GRADIENT REPRESENTATION FOR CAD-LAG SOLUTIONS OF ...

ˆ Dedicated to Dr. Constantin VARSAN 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 systems, 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 CauchyKowalewska 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 piecewise constant process (see Chapter 4 of [4]). As a natural extension, the analysis is now focused on getting gradient representations of cad-lag solutions 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 generalized processes valued in a dual space [C2g (Rd ; Z)]∗ of second order differentiable functions, involving three “ingredients”: a 2-dimensional piecewise constant b z) process (p(t), pb(t)), a continuous process zb(t, x) and a smooth mapping G(y; MATH. REPORTS 12(62), 3 (2010), 261–276

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(a composition of global flows) (see Definition 1). The second view (see Definition 2) is necessary and restricting the class of functions G ∈ C2g (Rd ; 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), . . . , Wm (t)), t > 0 be a standard m-dimensional Wiener process over a complete filtered probability space {Ω, {Ft }, F, P }, where the filtration {Ft ; t > 0} satisfied the usual conditions. Assume that {y(t) = (y1 (t), . . . , yd (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; Rd )), such that it doesn’t jump at the initial time t = 0 and has finitely many jumps in every finite time interval. Let {tj (ω); j > 1} the jump times of the process, where tj → ∞, P a.s., and each tj is a stopping time with respect to the given filtration. The process y(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 (1)

z(t) = x +

t

f0 (z(s))ds + 0

+

d X X

m Z X i=1

t

fi (z(s))dWi (s)

0

gk (z(s−))∆yk (s),

t > 0,

k=1 0 0 and v(t−) := lims%t v(s). It can be solved on each interval determined by successive jump times, (2)

dz(t) = f0 (z(t))dt +

m X

fi (z(t))dWi (t)

i=1

+

d X

gk (z(tj −))∆yk (tj ),

t ∈ [tj , tj+1 ).

k=1

We assume that each vector field h(z) ∈ {fi (z), gk (z)} is a Lipschitz continuous function.

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Proposition 1. Under our assumptions, there exists a unique adapted cad-lag solution {z(t); t > 0} of (1), which satisfies Z t d X z(t) = z(tj −) + gk (z(tj −))∆yk (tj ) + (3) f0 (z(s))ds tj

k=1

+

m Z X i=1

t

∀t ∈ [tj , tj+1 ).

fi (z(s))dWi (s),

tj

Proof. The SDE Z z0 (t) = x +

t

f0 (z0 (s))ds + 0

m Z X i=1

t

fi (z0 (s))dWi (s)

0

has a unique continuous and adapted solution z0 (t), defined for t ∈ [0, ∞). It follows that z(t) = z0 (t), for each t ∈ [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 1A the indicator function of the set A. The jump of z(t) occurring at the first jump time t1 of the process y(t) P is given by ∆z(t1 ) = z(t1 ) − z(t1 −) = dk=1 gk (z(t1 −))∆yk (t1 ), and so z(t1 ) = z0 (t1 ) +

d X

gk (z0 (t1 ))∆yk (t1 ).

k=1

On the time interval [t1 (ω), t2 (ω)), z(t) satisfies the SDE Z t m Z t X z(t) = z(t1 ) + f0 (z0 (s))ds + fi (z0 (s))dWi (s). t1

Denote by

t1

i=1

Zts,x ,

t > s, the flow generated by the SDEs Z t m Z t X Z(t) = x + f0 (Z(r))dr + fi (Z(r))dWi (r). s

i=1

s

It is well known that the random field Zts,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 t ,z(t ) have z(t) = Zt 1 1 , for t1 6 t < t2 , and thus t ,z(t1 )

z(t)1[t1 ,t2 ) (t) = Zt 1

t ∧t,z(t1 ∧t)

1[t1 ,t2 ) (t) = Zt 1

1[t1 ,t2 ) (t).

