GRADIENT REPRESENTATION FOR CAD-LAG SOLUTIONS OF SDEs WITH JUMPS

on the occasion of his 70th birthday

GRADIENT REPRESENTATION

FOR CAD-LAG SOLUTIONS OF SDEs WITH JUMPS

B. IFTIMIE, M. MARINESCU and I. MOLNAR

We obtain gradient representations for piecewise continuous cad-lag solutions of SDEs with jumps driven by nonlinear vector fields, under the assumption that the the vectors fields multiplying the jumps are commuting.

AMS 2000 Subject Classification: 60H05, 35A10.

Key words: gradient representation, stochastic differential equations with jumps, piecewise continuous process, commuting vector fields.

1. INTRODUCTION

The study of gradient representations for solutions of differential sys- tems, in the case of finite dimensional Lie algebra generated by a system of vector fields entering the studied equations, was brought in a new light by a monograph of Vˆarsan ([6]). Since then, various results were obtained by a group working with Vˆarsan, such as representations for SPDEs of Cauchy- Kowalewska and parabolic type (see [1]) and continuous integral equations (see [4]).

Recently, the interest was focused on obtaining gradient representations for the solutions of systems of differential equations with jumps. A first step in this direction was made for systems of impulsive ODEs involving a piece- wise constant process (see Chapter 4 of [4]). As a natural extension, the analysis is now focused on getting gradient representations of cad-lag solu- tions of systems of SDEs with jumps appearing under a multiplicative form, in the case of commuting vector fields which multiply the jumps. It can be viewed as a differential-integral representation of the solution using generali- zed processes valued in a dual space [C²_g(R^d;Z)]^∗ of second order differentiable functions, involving three “ingredients”: a 2-dimensional piecewise constant process (p(t),p(t)), a continuous processb bz(t, x) and a smooth mappingG(y;b z)

MATH. REPORTS12(62),3 (2010), 261–276

(a composition of global flows) (see Definition 1). The second view (see Defi- nition 2) is necessary and restricting the class of functions G∈C²_g(R^d;Z) to the finite composition of the global flows generated by the nonlinear vector fields in the impulsive part, we present the results in a more attractive way (see Lemmas 1, 2 and Theorems 1, 2).

2. CAD-LAG SOLUTIONS FOR SDEs CONTAINING JUMPS

Let W(t) = (W₁(t), . . . , W_m(t)), t > 0 be a standard m-dimensional Wiener process over a complete filtered probability space {Ω,{F_t},F, P}, where the filtration {F_t;t>0} satisfied the usual conditions.

Assume that{y(t) = (y₁(t), . . . , y_d(t)); t>0}is an adapted and bounded piecewise constant d-dimensional process, y(0) = 0, P a.s., right continuous and possessing left hand limits (in L²(Ω;R^d)), such that it doesn’t jump at the initial timet= 0 and has finitely many jumps in every finite time interval.

Let {t_j(ω);j >1} the jump times of the process, where tj → ∞,P a.s., and eacht_j is a stopping time with respect to the given filtration. The processy(t) is called a pure jump process. We further assume that the processes (W(t)) and (y(t)) are independent.

We consider the system of SDEs with jumps z(t) =x+

Z t 0

f₀(z(s))ds+

m

X

i=1

Z t 0

f_i(z(s))dW_i(s) (1)

+

d

X

k=1

X

0<s6t

g_k(z(s−))∆y_k(s), t>0,

where we set ∆y_k(s) := y_k(s)− y_k(s−) the jump of y_k(·) at time s > 0 and v(t−) := lim_s%tv(s). It can be solved on each interval determined by successive jump times,

dz(t) =f₀(z(t))dt+

m

X

i=1

f_i(z(t))dW_i(t) (2)

+

d

X

k=1

g_k(z(t_j−))∆y_k(t_j), t∈[t_j, t_j+1).

We assume that each vector field h(z)∈ {f_i(z), g_k(z)} is a Lipschitz continu- ous function.

Proposition 1. Under our assumptions, there exists a unique adapted cad-lag solution {z(t); t>0} of (1), which satisfies

z(t) =z(tj−) +

d

X

k=1

gk(z(tj−))∆y_k(tj) + Z t

tj

f0(z(s))ds (3)

+

m

X

i=1

Z t tj

fi(z(s))dWi(s), ∀t∈[tj, tj+1).

