An extension of the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps

We extend the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps, in the special case where small jumps are summable.

Brownian motion W and Poisson random measure µ dX t = b(t, X t− ) dt + σ(t, X t− ) dW t (1) where h : [0, ∞) → [0, ∞) is continuous and nondecreasing, h(0) = 0, h(x) > 0 for x > 0, and (4) (0,ε) h −2 (v) dv = ∞ for every ε > 0 , together with a Lipschitz condition on the drift b(t, x). It is interesting to ask for extensions of this . This question has already been raised by Bass (see the remarks on p. 9 in [B 04], and following (1.2) in [B 02]). Theorem 1.1 in [B 02] (see also [BBC 04], and [Z 02]) proves pathwise uniqueness for solutions of (5) dX t = F (X t − ) dS α t driven by a symmetric stable process S α of index 1 < α < 2, where F (·) is bounded and satisfies a continuity condition (3) with h(·) such that The proof of this result relies heavily on particular properties of the stable driving process of index We prove another type of extension of the Yamada-Watanabe condition for pathwise uniqueness of solutions of (1). Our result -of limited generality since we assume summability of small jumps of the process X -combines the original Yamada-Watanabe conditions (3)+(4) for the diffusive part with a simple Lipschitz condition concerning the small jumps of µ. Big jumps of µ are irrelevant in view of pathwise uniqueness. As an example, together with a Cox-Ingersoll-Ross type diffusion coefficient, the jump part can be as in (5) with F (·) Lipschitz and 0 < α < 1.
This note is organized as follows: i) we recall the general semimartingale setting (as in Jacod and Shiryaev [JS 87] or Métivier [M 82]) needed to deal with solutions of equation (1); ii) we state the result (theorem 1); iii) we give the proofs together with some related remarks, and point out at which stage the need for summability of small jumps in theorem 1 did arise. (y ∧ 1) 2 ν(dy) < ∞ .
Throughout, we make the following assumptions i)+ii) on the coefficients in equation (1) (3) and (4) above, whereas the drift is Lipschitz with some constant K; ii) the functions (t, x, y) → f i (t, x, y) are measurable for i = 1, 2; the function f 2 (·, ·, ·) is such that whenever we are interested in summability of small jumps of solutions to (1), we strengthen this to A solution to equation (1) is any process X = (X t ) t≥0 on (Ω, A, P ) satisfying iii)-v) below: iii) X is IF -adapted and càdlàg; iv) the following process is locally integrable: whenever (9) is assumed, we strengthen this to local integrability of These are general conditions needed to deal with solutions of SDE with jumps. In the restricted setting (9) where small jumps are summable, we have the following result.

Proofs and some associated results
We start in the general setting i)-v), without assuming summability of small jumps. The first lemma -essentially well known as seen from the remarks preceding ( If for arbitrary S ∈ T we can prove pathwise uniqueness for equation (11) , and write (S m,j ) j≥1 for the sequence of jump times of (µ((0, t]×{c m+1 < |y| ≤ c m }) t≥0 , for every m ≥ 0. The are mutually disjoint up to an evanescent set, and support the jumps of X.
2) Let us consider two solutions X ′ , X ′′ of equation (1) with respect to the same pair (µ, W ), starting at time 0 in the same initial condition X ′ 0 = X ′′ 0 , and let us prove -under the assumption of the lemma -that a.s. the paths of X ′ , X ′′ coincide up to time ∞.
This implies  (1) gives since all jumps of µ on ]]T 1 , T 2 [[ are small jumps. The same holds for X ′′ in place of X ′ . Now we put S := T 1 and consider the filtrationǏ F := IF T 1 , theǏ F -Brownian motionW := W T 1 , and theǏ F -Poisson random measureμ := µ T 1 , in the notation as above:W andμ are necessarily independent.
For s ≥ 0, putX ′ s := X ′ T 1 +s andX ′′ s := X ′′ T 1 +s . Then (13) shows that before timeŤ 1 := T 2 − T 1 of the first big jump ofμ,X ′ andX ′′ areǏ F -adapted solutions to equation (11) with S = T 1 , starting from initial valuesX ′ 0 andX ′′ 0 which coincide a.s. by (12). By our assumption, pathwise uniqueness holds for equation (11) with S = T 1 . This show that we havě Changing time back and putting this together with step i), we have pathwise uniqueness of solutions to (1) before time T 2 . At time T 2 , we have as above. This gives pathwise uniqueness for solutions to (1)  where S ∈ T , according to lemma 1. In equations (11), big jumps are absent.
I) First, for ease of notation, we consider the particular case S = 0 in equation (11).
2) We start without assuming summability of small jumps. By localization, it is sufficient to prove pathwise uniqueness on intervals [0, T ] (T deterministic) for solutions X to (1) satisfying with respect to the same pair (W, µ) and the same initial condition. Then has initial value D 0 = 0 and a representation s− , y)} ψ ′ n (D s− ) ] µ(ds, dy) .
The third and fourth terms on the right hand side are martingales (recall |ψ ′ n (·)| ≤ 1); all terms on the right hand side are integrable.
3) The second term on the right hand side of the Ito formula can be treated without any changes as [KS 91], using assumptions (3)+(4): we have where ψ ′′ n = ρ n and ρ n ≤ 2 n h 2 : for this term, we have the bound 4) Taylor formula with remainder terms written in form s− , y)} will be used to consider the fifth term on the right hand side of the Ito formula.
i) A first idea is to approximate (16) by s− , y)} 2 (17) and would allow to use -instead of our Lipschitz assumption (10) -a much weaker assumption (17) is bounded by t n in analogy to (15) above. With this approach however I was unable to control remainder terms which involve the heavily fluctuating derivative ρ ′ n (·).
ii) In the more restrictive setting of summability (9) of small jumps as assumed in the theorem, together with the Lipschitz condition (10) on small jumps, remainder terms do not present any difficulty. The localization step in the beginning of 2) now takes the form in accordance with (9). Write the term (16) as s− , y)| for the fifth term (16) on the right hand side of the Ito formula, which by (10) is smaller than iii) We note the following: as long as D s− might take values in the support (a n , a n−1 ) of ρ n (·), we have to account in (20) above for values ofr which are arbitrarily close to 0. 5) Since |ψ ′ n (·)| ≤ 1 on IR, we use assumption (8) to write the first term on the right hand side of the Ito formula as exactly as in [KS 91].
We add a remark on the case where the heigth of small jumps of the solution process X does not depend on the present state of X.
Proof: This is a variant of the proof of theorem 1, which does not require the restrictive condition on summability of small jumps used in step 4) of the preceding proof. According to lemma 1, we have to prove pathwise uniqueness for all equations (11) dX S s = b(S+s, X S s− ) ds + σ(S+s, X S s− ) dW S s + {|y|≤c} f 2 (S+s, X S s− , y) µ S (ds, dy) , s ≥ 0 where S ∈ T . Consider solutions X (1) , X (2) of (11) starting at the same point. In case where the function f 2 (t, x, y) does not depend on the space variable x, the difference D := X (1) − X (2) is a process with continuous paths Assuming (3)+(4)+(8) and localizing as in the beginning of step 2) above, (15)+(22) conclude the proof, exactly as in the original Yamada-Watanabe argument for the continuous process (2).