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

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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, ∞ X z(t) = z(t ∧ tj )1[tj ,tj+1 ) (t), j=0

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 X X zd (t) := gk (z(tj −))∆yk (tj ). 16k6d 0 0, where the continuous component zc (t) is the unique solution of the SDE Z t m Z t X zc (t) = x + f0 (zc (s) + zd (s))ds + fi (zc (s) + zd (s))dWi (s). 0

i=1

0

Next statement is the multidimensional version of the Itˆo-Doeblin formula for jump processes (the one dimensional formula is given for instance in Shreve [7], Theorem 11.5.1, page 484). n Proposition 2. Let ϕ ∈ C1,2 p ([0, ∞) × R ), the set of functions ϕ which are continuously differentiable of first order with respect to t and second order with respect to z, satisfying the polynomial growth conditions

|ϕ(t, z)|, |∂t ϕ(t, z)|, |∂zi ϕ(t, z)|, |∂z2i zj ϕ(t, z)| 6 CT (1+|z|N ), t ∈ [0, T ], z ∈ Rn . Then (4)

ϕ(t, z(t)) = ϕ(0, x) +

Z th

∂t ϕ(s, z(s)) + h∇z ϕ(s, z(s)), f0 (z(s))i

0 m i 1X 2 hDz ϕ(s, z(s))fi (z(s); y(s), fi (z(s))i ds 2 i=1 m Z t X + h∇z ϕ(s, z(s)), fi (z(s))idWi (s)

+

i=1

+

X

0

[ϕ(tj , z(tj )) − ϕ(tj , z(tj −))],

0 0,

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Gradient representation for cad-lag solutions of SDEs with jumps

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where ∇z ϕ(z, t) and Dz2 ϕ(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 C2g (Rd ; Z) be the subspace of functions G(y; z) : Rd × 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 ∈ Rd | |y| < R}. We set t0 := 0 and by h(0−) we mean h(0), where h(t) is a function (process) of t > 0. b ∈ C2g (Rd ; Z), a pair of piecewise Definition 1. We say that a function G constant processes {¯ p(t) = (p(t), pb(t)) = (p(tj ), pb(tj )); t ∈ [tj , tj+1 ), j > 0} and a continuous process {b z (t, x); t > 0, zb(0, x) = x ∈ Rn } define a gradib ent representation for the jump process (z(t, x)) if z(t, x) = [¯ p(t), zb(t, x)](G), where [¯ p(t), zb(t, x)] is a piecewise continuous process valued in the dual space [C2g (Rd ; Z)]∗ , for each ω ∈ Ω, s.t. for t ∈ [tj , tj+1 ) we have [¯ p(tj ), zb(t, x)](G) := G(b p(tj −); zb(t, x)) + ∇y G(b p(tj −); zb(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 b ∈ C2 (Rd ; Z), particular G g (5)

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

is used, where Gi (yi )(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. b ∈ C2g (Rd ; Z) of the form given Definition 2. We say that a mapping G in (5), a pair of piecewise constant processes {(p(t), pb(t)) = (p(tj ), pb(tj )) ∈ B(0, R) × B(0, R); t ∈ [tj , tj+1 )} and a continuous process {b z (t, x)} define a gradient representation for {z(t, x)} if  d X    b b p(tj −); zb(tj )))∆yk (tj ), t ∈ [tj , tj+1 ); z(t) = G(b p(tj −); zb(t)) + gk (G(b    k=1 (6) d  X   b b p(tj −); zb(tj )))∆yk (tj ).  p(tj −); zb(tj ))(p(tj ) − p(tj −)) = gk (G(b   ∇y G(b k=1

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Remark 3. There are several constraints which must be satisfied by the processes (p(t), pb(t)) and {b z (t, x)}. In addition, the second equations (6) will lead us to the solution (p(tj ) − b ∈ p(tj −)) provided pb(tj −) and zb(tj ) are known and the smooth mapping G C2g (Rd ; Z) (see (5)) has the property (A1): there exist smooth vector fields b z)qk (p) = gk (G(p; b z)), for qk (p), qk (0) = ek , k = {1, . . . , d}, such that ∇y G(p; d n d any p ∈ R , z ∈ R , q1 (p) = e1 ∈ R . Assuming (A1), p(t) is found according to (7)

p(tj ) − p(tj −) =

d X

qk (b p(tj −))(yk (tj ) − yk (tj −)),

k=1

and we impose p(0) = 0, pb(0) = 0. 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 fields g1 , . . . , gd commute and in this case the vectors qj (p) are given by the vectors of the canonical basis in Rd , and thus p(t) = y(t). We rewrite the first equations in (6), for t = tj (8)