Proof. The SDE z₀(t) =x+

Z t 0

f₀(z₀(s))ds+

m

X

i=1

Z t 0

f_i(z₀(s))dW_i(s)

has a unique continuous and adapted solution z0(t), defined for t∈[0,∞). It follows that z(t) = z₀(t), for eacht∈[0, t₁(ω)) and z(t₁−) =z₀(t₁). Clearly, z(t)1_[0,t1)(t) = z₀(t)1_[0,t1)(t) = z₀(t∧t₁)1_[0,t1)(t), and since t₁ is a stopping time, then the stopped process z0(t∧t1) is adapted and thus z(t)1_[0,t1)(t) is also adapted. We denoted by 1_A the indicator function of the set A.

The jump ofz(t) occurring at the first jump timet₁ of the process y(t) is given by ∆z(t1) =z(t1)−z(t1−) =Pd

k=1gk(z(t1−))∆y_k(t1), and so z(t1) =z0(t1) +

d

X

k=1

g_k(z0(t1))∆y_k(t1).

On the time interval [t1(ω), t2(ω)), z(t) satisfies the SDE z(t) =z(t1) +

Z t t1

f0(z0(s))ds+

m

X

i=1

Z t t1

fi(z0(s))dWi(s).

Denote by Z_t^s,x,t>s, the flow generated by the SDEs Z(t) =x+

Z t s

f0(Z(r))dr+

m

X

i=1

Z t s

fi(Z(r))dWi(r).

It is well known that the random fieldZ_t^s,x(ω) is jointly continuous with respect to (s, x, t), and for each s> 0 and x ∈ Rⁿ, the process Z·^s,x is adapted. We have z(t) =Z_t^t1,z(t1), fort1 6t < t2, and thus

z(t)1_[t1,t2)(t) =Z_t^t1,z(t1)1_[t1,t2)(t) =Z_t^{t1∧t,z(t1∧t)}1_[t1,t2)(t).

The random variable z(t1 ∧t) is F_t-measurable, and it easily follows that Z_t^{t1∧t,z(t1∧t)} has the same property. The indicator function 1_[t1,t2)(t) is clearly F_t-measurable, sincet1 and t2 are stopping times of the filtration{F_t}. This

shows the adaptivity of the process (z(t)) on [t1(ω), t2(ω)). The proof is now complete by an induction argument.

Another way to prove the statement consists in writing, for any t>0, z(t) =

∞

X

j=0

z(t∧tj)1_[tj,tj+1)(t),

with the convention t₀ = 0. It is obvious that the sum in the r.h.s. is finite and each term in the sum is easily seen to be adapted.

Remark 1. The first decomposition formula of the jump process (z(t)) as the sum of a continuous process and a piecewise constant process is easily obtained. Define the piecewise constant component as

z_d(t) := X

16k6d

X

0<tj6t

g_k(z(tj−))∆y_k(tj).

Then z(t) =z_c(t) +z_d(t),t>0, where the continuous component z_c(t) is the unique solution of the SDE

zc(t) =x+ Z t

0

f0(zc(s) +z_d(s))ds+

m

X

i=1

Z t 0

fi(zc(s) +z_d(s))dWi(s).

Next statement is the multidimensional version of the Itˆo-Doeblin for- mula for jump processes (the one dimensional formula is given for instance in Shreve [7], Theorem 11.5.1, page 484).

Proposition2. Let ϕ∈C^1,2p ([0,∞)×Rⁿ), the set of functions ϕwhich are continuously differentiable of first order with respect to tand second order with respect to z, satisfying the polynomial growth conditions

|ϕ(t, z)|,|∂_tϕ(t, z)|,|∂_ziϕ(t, z)|,|∂_z²

izjϕ(t, z)|6CT(1+|z|^N), t∈[0, T], z ∈Rⁿ. Then

ϕ(t, z(t)) =ϕ(0, x) + Z t

0

h

∂tϕ(s, z(s)) +h∇_zϕ(s, z(s)), f0(z(s))i (4)

+1 2

m

X

i=1

hD_z²ϕ(s, z(s))f_i(z(s);y(s), f_i(z(s))ii ds +

m

X

i=1

Z t 0

h∇_zϕ(s, z(s)), fi(z(s))idW_i(s)

+ X

0<tj6t

[ϕ(tj, z(tj))−ϕ(tj, z(tj−))], t>0,

where ∇_zϕ(z, t) and D_z²ϕ(z, t) denote the gradient and the hessian matrix of ϕ with respect to z.

