(µ)k.

(b) For anyγ as above, lim

t↑∞exp(γt)k˜z(t, x)k² = 0 P-a.s.

3. SOLUTIONS OF PROBLEM C

Assume that there existq₁, q₂, . . . , q_d∈Rⁿ,d6n, and symmetricd×d- matrices Bi(µ) such that

Q^∗gi(z;µ) =Bi(µ)y, i∈ {0,1, . . . , m}, µ∈S, z ∈Rⁿ,

where gi(z;µ) =ai(µ) +Ai(µ)z, Q= [q1, q2, . . . , q_d] and y=col(q₁^∗z, q^∗₂z, . . . , q^∗_dz). Associate the linear stochastic system

(36)







˜

z(t) =B0(µ(t))˜z(t)t+

m

X

j=1

Bj(µ(t))˜z(t)dWj(t), t>0,

˜

z(0) =Q^∗x, x∈Rⁿ.

Moreover, assume the matrices B_i(µ) are bounded, continuous and mutually commuting, i.e.,

(a)B_i(µ)B_j(µ) =B_j(µ)B_i(µ)for any µ∈S; (b)B(µ) = 2B0(µ) +

m

P

j=1

B_j²(µ) is a constant matrix.

Theorem 1. Let z(t)˜ be the unique solution of system (36) and γ < 0 such that |γ|>4

h

kB₀k+¹₂Pm

j=1kB_jk²i . Then

(37) lim

t↑∞exp(γt)E

k˜z(t, x)k²

= 0, for anyx∈Rⁿ.

Proof. Let ϕ_γ ∈ P₂(z;µ) be the positive dened quadratic functional ϕ_γ(y;µ) = hR_γ(µ)y, yi such that the conclusions in Lemma 3 hold. Here, z∈Rⁿis replaced by y∈R^d and the equations mentioned are replaced by (38)







(a) γϕ_γ(y;µ) +kyk²+L(ϕ_γ)(y;µ) = 0, ∀(y, µ)∈Rⁿ×S, (b) δ(γ)kyk² 6ϕ_γ(y;µ)6 1

C(γ)kyk², where

L(ϕγ)(y;µ) =h∇_yϕ(y, µ), B0(µ)yi+1 2

m

X

j=1

h∇_y∇_yϕ(y;µ)Bj(µ)y, Bj(µ)yi, C(γ), δ(γ) are positive constants, Rγ(µ) = [|γ|I_d−B(µ)]⁻¹ and B(µ) = 2B0(µ) +Pm

j=1B_j²(µ).

SetUb_γ(t, x) = exp(γt)ϕ_γ(˜z(t, x);µ(t)),t>0,x∈Rⁿ. Then the piecewise continuous process {Ub_γ(t, x);t>0} fulls all the necessary conditions needed

to apply Lemma 4. As a consequence, Ub_γ(t, x) is a solution of the integral equation

(39) Ub_γ(t, x) =ϕ_γ(˜z(0, x);µ(0)) +Db_y^γ(t, x) +

Z t 0

h

−C(γ)Ub_γ(s, x) + exp(γs)βγ(˜z(s, x);µ(s)) i

ds +

m

X

j=1

Z t 0

exp(γs)h∇_yϕγ(˜z(s, x);µ(s)), Bj(µ(s))˜z(s, x)idW_j(s), where







βγ(y;µ) =C(γ)ϕγ(y;µ)− kyk²60, ∀(y, µ)∈Rⁿ×S, Db^γy(t, x) = X

0<tk6t

exp(γtk) [ϕγ(˜z(tk, x);µ(tk))−ϕγ(˜z(tk, x);µ(tk−1))]. Notice that the last term in (39) is a martingale on the ltered probability space {Ω,F,{F_t}_t>0,P}. Taking now the expectation in equation (39) and setting v(t, x) =EUb_γ(t, x), we get

(40) v(t, x) =v(0, x) +D(t, x) + Z t

0

[−C(γ)v(s, x) + exp(γs)β(s, x)] ds, where

D(t, x) = X

0<tk6t

exp(γt_k)E{h[R_γ(µ(t_k))−R_γ(µ(tk−1))] ˜z(t_k, x),z(t˜ _k, x)i}

and

β(s, x) =E[βγ(˜z(s, x);µ(s))]b 60 for anys>0, x∈Rⁿ. Notice that

R_γ(µ) = 1

|γ|

∞

X

k=0

B(µ)

|γ|

k

, whereB(µ) = 2B0(µ)+

m

P

j=1

B²_j(µ). In addition, sinceB(µ)is a constant matrix, we easily get that R_γ(µ) is also constant for any µ∈S and γ <0. Therefore, the piecewise constant component{D(t, x); t>0}vanishes and by a standard integral representation formula we have

(41) 06v(t, x) = exp[−C(γ)t]

v(0, x) + Z t

0

exp(C(γ)s) exp(γs)β(s, x)ds

6exp(−C(γ)t)v(0, x) for anyt>0, x∈Rⁿ. Notice that

exp(γt)Ekz(t, x)k˜ ² 6 1

δ(γ)v(t, x).

Using (41) we get (37).

Theorem 2. Let the assumptions of Theorem1hold. Moreover, letB_j(µ) = θj(µ)Id, j= 1, . . . , m, and assume µj(µ) continuous and bounded.

Dene the exponential martingale (42) ξ(t) = exp



2

m

X

j=1

Z t 0

θb_j(s)dW_j(s)− Z _t

0

(θb_j(s))²ds



, t>0, where θbj(t) =θj(µ(t)). Then

exp(γt)k˜z(t, x)k² 6exp[−C(γ)t]ξ(t)

k˜z(0, x)k² δ(γ)

.

Proof. By hypothesis,Ub_γ(t, x)dened in Theorem 1, satises the integral equation (27). Since the matrix R_γ(µ) is constant, we get that Db^γ_y(t, x) = 0 for any t>0,x∈Rⁿ. Hence, (27) becomes

(43) Ub_γ(t, x) =ϕγ(˜z(0, x);µ(0)) +

Z t 0

h

−C(γ)Ub_γ(s, x) + exp(γs)βγ(˜z(s, x);µ(s)) i

ds +2

m

X

j=1

Z t 0

Ub_γ(s, x)bθ_j(s)dW_j(s).

A standard integral representation formula allows us to write (44) 06Ub_γ(t)6exp[−C(γ)t]ξ(t) [ϕ_γ(˜z(0, x), µ(0))], ∀t>0.

Notice that

exp(γt)k˜z(t, x)k²6 1

δ(γ)Ub_γ(t, x), t>0, x∈Rⁿ and estimate (44) leads us to the conclusion.

Acknowledgement. The authors were supported by Contract 2-CEx06-11-18/2006.

REFERENCES

[1] R. Cont and P. Tankov, Financial Modelling with Jump Processes. Chapman and Hall, 2004.

[2] I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, 2^nd Edition.

Springer Verlag, 1991.

[3] D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance.

Chapman and Hall, 1996.

[4] M. Musiela and M. Rutkovski, Martingale Methods in Financial Modelling. Springer Verlag, 1998.

[5] B. Oksendal, Stochastic Dierential Equations. An Introduction with Applications, 5^th Edition. Springer, 2000.

Received 12 November 2007 B. Iftimie

Academy of Economic Studies Department of Mathematics

Piaµa Roman , 6 010374 Bucharest, Romania

and

I. Molnar, C. Vârsan Romanian Academy

S. Stoilow Institute of Mathematics P.O. Box 1-764

014700 Bucharest, Romania