z(tj −) +

d X

gk (z(tj −))∆yk (tj ) =

k=1

b p(tj −); zb(tj )) + = G(b

d X

b p(tj −); zb(tj )))∆yk (tj ). gk (G(b

k=1

These equations are solved provided zb(tj ) is known and we need to find pb(tj −) = pb(tj−1 ) such that (9)

b p(tj−1 ); zb(tj )) = z(tj −). G(b

This algorithm requires a precise order in solving the equations. Step 1. Find the values of zb(t) for t ∈ [0, t1 ) such that it fulfills (6) for b z) = z and j = 0. We get z(t) = zb(t), t ∈ [0, t1 ) (see pb(0−) = pb(0) = 0, G(0; y(0−) = y(0) = 0). Next, find {b z (t); t ∈ [t1 , t2 )} s.t. (10)

z(t) = zb(t) +

d X k=1

gk (b z (t1 ))yk (t1 ),

t ∈ [t1 , t2 ).

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Gradient representation for cad-lag solutions of SDEs with jumps

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Since we are looking for a continuous process zb(t), its value at time t1 is given by zb(t1 ) = zb(t1 −). Actually, (10) is the first equation we have to solve in order to get a solution of (9) for j = 2, pb(t2 −) = pb(t1 ). 4. GRADIENT REPRESENTATION IN THE CASE OF COMMUTING VECTOR FIELDS g1 , . . . , gd

We make some additional assumptions. We suppose that each gk together with its first and second order partial derivatives are bounded and that y(t) has finite total variation, i.e.,  |g (z)|, |∂zi gk (z)|, |∂z2i zj gk (z)| 6 C2 , ∀z ∈ Rn ,    k ∞ X (11)  V := |y(tj+1 − y(tj ))| 6 ρ.   y j=0

In addition, assume that (12)

{g1 , . . . , gd } ⊂ C2b (Rn ; Rn ) are commuting,

i.e., all the Lie brackets [gi , gj ] = 0, where [gi , gj ](z) := ∇z gi (z)gj (z) − ∇z gj (z)gi (z). Associate the reduced system with jumps  d  X   dh(t; zb) = gk (h(t−; zb))dyk (t), (13) k=1    h(0, zb) = zb,

t > 0,

for which the solution of (13) is given by the piecewise constant process (14)

h(t; zb) = h(tj ; zb) = h(tj −; zb) +

d X

gk (h(tj −; zb))∆yk (tj ),

k=1

for t ∈ [tj , tj+1 ). We can also write h(t; zb) = zb +

d X X

gk (h(s−; zb))∆yk (s).

k=1 0 0, and considering a continuous process {b z (t, zb) = zb; t > 0}, we construct a piecewise constant process {b y (t) = yb(tj ); t ∈ [tj , tj+1 )}, yb(t) = 0, t ∈ [0, t1 ), such that (18)

b y (tj −); zb) + h(tj ; zb) = G(b

d X

b y (tj −); zb))∆yk (tj ), gk (G(b

j > 0.

k=1

The unique solution yb(tj −) = yb(tj−1 ) of (18) is obtained by solving the corresponding equations (19)

b y (tj−1 ); zb) = h(tj−1 ; zb), G(b

j > 1,

b y (tj ); zb) = h(tj ; zb), for any j > 1. We write which imply yb(0) = 0 and G(b h(t1 ; zb) = zb +

m X

gk (b z )yk (t1 )

k=1

and Z b y (t1 ); zb) = zb + G(b = zb +

1

∂y G(θb y (t1 ); zb)dθ 0 d Z 1 X b

k=1

0

yb(t1 )

b y (t1 ); zb))dθ ybk (t1 ) gk (G(θb

and we assume that each gk ∈ C2b (Rn ; Rn ) has the following structure which agrees with (11) and (12), (20)

gk (z) = αk (z)bk .