3. DEFINITION OF THE GRADIENT REPRESENTATION FOR CAD-LAG SOLUTIONS

Denote Z = C²(Rⁿ;Rⁿ) and let C²_g(R^d; Z) be the subspace of functions G(y;z) :R^d×Rⁿ → Rⁿ fulfilling G(0;z) = z and such that the map G(y;·) is invertible, for y in some open ball B(0, R) := {y ∈R^d | |y|< R}. We set t0 := 0 and byh(0−) we meanh(0), whereh(t) is a function (process) oft>0.

Definition 1. We say that a function Gb ∈C²_g(R^d; Z), a pair of piecewise constant processes {¯p(t) = (p(t),p(t)) = (p(tb _j),p(tb _j)); t ∈ [t_j, t_j+1), j > 0}

and a continuous process {bz(t, x); t > 0, bz(0, x) = x ∈ Rⁿ} define a gradi- ent representation for the jump process (z(t, x)) if z(t, x) = [¯p(t),bz(t, x)](G),b where [¯p(t),z(t, x)] is a piecewise continuous process valued in the dual spaceb [C²_g(R^d;Z)]^∗, for each ω∈Ω, s.t. fort∈[t_j, t_j+1) we have

[¯p(tj),bz(t, x)](G) :=G(p(tb j−);bz(t, x)) +∇_yG(p(tb j−);bz(tj, x))(p(tj)−p(tj−)).

Remark 2. The above given definition is too abstract for the construction of the gradient representation and we mention the real constraints which are contained in it. First of all, notice that in the analysis which follows only the particular Gb∈C²_g(R^d; Z),

(5) G(y;b z) :=G1(y1)◦ · · · ◦Gd(yd)(z),

is used, where G_i(y_i)(z) stands for the global flow generated by the vector field gi ∈ C²(Rⁿ;Rⁿ). In addition, the basic equation in Definition 1 can be separated into two parts.

Definition 2. We say that a mapping Gb ∈ C²_g(R^d; Z) of the form given in (5), a pair of piecewise constant processes {(p(t),p(t)) = (p(tb _j),p(tb _j)) ∈ B(0, R)×B(0, R);t ∈ [tj, tj+1)} and a continuous process {z(t, x)}b define a gradient representation for {z(t, x)} if

(6)















z(t) =G(b p(tb j−);z(t)) +b

d

X

k=1

gk(G(b p(tb j−);bz(tj)))∆yk(tj), t∈[tj, tj+1);

∇_yG(b p(tb j−);bz(tj))(p(tj)−p(tj−)) =

d

X

k=1

g_k(G(b p(tb j−);bz(tj)))∆y_k(tj).

Remark 3. There are several constraints which must be satisfied by the processes (p(t),p(t)) andb {z(t, x)}.b

In addition, the second equations (6) will lead us to the solution (p(t_j)− p(t_j−)) provided p(tb _j−) and z(tb _j) are known and the smooth mapping Gb ∈ C²_g(R^d;Z) (see (5)) has the property (A1): there exist smooth vector fields qk(p), qk(0) =ek, k={1, . . . , d}, such that ∇_yG(p;b z)qk(p) =gk(G(p;b z)), for any p∈R^d, z∈Rⁿ, q₁(p) =e₁ ∈R^d.

Assuming(A1),p(t) is found according to (7) p(tj)−p(tj−) =

d

X

k=1

qk(p(tb j−))(y_k(tj)−yk(tj−)), and we imposep(0) = 0, p(0) = 0.b

Since the gradient representation formula is nonsingular only locally in the case of non-commuting vector fields generating some finite dimensional Lie algebra, we shall assume here that the vector fieldsg1, . . . , gdcommute and in this case the vectors q_j(p) are given by the vectors of the canonical basis in R^d, and thusp(t) =y(t).

We rewrite the first equations in (6), fort=t_j z(t_j−) +

d

X

k=1

g_k(z(t_j−))∆y_k(t_j) = (8)

=G(b p(tb j−);bz(tj)) +

d

X

k=1

g_k(G(b p(tb j−);bz(tj)))∆y_k(tj).

These equations are solved provided z(tb j) is known and we need to find p(tb j−) =p(tb j−1) such that

(9) G(b p(tb j−1);bz(t_j)) =z(t_j−).

This algorithm requires a precise order in solving the equations.

Step 1. Find the values of bz(t) for t ∈[0, t₁) such that it fulfills (6) for j = 0. We get z(t) = bz(t), t∈ [0, t1) (seep(0−) =b bp(0) = 0, G(0;b z) = z and y(0−) =y(0) = 0).

Next, find{z(t);b t∈[t₁, t₂)} s.t.

