An approximate martingale problem

3.4 An approximate martingale problem

We now derive the approximate martingale problem. In short, the idea is to express the integral of ξ t against time-dependent test functions as the sum of a martingale, average drift terms and fluctuation error terms. Take a test function φ : [0, ∞ × N −1 Z → R with t 7→ φ t x continuously differentiable and satisfying Z T 〈|φ s | + φ 2 s + |∂ s φ s |, 1〉ds ∞ 25 this ensures that the following integration and summation are well-defined. We apply integration by parts to ξ t x φ t x, sum over x and multiply by 1 N , to obtain for t ≤ T recall the definition of ν t from 3 and that 〈ξ t , φ〉 = 〈ν t , φ〉 〈ν t , φ t 〉 =〈ν , φ 〉 + Z t 〈ν s , ∂ s φ s 〉ds 26 + 1 N X x X y ∼x Z t ξ s − y φ s x − φ s y d P s x; y 27 + 1 N X x X y ∼x Z t ξ s − x φ s x d P s y; x − d P s x; y 28 + X k=0,1 1 − 2k X m ≥2,i, j=0,1 1 N X x X y 1 ,..., y m ∼x X z Z t 1 ξ s − x = k 1 ξ s − y 1 = j × m Y l=2 1 ξ s − y l = 1 − k 1 ξ s − z = i φ s xdQ m,i, j,k s x; y 1 , . . . , y m ; z. 29 The main ideas for analyzing terms 27 and 28 will become clear once we analyze term 29 in detail. The latter is the only term where calculations changed seriously compared to [13]. Hence, we shall only summarize the results for terms 27 and 28 in what follows. We break term 27 into two parts, an average term and a fluctuation term and after proceeding as for term 3.1 in [13] we obtain 27 = Z t 〈ν s − , ∆ φ s 〉ds + E 1 t φ, where E 1 t φ ≡ 1 N X x X y ∼x Z t ξ s − y φ s x − φ s y d P s x; y − d〈Px; y〉 s . We have suppressed the dependence on N in E 1 t φ. E 1 t φ is a martingale recall that if N ∼ Pois λ, then N t − λt is a martingale with quadratic variation 〈N〉 t = λt with predictable brackets process given by d E 1 φ t ≤ D φ t , 1 p N 2 λ 〈1, e −2λ 〉d t. 30 633 Alternatively we also obtain the bound d E 1 φ t ≤ 4 kφ t k 〈 φ t , 1 〉d t 31 with kφ t k = sup x |φ t x|. The second term 28 is a martingale which we shall denote by M N t φ in what follows we shall drop the superscripts w.r.t. N and write M t φ. It can be analyzed similarly as the martingale Z t φ of 3.3 in [13]. We obtain in particular that 〈Mφ〉 t = 2 N − θ N N ¨Z t 〈ξ s − , φ 2 s 〉ds − Z t 〈A ξ s − φ s , ξ s − φ s 〉ds « . 32 Using that A ξ s − φ s x ≡ 1 2cN N 1 2 X y ∼x ξ s − y φ s y ≤ sup y ∼x |φ s y| we can further dominate 〈Mφ〉 t by 〈Mφ〉 t ≤ Cλ Z t € kφ s k 2 λ 〈1, e −2λ 〉 Š ∧ kφ s k 〈ξ s − , |φ s |〉 ds. 33 We break the third term 29 into two parts, an average term and a fluctuation term. Recall Notation 2.9 and observe that if we only consider a ∈ {0, 1} we have F k a = 1a = k. We can now rewrite 29 to X k=0,1 1 − 2k X m ≥2,i, j=0,1 1 N X x X y 1 ,..., y m ∼x X z Z t 1 ξ s − x = k 1 ξ s − y 1 = j 34 × m Y l=2 1 ξ s − y l = 1 − k 1 ξ s − z = i φ s x q k,m,N i j 2cN m N m 2 pN x − zds + E 3 t φ = X k=0,1 1 − 2k X m ≥2,i, j=0,1 q k,m,N i j Z t 1 N X x 1 2cN N 1 2 X y 1 ∼x 1 ξ s − y 1 = j × m Y l=2 1 2cN N 1 2 X y l ∼x 1 ξ s − y l = 1 − k X z pN x − z1 ξ s − z = i × 1 ξ s − x = k φ s xds + E 3 t φ = X k=0,1 1 − 2k X m ≥2,i, j=0,1 q k,m,N i j Z t 1 N X x F j A ξ s − x × F 1 −k A ξ s − x m −1 F i p N ∗ ξ s − x1 ξ s − x = k φ s xds + E 3 t φ = X k=0,1 1 − 2k X m ≥2,i, j=0,1 q k,m,N i j Z t 〈 € F j ◦ Aξ s − Š × F 1 −k ◦ Aξ s − m −1 € F i ◦ p N ∗ ξ s − Š 1 ξ s − · = k , φ s 〉ds + E 3 t φ, 634 where for x ∈ ZN we set € p N ∗ f Š x ≡ X z ∈ZN pN x − z f z 35 and E 3 t φ ≡ X k=0,1 1 − 2k X m ≥2,i, j=0,1 1 N X x X y 1 ,..., y m ∼x X z Z t 1 ξ s − x = k × 1 ξ s − y 1 = j m Y l=2 1 ξ s − y l = 1 − k 1 ξ s − z = i φ s x ×    dQ m,i, j,k s x; y 1 , . . . , y m ; z − q k,m,N i j 2cN m N m 2 pN x − zds    . We have suppressed the dependence on N in E 3 t φ. Here, E 3 t φ is a martingale with predictable brackets process given by E 3 φ t ≤ X m ≥2,i, j,k=0,1 q k,m,N i j 1 N 2 X x m Y l=0 X y l ∼x 1 2cN N 1 2 X z pN x − z Z t φ 2 s xds 36 ≤ 1 N X m ≥2,i, j,k=0,1 q k,m,N i j Z t kφ s k 2 λ 〈e −2λ , 1 〉ds. Taking the above together we obtain the following approximate semimartingale decomposition from 26. 〈ν t , φ t 〉 =〈ν , φ 〉 + Z t 〈ν s , ∂ s φ s 〉ds + Z t 〈ν s − , ∆ φ s 〉ds + E 1 t φ + M t φ + E 3 t φ 37 + X k=0,1 1 − 2k X m ≥2,i, j=0,1 q k,m,N i j Z t 〈 € F j ◦ Aξ s − Š × F 1 −k ◦ Aξ s − m −1 € F i ◦ p N ∗ ξ s − Š 1 ξ s − · = k , φ s 〉ds. Remark 3.3. Note that this approximate semimartingale decomposition provides the link between our approximate densities and the limiting SPDE in 21 for the case with no short-range competition. Indeed, uniqueness of the limit u t of A ξ N t will be derived by proving that u t solves the martingale problem associated with the SPDE 21.

3.5 Green’s function representation