Here {b1 , . . . , bd } ⊂ Rd are fixed and αk (·) ∈ C2b (Rn ; R) satisfying 0 < δ 6 αk (z) 6 M agree with the commuting property in (12), i.e., αj (z)h∂z αi (z), bj i − αi (z)h∂z αj (z), bi i = 0, z ∈ Rn , i, j = 1, . . . , d. P It follows h(t1 ; zb) = zb + dk=1 (αk (b z )yk (t1 ))bk and we solve the equation (21)

b y (t1 ); zb) = h(t1 ; zb), G(b

∀b z ∈ Rn ,

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Gradient representation for cad-lag solutions of SDEs with jumps

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using a nonlinear contractive mapping. Rewrite (22)

b y ; zb) = zb + G(b

d X

(αk (b z )b yk )bk +

k=1

d X

αk (b z )b yk

X d

 βkj (b y ; zb)b yj bk ,

j=1

k=1

where Z 1 Z 1 1 b 1 θb b 1 θb hαj (G(θ y ; zb))∂z αk (G(θ y ; zb)), bj idθ1 θdθ βkj (b y ; zb) := αk (b z) 0 0 is a continuous and bounded function, fulfilling ( b yb ∈ Rd , zb ∈ Rn , |βkj (b y ; zb)| 6 C, (23) b y 0 − yb00 |, ∀b |βkj (b y 0 ; zb) − βkj (b y 00 )| 6 L|b y 0 , yb00 ∈ Rd , zb ∈ Rn . b C), b ρ1 = ρ M and Lρ := (1 + 2ρ1 )L. Choose ρ > 0 sufficiently Set L = max(L, 1 δ small such that d X 1 1 (24) γ := 4ρ1 Lρ1 6 , y ∈ B(0, 2ρ1 ), zb ∈ Rn . |βkj (b y ; zb)b yj | 6 , ∀b 2 2 j=1

Define a map T (b y ) : B(0, 2ρ1 ) → C(Rn ; Rd ) by (25)

Tk (b y )(z) = 1 +

d X

βkj (b y ; z)b yj

j=1

and associate the nonlinear operator U (b y ) : B(0, 2ρ1 ) → C(B(0, ρ) × R2n ; Rd ), (26)

Uk (b y )(y, z) := yk [Tk (b y )(z2 )]−1

αk (z1 ) , αk (z2 )

z = (z1 , z2 ) ∈ R2n .

Using (23) and the second inequality in (24) we notice that  |[Tk (b y )(z2 )]−1 | 6 2, ∀b y ∈ B(0, 2ρ1 ), z2 ∈ Rn ,    (27) |[Tk (b y 0 )(z2 )]−1 − [Tk (b y 00 )(z2 )]−1 | 6 4|Tk (b y 0 )(z2 ) − Tk (b y 00 )(z2 )|    6 4(1 + 2ρ1 )L|b y 0 − yb00 | = 4Lρ1 |b y 0 − yb00 |, ∀b y 0 , yb00 ∈ B(0, 2ρ1 ). Using (27) and (24) we get that {U (b y ); yb ∈ B(0, 2ρ1 )} is a Lipschitz continuous operator with a Lipschitz constant γ and M (28) |Uk (b y 0 )(z2 )]−1 − [Tk (b y 00 )(z2 )]−1 | y 0 )(y, z) − Uk (b y 00 )(y, z)| 6 |yk | |[Tk (b δ 6 ρ1 (4Lρ1 )|b y 0 − yb00 | = γ|b y 0 − yb00 |, ∀b y 0 , yb00 ∈ B(0, 2ρ1 ), y ∈ B(0, ρ). In addition, equation (21) for the unknown yb(t1 ) is replaced by the functional nonlinear equations (29)

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

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The unique solution of (29) is constructed as the limit of the Cauchy sequence {b y j } in the complete metric space Yb ,  0 yb = {0},         α (z ) α (z ) 1 1 1 d  2n 1  yb := U (0) = , : (z1 , z2 ) ∈ R , . . . , yd y1 α1 (z2 ) αd (z2 ) (30)  X   j   M 1 2M  j+1 j j+1 1 i  := U (b y ), kb y (y)k 6 kb y (y)k γ 6 |y| 6 |y|,  yb δ 1−γ γ i=0

for any y ∈ B(0, ρ), where kb y (y)k := supz∈R2n |b y (y, z)|. 0 00 Using the metric d(b y , yb ) := sup(y,z)∈B(0,ρ)×R2n |b y 0 (y, z) − yb00 (y, z)|, an induction argument leads us to |b y j+1 (y, z) − ybj (y, z)| 6 |b y 1 (y, z)|γ j . As a consequence the estimate 2M M 1 |y| 6 |y| δ 1−γ δ is valid. Denote f (y, z) = lim ybj (y, z), which is an element of Yb . The above kb y j+1 (y)k 6 kb y 1 (y)k(1 + γ + · · · + γ j ) 6 j→∞

computations are restated as Lemma 1. Consider the equations (31)

b y ; z2 ) = z2 + G(b

d X

gk (z1 )yk .