(10) z(t) =bz(t) +

d

X

k=1

g_k(bz(t1))y_k(t1), t∈[t1, t2).

Since we are looking for a continuous processbz(t), its value at time t1 is given by bz(t₁) = bz(t₁−). Actually, (10) is the first equation we have to solve in order to get a solution of (9) for j= 2,p(tb ₂−) =p(tb ₁).

4. GRADIENT REPRESENTATION IN THE CASE OF COMMUTING VECTOR FIELDSg1, . . . , gd

We make some additional assumptions. We suppose that eachg_ktogether with its first and second order partial derivatives are bounded and that y(t) has finite total variation, i.e.,

(11)











|g_k(z)|,|∂_zig_k(z)|,|∂²_zizjg_k(z)|6C₂, ∀z∈Rⁿ, Vy :=

∞

X

j=0

|y(t_j+1−y(tj))|6ρ.

In addition, assume that

(12) {g₁, . . . , g_d} ⊂C²_b(Rⁿ;Rⁿ) are commuting, i.e., all the Lie brackets [g_i, g_j] = 0, where

[gi, gj](z) :=∇_zgi(z)gj(z)− ∇_zgj(z)gi(z).

Associate the reduced system with jumps

(13)











dh(t;z) =b

d

X

k=1

gk(h(t−;bz))dyk(t), t>0, h(0,z) =b bz,

for which the solution of (13) is given by the piecewise constant process (14) h(t;z) =b h(tj;z) =b h(tj−;z) +b

d

X

k=1

gk(h(tj−;z))∆yb k(tj), for t∈[tj, tj+1). We can also write

h(t;z) =b zb+

d

X

k=1

X

0<s6t

g_k(h(s−;bz))∆y_k(s).

It may be easily proved that a gradient representation is valid forbh(t). In this respect consider Gb∈C²_g(R^d; Z) given by

(15) G(y;b z) :=G1(y1)◦ · · · ◦Gd(yd)(z), y= (y1, . . . , yd)∈R^d, G(0, z) =b z,

where Gk(yk)(z), yk ∈ R, z ∈ Rⁿ, is the global flow generated by gk. Set H(y;b ·) := [G(y;b ·)]⁻¹ =G(−y;b ·) and it is true that

(16) ∇_yG(y;b z) = (g₁(G(y;b z)), . . . , g_d(G(y;b z))), ∀y∈R^d, z∈Rⁿ. The assumption (A1) is satisfied when (12) is assumed. Using (16), we get q_k(y) = e_k, where {e₁, . . . , e_d} is the canonical basis of R^d. It implies that (p(t)) satisfies

(17) p(tj)−p(tj−) =y(tj)−y(tj−), p(t) =y(t), t∈[0, t1).

Therefore, p(t) = y(t), for any t > 0, and considering a continuous pro- cess {bz(t,bz) = bz;t > 0}, we construct a piecewise constant process {y(t) =b y(tb j);t∈[tj, tj+1)},by(t) = 0,t∈[0, t1), such that

(18) h(tj;bz) =G(b y(tb j−);z) +b

d

X

k=1

g_k(G(b y(tb j−);z))∆yb _k(tj), j >0.

The unique solution y(tb j−) =y(tb j−1) of (18) is obtained by solving the cor- responding equations

(19) G(b y(tb j−1);bz) =h(tj−1;z),b j >1,

which imply y(0) = 0 andb G(b y(tb j);z) =b h(tj;z), for anyb j>1. We write h(t1;z) =b bz+

m

X

k=1

gk(z)yb k(t1) and

G(b y(tb 1);z) =b bz+ Z 1

0

∂yG(θb by(t1);z)dθb by(t1)

=bz+

d

X

k=1

Z 1 0

g_k(G(θb by(t₁);z))dθb yb_k(t₁)

and we assume that each g_k ∈ C²_b(Rⁿ;Rⁿ) has the following structure which agrees with (11) and (12),

(20) g_k(z) =α_k(z)b_k.

Here {b₁, . . . , b_d} ⊂ R^d are fixed and α_k(·) ∈ C²_b(Rⁿ;R) satisfying 0 < δ 6 αk(z)6M agree with the commuting property in (12), i.e.,

αj(z)h∂_zαi(z), bji −αi(z)h∂_zαj(z), bii= 0, z∈Rⁿ, i, j= 1, . . . , d.