k=1

There exists a unique continuous and bounded function yb = f (y, z1 , z2 ) : B(0, ρ) × Rn × Rn → B(0, 2ρ1 ) satisfying (31) and 2M |y| 6 2ρ1 . kf (y)k = sup |f (y, z1 , z2 )| 6 γ z1 ,z2 ∈Rn Equation (21) is a particular case of equation (31) in Lemma 1 and define (32)

yb(t1 ) = 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 yb(tj−1 ) = yb(tj −) fulfills (33)

b y (tj −); zb) = h(tj −; zb), G(b

∀b z ∈ Rn ,

where |b y (tk ) − yb(tk −)| 6 2M δ |y(tk ) − y(tk −)|, for any 1 6 k 6 j − 1 and j > 2 is fixed. Therefore, the equations (34)

b y (tj ); zb) = h(tj ; zb) G(b

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Gradient representation for cad-lag solutions of SDEs with jumps

can be solved and |∆b y (tj )| := |b y (tj ) − yb(tj −)| 6 (35)

2M δ |∆y(tj )|.

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Rewrite (34) as

b y (tj ); zb) = G((b b y (tj ) − yb(tj −)); h(tj−1 , zb)) G(b = h(tj−1 , zb) +

d X

gk (h(tj−1 , zb))∆yk (tj ).

k=1

Now replace equation (35) for the unknown yb = yb(tj )−b y (tj −) and h(tj−1 ; zb) := b h by (36)

b y; b G(b h) = b h+

d X

gk (b h)∆yk (tj ),

b h ∈ Rn .

k=1

Using hypothesis (20) we construct the unique solution of (36) (37)

yb = f (∆y(tj ), b h, b h),

where the continuous and bounded function f (y, z1 , z2 ) : B(0, ρ) × Rn × Rn → B(0, 2ρ1 ) is the unique solution of the functional equation (29). Then yb(tj ) = yb(tj−1 ) + f (∆y(tj ), h(tj−1 ; zb), h(tj−1 ; zb)) and it satisfies the equation (34). The above computations allow us to state Theorem 1. There exists a piecewise constant and bounded process {b y (t) = yb(tj ); t ∈ [tj , tj+1 )} with yb(0) = 0, such that b y (tj −); zb) + h(tj ; zb) = G(b

d X

b y (tj −); zb))(yk (tj ) − yk (tj −)). gk (G(b

k=1

Remark 4. This theorem provides the gradient representation of the piecewise continuous process (h(t; zb)). Step 2. The first significant equation for the unknown yb(tj −) ∈ Rd appears for j = 2, yb(t2 −) = yb(t1 ), and using zb(t2 ) = zb(t2 −). We write it for j = 2 and t = t2 (38)

b y (t1 ); zb(t2 )) + z(t2 ) = G(b

d X

b y (t1 ); zb(t2 )))(yk (t2 ) − yk (t2 −)), gk (G(b

k=1

where (39)

z(t2 ) = z(t2 −) +

d X k=1

gk (z(t2 −))(yk (t2 ) − yk (t2 −)).

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Both equations (38) and (39) are fulfilled provided yb(t1 ) is found such that  b y (t1 ); zb(t2 )) = z1 (t2 −),  G(b    d d (40) X X   gk (b z (t1 ))yk (t1 ) = zb(t2 ) + gk (b z1 (t1 ))yk (t1 ).   z(t2 ) = zb(t2 ) + k=1

k=1

Notice that (40) are solved if we find first yb the solution of the equation (41)

b y ; z2 ) = z2 + G(b

d X

gk (z1 )yk (t1 ),

y(t1 ) ∈ B(0, ρ), z1 , z2 ∈ Rn .

k=1

By hypothesis, equation (41) fulfills the assumptions of Lemma 1 and let yb = f (y, z1 , z2 ) : B(0, ρ) × Rn × Rn → Rd 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)k 6 2M δ |y|, ∀y ∈ B(0, ρ). Define yb(t1 ) := f (y(t1 ), zb(t1 ), zb(t2 )), which satisfies |b y (t1 )| 6 2ρ1 .