It follows h(t1;z) =b bz+P_d

k=1(α_k(bz)y_k(t1))b_k and we solve the equation (21) G(b y(tb 1);bz) =h(t1;z),b ∀bz∈Rⁿ,

using a nonlinear contractive mapping. Rewrite (22) G(b y;b bz) =zb+

d

X

k=1

(α_k(z)byb_k)b_k+

d

X

k=1

α_k(bz)yb_k d

X

j=1

β_kj(y;b bz)ybj

b_k, where

β_kj(by;z) :=b 1 α_k(bz)

Z 1 0

θdθ Z 1

0

hα_j(G(θb ₁θby;bz))∂_zα_k(G(θb ₁θy;b z)), bb _jidθ₁ is a continuous and bounded function, fulfilling

(23)

(|β_kj(y;b bz)|6C,b by∈R^d,zb∈Rⁿ,

|β_kj(yb⁰;z)b −β_kj(by⁰⁰)|6L|byb⁰−yb⁰⁰|, ∀yb⁰,yb⁰⁰∈R^d,zb∈Rⁿ.

SetL= max(L,b C),b ρ₁ =ρ^M_δ andL_ρ1 := (1 + 2ρ₁)L. Chooseρ >0 sufficiently small such that

(24) γ := 4ρ1Lρ1 6 1 2,

d

X

j=1

|β_kj(by;z)bybj|6 1

2, ∀by∈B(0,2ρ1),zb∈Rⁿ. Define a map T(y) :b B(0,2ρ₁)→C(Rⁿ;R^d) by

(25) T_k(y)(z) = 1 +b

d

X

j=1

β_kj(by;z)yb_j

and associate the nonlinear operator U(y) :b B(0,2ρ1)→C(B(0, ρ)×R²ⁿ;R^d), (26) Uk(by)(y, z) :=yk[Tk(by)(z2)]⁻¹αk(z1)

α_k(z2), z= (z1, z2)∈R²ⁿ. Using (23) and the second inequality in (24) we notice that

(27)











|[T_k(y)(zb 2)]⁻¹|62, ∀yb∈B(0,2ρ1), z2 ∈Rⁿ,

|[T_k(yb⁰)(z2)]⁻¹−[T_k(yb⁰⁰)(z2)]⁻¹|64|T_k(by⁰)(z2)−T_k(yb⁰⁰)(z2)|

64(1 + 2ρ₁)L|yb⁰−yb⁰⁰|= 4L_ρ1|yb⁰−by⁰⁰|, ∀yb⁰,by⁰⁰∈B(0,2ρ₁).

Using (27) and (24) we get that{U(by); by∈B(0,2ρ1)}is a Lipschitz continuous operator with a Lipschitz constant γ and

|U_k(yb⁰)(y, z)−U_k(yb⁰⁰)(y, z)|6|y_k|M

δ |[T_k(yb⁰)(z₂)]⁻¹−[T_k(yb⁰⁰)(z₂)]⁻¹| (28)

6ρ₁(4L_ρ1)|yb⁰−by⁰⁰|=γ|yb⁰−yb⁰⁰|,

∀by⁰,by⁰⁰ ∈B(0,2ρ₁), y ∈ B(0, ρ). In addition, equation (21) for the unknown y(tb 1) is replaced by the functional nonlinear equations

(29) ybk=Uk(by), for by∈Yb := C(B(0, ρ)×R²ⁿ;B(0,2ρ1)).

The unique solution of (29) is constructed as the limit of the Cauchy sequence {by^j}in the complete metric space Yb,

(30)



















yb⁰={0}, yb¹:=U(0) =

y1

α₁(z₁)

α1(z2), . . . , yd

α_d(z₁) α_d(z2)

: (z1, z2)∈R²ⁿ

,

yb^j+1 :=U(yb^j), kby^j+1(y)k6kby¹(y)k ^j

X

i=0

γⁱ

6|y|M δ

1

1−γ 6 2M γ |y|, for any y ∈B(0, ρ), whereky(y)kb := sup_z∈R2n|y(y, z)|.b

Using the metric d(by⁰,yb⁰⁰) := sup(y,z)∈B(0,ρ)×R²ⁿ|yb⁰(y, z)−yb⁰⁰(y, z)|, an induction argument leads us to

|by^j+1(y, z)−yb^j(y, z)|6|yb¹(y, z)|γ^j. As a consequence the estimate

kby^j+1(y)k6kby¹(y)k(1 +γ+· · ·+γ^j)6 M δ

1

1−γ|y|6 2M δ |y|

is valid. Denote f(y, z) = lim

j→∞yb^j(y, z), which is an element of Yb. The above computations are restated as

Lemma 1. Consider the equations (31) G(b y;b z₂) =z₂+

d

X

k=1

g_k(z₁)y_k.

There exists a unique continuous and bounded function yb = f(y, z₁, z₂) : B(0, ρ)×Rⁿ×Rⁿ→B(0,2ρ₁) satisfying (31) and

kf(y)k= sup

z1,z2∈Rⁿ

|f(y, z1, z2)|6 2M

γ |y|62ρ1.

Equation (21) is a particular case of equation (31) in Lemma 1 and define (32) y(tb 1) =f(y(t1), z1, z2), y(t1)∈B(0, ρ).

Using an induction argument we prove that equations (19) can be solved. In this respect, assume that y(tb j−1) =y(tb j−) fulfills

(33) G(b y(tb j−);bz) =h(tj−;z),b ∀bz∈Rⁿ,

where |y(tb k)−y(tb k−)|6 ^2M_δ |y(t_k)−y(tk−)|, for any 16k6j−1 and j>2 is fixed. Therefore, the equations

(34) G(b by(tj);bz) =h(tj;bz)

can be solved and |∆y(tb j)|:=|y(tb j)−y(tb j−)|6 ^2M_δ |∆y(t_j)|. Rewrite (34) as G(b y(tb _j);z) =b G((b by(t_j)−y(tb _j−));h(tj−1,z))b

(35)

=h(tj−1,bz) +

d

X

k=1

gk(h(tj−1,z))∆yb k(tj).

Now replace equation (35) for the unknownby=y(tb j)−y(tb j−) andh(tj−1;bz) :=

bh by

(36) G(b by;bh) =bh+

d

X

k=1

gk(bh)∆yk(tj), bh∈Rⁿ. Using hypothesis (20) we construct the unique solution of (36) (37) by=f(∆y(tj),bh,bh),

where the continuous and bounded functionf(y, z₁, z₂) :B(0, ρ)×Rⁿ×Rⁿ→ B(0,2ρ1) is the unique solution of the functional equation (29).

Thenby(tj) =y(tb j−1) +f(∆y(tj), h(tj−1;z), h(tb j−1;z)) and it satisfies theb equation (34). The above computations allow us to state

Theorem1. There exists a piecewise constant and bounded process{y(t) =b y(tb _j);t∈[t_j, t_j+1)} withy(0) = 0, such thatb

h(t_j;z) =b G(b y(tb _j−);z) +b

d

X

k=1

g_k(G(b y(tb _j−);bz))(y_k(t_j)−y_k(t_j−)).

Remark 4. This theorem provides the gradient representation of the piecewise continuous process (h(t;bz)).

Step 2. The first significant equation for the unknown y(tb j−) ∈ R^d ap- pears for j = 2, by(t₂−) = y(tb ₁), and using bz(t₂) = z(tb ₂−). We write it for j = 2 andt=t2

(38) z(t₂) =G(b by(t₁);z(tb ₂)) +

d

X

k=1

g_k(G(b y(tb ₁);bz(t₂)))(y_k(t₂)−y_k(t₂−)), where

(39) z(t2) =z(t2−) +

d

X

k=1

g_k(z(t2−))(y_k(t2)−y_k(t2−)).

Both equations (38) and (39) are fulfilled provided by(t1) is found such that

(40)











G(b y(tb ₁);bz(t₂)) =z₁(t₂−), z(t₂) =bz(t₂) +

d

X

k=1

g_k(bz(t₁))y_k(t₁) =bz(t₂) +

d

X

k=1

g_k(bz₁(t₁))y_k(t₁).

Notice that (40) are solved if we find first ybthe solution of the equation (41) G(b y;b z2) =z2+

d

X

k=1

gk(z1)yk(t1), y(t1)∈B(0, ρ), z1, z2 ∈Rⁿ. By hypothesis, equation (41) fulfills the assumptions of Lemma 1 and let yb = f(y, z1, z2) : B(0, ρ)×Rⁿ ×Rⁿ → R^d be the unique continuous and bounded solution for (41), such that f(y, z₁, z₂) ∈ B(0,2ρ₁), ∀y ∈ B(0, ρ), z1, z2 ∈Rⁿ and kf(y)k6 ^2M_δ |y|,∀y∈B(0, ρ).