(42)

Step 3. We find {b z (t); t ∈ [t2 , t3 )} such that zb(t2 ) = zb(t2 −) fulfilling  d  X   b y (t1 ); zb(t2 )))(yk (t2 ) − yk (t2 −)), b y (t1 ); zb(t)) +  z(t) = G(b gk (G(b k=1     zb(t) = G(−b b y (t1 ); z¯(t)),

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

d X

b y (t1 ); zb(t2 )))(yk (t2 ) − yk (t2 −)). gk (G(b

k=1

Step 4. We are in position to stipulate what is necessary for getting the representation formula b y (tj −); zb(t)) + (43) z(t) = G(b

d X

b y (tj −); zb(tj )))∆yk (tj ), gk (G(b

t ∈ [tj , tj+1 ),

k=1

by using an induction argument. For some j > 2 the algorithm requires as known the following items: (b z (t, x)), for t ∈ [0, tj+1 ) and {b y (t) = yb(tk ); t ∈ [tk , tk+1 ), 0 6 k 6 j − 1}, fulfilling (44)

|∆b y (tk )| 6

2M |∆y(tk )|, δ

0 6 k 6 j − 1.

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273

In particular, the equations (45)

b y (tk −); zb(tk )) = z(tk −), G(b

06k6j

are fulfilled. Assuming (44) and (45), we must find ∆b y (tj ) = yb(tj ) − yb(tj −) and a continuous process {b z (t); t ∈ [tj+1 , tj+2 ), zb(tj+1 ) = zb(tj+1 −)}, such that |∆b y (tj )| 6

(46)

2M |∆y(tj )|, δ

and b y (tj ); zb(tj+1 )) = z(tj+1 −). G(b

(47)

Using the equality yb(tj+1 −) = yb(tj ) = yb(tj −) + ∆b y (tj ), define zbj+1 (t) for t ∈ [tj+1 , tj+2 ) as the solution of the equation b y (tj ); zb(t)) + (48) z(t) = G(b

d X

b y (tj ); zb(tj+1 )))(yk (tj+1 ) − yk (tj+1 −)), gk (G(b

k=1

for t ∈ [tj+1 , tj+2 ). We are in position to state Lemma 2. Consider the piecewise constant process {b y (t) = yb(tk ); t ∈ [tk , tk+1 ), 0 6 k 6 j − 1} and the continuous process {b z (t, x); t ∈ [tk , tk+1 ), 0 6 k 6 j} such that they satisfy (43), (44) and (45). Then there exists yb(tj ) = yb(tj+1 −) such that the equations (46) and (47) are satisfied and {b z (t); t ∈ 2M [tj+1 , tj+2 )} is defined in (48). In addition, |∆b y (tj )| 6 δ |∆y(tj )|, for any P 2M |∆b y (t )| 6 j > 1 and Vyb := ∞ ρ. j j=1 δ b ·), y ∈ Rd , we get Proof. Using the group property of G(y; (49)

b y (tj ); z) = G(∆b b y (tj ); G(b b y (tj −); z)), G(b

∀z ∈ Rn .

On the other hand, (50)

b y (tj −); zb(tj+1 )) + z(tj+1 −) = G(b

d X

b y (tj −); zb(tj )))∆yk (tj ). gk (G(b

k=1

In order to solve (47), we use (49) and (50) and rewrite (47) as (51)

b y (tj ); zb2 ) = zb2 + G(b

d X

gk (b z1 )∆yk (tj ),

k=1

b y (tj −); zb(tj+1 )) and zb1 := G(b b y (tj −); zb(tj )). where zb2 := G(b

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14

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

b y ; z 2 ) = z2 + G(b

(52)

d X

gk (z1 )yk ,

∀z1 , z2 ∈ Rn , y ∈ B(0, ρ).