Definey(tb 1) :=f(y(t1),bz(t1),z(tb 2)), which satisfies |y(tb 1)|62ρ1. Step 3. We find{z(t);b t∈[t₂, t₃)} such thatz(tb ₂) =z(tb ₂−) fulfilling

(42)











z(t) =G(b y(tb ₁);bz(t)) +

d

X

k=1

g_k(G(b by(t₁);z(tb ₂)))(y_k(t₂)−y_k(t₂−)),

z(t) =b G(−b y(tb ₁); ¯z(t)),

for t∈[t₂, t₃), where G(−y;b ·) = [G(y;b ·)]⁻¹ and

¯

z(t) :=z(t)−

d

X

k=1

g_k(G(b y(tb ₁);bz(t₂)))(y_k(t₂)−y_k(t₂−)).

Step 4. We are in position to stipulate what is necessary for getting the representation formula

(43) z(t) =G(b by(t_j−);bz(t)) +

d

X

k=1

g_k(G(b y(tb _j−);bz(t_j)))∆y_k(t_j), t∈[t_j, t_j+1), by using an induction argument.

For some j > 2 the algorithm requires as known the following items:

(bz(t, x)), for t ∈ [0, tj+1) and {y(t) =b y(tb _k);t ∈ [t_k, t_k+1),0 6 k 6 j−1}, fulfilling

(44) |∆y(tb _k)|6 2M

δ |∆y(t_k)|, 06k6j−1.

In particular, the equations

(45) G(b by(t_k−);bz(t_k)) =z(t_k−), 06k6j

are fulfilled. Assuming (44) and (45), we must find ∆by(tj) = by(tj)−by(tj−) and a continuous process{bz(t); t∈[t_j+1, t_j+2),z(tb _j+1) =bz(t_j+1−)}, such that

(46) |∆by(tj)|6 2M

δ |∆y(t_j)|, and

(47) G(b y(tb _j);bz(t_j+1)) =z(t_j+1−).

Using the equality y(tb j+1−) = y(tb j) = y(tb j−) + ∆y(tb j), define bzj+1(t) for t∈[tj+1, tj+2) as the solution of the equation

(48) z(t) =G(b by(t_j);bz(t)) +

d

X

k=1

g_k(G(b y(tb _j);z(tb _j+1)))(y_k(t_j+1)−y_k(t_j+1−)), for t∈[t_j+1, t_j+2). We are in position to state

Lemma 2. Consider the piecewise constant process {y(t) =b y(tb _k); t ∈ [t_k, t_k+1), 0 6 k 6 j−1} and the continuous process {bz(t, x); t ∈ [t_k, t_k+1), 06k6j}such that they satisfy (43),(44)and(45). Then there existsy(tb j) = y(tb _j+1−) such that the equations (46) and (47) are satisfied and {z(t);b t ∈ [tj+1, tj+2)} is defined in (48). In addition, |∆y(tb j)| 6 ^2M_δ |∆y(t_j)|, for any j >1 and V_yb:=P∞

j=1|∆y(tb j)|6 ^2M_δ ρ.

Proof. Using the group property ofG(y;b ·),y∈R^d, we get (49) G(b by(tj);z) =G(∆b y(tb j);G(b y(tb j−);z)), ∀z∈Rⁿ. On the other hand,

(50) z(t_j+1−) =G(b y(tb _j−);bz(t_j+1)) +

d

X

k=1

g_k(G(b y(tb _j−);z(tb _j)))∆y_k(t_j).

In order to solve (47), we use (49) and (50) and rewrite (47) as (51) G(b y(tb j);zb2) =bz2+

d

X

k=1

gk(bz1)∆yk(tj), where bz2 :=G(b y(tb j−);bz(tj+1)) andzb1 :=G(b by(tj−);z(tb j)).

By hypothesis, the functional equations (51) fulfill the conditions of ap- plicability of Lemma 1 and letby=f(y, z₁, z₂) :B(0, ρ)×Rⁿ×Rⁿ→B(0,2ρ₁) be the unique continuous and bounded function satisfying

(52) G(b y;b z₂) =z₂+

d

X

k=1

g_k(z₁)y_k, ∀z₁, z₂∈Rⁿ, y∈B(0, ρ).

Define ∆by(tj) :=f(∆y(tj),bz1,bz2). In addition, (53) |∆by(t_j)|6kf(∆y(t_j)k= sup

z1,z2∈Rⁿ

|f(∆y(t_j),zb₁,zb₂)|6 2M

δ |∆y(t_j)|.

Set

(54) z(t) =¯ z(t)−

d

X

k=1

g_k(z(tj−))∆y_k(tj), t∈[tj, tj+1), j >0.