k=1

Define ∆b y (tj ) := f (∆y(tj ), zb1 , zb2 ). In addition, (53) |∆b y (tj )| 6 kf (∆y(tj )k =

sup z1 ,z2 ∈Rn

|f (∆y(tj ), zb1 , zb2 )| 6

2M |∆y(tj )|.  δ

Set (54)

z¯(t) = z(t) −

d X

gk (z(tj −))∆yk (tj ),

t ∈ [tj , tj+1 ), j > 0.

k=1

b ·)]−1 Taking into account the equation (43) and using the inverse mapping [G(y; b = G(−y; ·), we write (55)

b y (tj−1 ); z¯(t)), zb(t) = G(−b

t ∈ [tj , tj+1 ),

where (¯ z (t); t > 0) is a continuous process fulfilling the system of SDEs  d  X    d¯ z (t) = f (¯ z (t) + gk (z(tj −))∆yk (tj )dt 0 j     k=1  m d (56) X X   + f (¯ z (t) + gk (z(tj −))∆yk (tj ))dWi (t), t ∈ [tj , tj+1 ), i j     i=1 k=1    z¯(t ) = z(t −). j

j

Applying the standard rule of stochastic derivation for the function ϕ(y, z) = b G(−y, z), and the continuous process {(−b y (tj−1 ), z¯(t)); t ∈ [tj , tj+1 )}, we get  m X   db b x))dt + b x))dWi (t), t ∈ [tj , tj+1 ), z (t) = h0 (b z (t); λ(t, hi (b z (t); λ(t, (57) i=1   zb(tj ) = zb(tj −).

15

Gradient representation for cad-lag solutions of SDEs with jumps

275

b x)) ∈ Rn , zbj ∈ Rn , i = 0, 1, . . . , m, In addition, the vector fields hi (b z ; λ(t, b x)) := (b where λ(t, y (tj −), y(t), z(t, x)), are given by  b x)) = ∇z G(−b b y (tj−1 ); z¯(t))fi (G(b b y (tj−1 ), zb))  hi (b z ; λ(t,     d  X    + gk (z(tj −))∆yk (tj ), 1 6 i 6 m,    k=1 (58)  b b y (tj−1 ), z¯(t))f0 (G(b b y (tj−1 ), zb)) z ; λ(t, x)) = ∇z G(−b   h0 (b    d  X   c0 (λ(t, b x)),  + gk (z(tj −))∆yk (tj ) + h   k=1

for t ∈ [tj , tj+1 ). Here b h0 = (b h10 , . . . , b hn0 ) is defined as m

(59)

1 X 2 bk b hDz G (−b y (tj−1 ); z¯(t))fi (z(t)), fi (z(t))i. hk0 := 2 i=1

We are now in a position to state the main result of this paper Theorem 2. There exists a piecewise constant process {b y (t) = yb(tj ); t ∈ [tj , tj+1 )} and a continuous process {b z (t, x); t > 0} such that the jump process (z(t, x)) has the gradient representation (according to Definition 2) (60)

b y (tj−1 ); zb(t, x)) + z(t, x) = G(b

d X

b y (tj−1 ); zb(tj ; x)))∆yk (tj ), gk (G(b

k=1

for any t ∈ [tj , tj+1 ), j > 0. In addition, the process {b z (t, x); t > 0} fulfills the SDE (57). Final comment. In a future work we intend to obtain a gradient representation for the solution of (1) in the “non-commuting” case, when the vector fields g1 , . . . , gd are mutually in involution over R, i.e., when the Lie bracket [gi , gk ] = αgi + βgk , for some constants α, β depending on gi and gk . REFERENCES [1] B. Iftimie and C. Vˆ arsan, Evolution systems of Cauchy-Kowalewska and parabolic type with stochastic perturbations. Math. Reports 10(60) (2008), 3, 213–238. [2] B. Iftimie, I. Molnar and C. Vˆ arsan, 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.

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[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. Vˆ arsan, Gradient representation and positive cad-lag solutions for jump-differential equations. Preprint nr. 3, IMAR, 2008. [6] C. Vˆ arsan, Applications of Lie Algebras to Hyperbolic and Stochastic Differential Equations. Kluwer, 1999. [7] S. Shreve, Stochastic Calculus for Finance II. 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