Taking into account the equation (43) and using the inverse mapping [G(y;b ·)]⁻¹

=G(−y;b ·), we write

(55) z(t) =b G(−b y(tb j−1); ¯z(t)), t∈[t_j, t_j+1),

where (¯z(t);t>0) is a continuous process fulfilling the system of SDEs

(56)























d¯z(t) =f0(¯zj(t) +

d

X

k=1

g_k(z(tj−))∆y_k(tj)dt

+

m

X

i=1

f_i(¯z_j(t) +

d

X

k=1

g_k(z(t_j−))∆y_k(t_j))dW_i(t), t∈[t_j, t_j+1),

¯

z(tj) =z(tj−).

Applying the standard rule of stochastic derivation for the function ϕ(y, z) = G(−y, zb ), and the continuous process{(−y(tb j−1),z(t));¯ t∈[tj, tj+1)}, we get

(57)







dz(t) =b h₀(z(t);b λ(t, x))dtb +

m

X

i=1

h_i(bz(t);bλ(t, x))dW_i(t), t∈[t_j, t_j+1), z(tb j) =bz(tj−).

In addition, the vector fieldshi(z;b bλ(t, x))∈Rⁿ,bzj ∈Rⁿ,i= 0,1, . . . , m, where bλ(t, x)) := (by(t_j−), y(t), z(t, x)), are given by

(58)



























h_i(z;b bλ(t, x)) =∇_zG(−b y(tb j−1); ¯z(t))f_i(G(b y(tb j−1),bz)) +

d

X

k=1

gk(z(tj−))∆y_k(tj), 16i6m, h0(z;b bλ(t, x)) =∇_zG(−b y(tb j−1),z(t))f¯ 0(G(b y(tb j−1),bz))

+

d

X

k=1

g_k(z(t_j−))∆y_k(t_j) +ch₀(bλ(t, x)), for t∈[tj, tj+1). Here bh0= (bh¹₀, . . . ,bhⁿ₀) is defined as

(59) bh^k₀ := 1 2

m

X

i=1

hD²_zGb^k(−y(tb j−1); ¯z(t))f_i(z(t)), f_i(z(t))i.

We are now in a position to state the main result of this paper

Theorem2. There exists a piecewise constant process {by(t) =y(tb _j); t∈ [t_j, t_j+1)}and a continuous process{z(t, x);b t>0}such that the jump process (z(t, x)) has the gradient representation(according to Definition 2)

(60) z(t, x) =G(b by(tj−1);z(t, x)) +b

d

X

k=1

g_k(G(b y(tb j−1);bz(t_j;x)))∆y_k(t_j), for any t∈ [t_j, t_j+1), j > 0. In addition, the process {z(t, x);b t> 0} fulfills the SDE (57).

Final comment. In a future work we intend to obtain a gradient repre- sentation for the solution of (1) in the “non-commuting” case, when the vector fields g1, . . . , g_d are mutually in involution over R, i.e., when the Lie bracket [gi, gk] =αgi+βgk, for some constants α, β depending ongi andgk.

REFERENCES

[1] B. Iftimie and C. Vˆarsan,Evolution systems of Cauchy-Kowalewska and parabolic type with stochastic perturbations. Math. Reports10(60)(2008),3, 213–238.

[2] B. Iftimie, I. Molnar and C. Vˆarsan, Solutions of some elliptic equations associated with a piecewise continuous process. Rev. Roumaine Math. Pures Appl. 53(2008), 4, 323–338.

[3] I. Karatzas and S. Shreve,Brownian Motion and Stochastic Calculus. Springer Verlag, New York, 1988.

[4] M. Marinescu, Reprezent˘ari gradient pentru ecuat¸ii integrale continue ¸si cu salturi.

Ph.D. Thesis, IMAR, 2008.

[5] M. Marinescu, I. Molnar and C. Vˆarsan,Gradient representation and positive cad-lag solutions for jump-differential equations. Preprint nr. 3, IMAR, 2008.

[6] C. Vˆarsan,Applications of Lie Algebras to Hyperbolic and Stochastic Differential Equa- tions. Kluwer, 1999.

[7] S. Shreve,Stochastic Calculus for FinanceII. Springer Finance Series, 2004.

Received 18 May 2009 B. Iftimie, M. Marinescu

Academy of Economic Studies Department of Mathematics

6 Piat¸a Roman˘a 010374 Bucharest, Romania

[email protected] and I. Molnar

“Simion Stoilow” Institute of Mathematics of the Romanian Academy

PO Box 1-764 014700 Bucharest, Romania