Solution to selected problems.

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Poisson process, ⎡ C∑ t−s E[N t − N s |F s ] = E ⎣ =µ i=1 U i ⎤ ⎦ = ∞∑ kP (C t−s = k) = µλ(t − s) k=1 ⎡ ⎤ C ∞∑ ∑ t−s E ⎣ U i |C t−s = k⎦ P (C t−s = k) k=1 Therefore N t − µλt is a martingale and the claim is shown. i=1 (53) 16. By direct computation, 1 1 − e −λh λ(t) = lim P (t ≤ T < t + h|T ≥ t) = lim h→0 h h→0 h A case that T is Weibull is similar and omitted. = λ. (54) 17. Observe that P (U > s) = P (T > 0, U > s) = exp{−µs}. P (T ≤ t, U > s) = P (U > s) − P (T > t.U > s). Then ∫ t+h P (t ≤ T < t + h, t ≤ U) t e −µt (λ + θt)e −(λ+θt)u du P (t ≤ T < t + h|T ≥ t, U ≥ t) = = P (t ≤ T, t ≤ U) exp{−λt − µt − θt 2 } and = −1[exp{−λ(t + h) − θt(t + h)} − exp{−λt − θt2 }] exp{−λt − θt 2 } = 1 − exp{−(λ + θt)h} λ # 1 − exp{−(λ + θt)h} (t) = lim = λ + θt (56) h→0 h 18. There exists a disjoint sequence of predictable times {T n } such that A = {t > 0 : △〈M, M〉 t ≠ 0} ⊂ ∪ n [T n ]. (See the discussion in the solution of exercise 12 for details.) In addition, all the jump times of 〈M, M〉· is a predictable time. Let T be a predictable time such that 〈M, M〉 T ≠ 0. Let N = [M, M] − 〈M, M〉. Then N is a martingale with finite variation since 〈M, M〉 is a compensator of [M, M]. Since T is predictable, E[N T |F T − ] = N T − . On the other hand, since {F t } is a quasi-left-continuous filtration, N T − = E[N T |F T − ] = E[N T |F T ] = N T . This implies that △〈M, M〉 T = △[M, M] T = (△M T ) 2 . Recall that M itself is a martingale. So M T = E[M T |F T ] = E[M T |F T − ] = M T − and △M T = 0. Therefore △〈M, M〉 T = 0 and 〈M, M〉 is continuous. 19. By theorem 36, X is a special semimartingale. Then by theorem 34, it has a unique decomposition X = M + A such that M is local martingale and A is a predictable finite variation process. Let X = N + C be an arbitrary decomposition of X. Then M − N = C − A. This implies that A is a compensator of C. It suffices to show that a local martingale with finite variation is locally integrable. Set Y = M − N and Z = ∫ t 0 |dY s|. Let S n be a fundamental sequence of Y and set T n = S n ∧ n ∧ inf{t : Z t > n}. Then Y Tn ∈ L 1 (See the proof of theorem 38) and Z Tn ≤ n + |Y Tn | ∈ L 1 . Thus Y = M − N is has has a locally integrable variation. Then C is a sum of two process with locally integrable variation and the claim holds. 22 (55)
20. Let {T n } be an increasing sequence of stopping times such that X Tn is a special semimartingale as shown in the statement. Then by theorem 37, Xt∧T ∗ n = sup s≤t |X T n s | is locally integrable. Namely, there exists an increasing sequence of stopping times {R n } such that (Xt∧T ∗ n ) R n is integrable. Let S n = T n ∧ R n . Then S n is an increasing sequence of stopping times such that (Xt ∗ ) Sn is integrable. Then Xt ∗ is locally integrable and by theorem 37, X is a special semimartingale. 21. Since Q ∼ P, dQ/dP > 0 and Z > 0. Clearly M ∈ L 1 (Q) if and only if MZ ∈ L 1 (P). By generalized Bayes’ formula. E Q [M t |F s ] = E P[M t Z t |F s ] E P [Z t |F s ] Thus E P [M t Z t |F s ] = M s Z s if and only if E Q [M t |F s ] = M s . = E P[M t Z t |F s ] Z s , t ≥ s (57) 22. By Rao’s theorem. X has a unique decomposition X = M + A where M is (G, P)-local martingale and A is a predictable process with path of locally integrable variation. E[X ti+1 − X ti |G ti ] = E[A ti+1 − A ti |G ti ]. So [ n∑ ] sup E |E[A ti − A ti +1|G ti ]| < ∞ (58) τ i=0 Y t = E[X t |F t ] = E[M t |F t ] + E[A t |F t ]. Since E[E[M t |F t ]|F s ] = E[M t |F s ] = E[E[M t |G s ]|F s ]] = E[M s | fil s ], E[M t |F t ] is a martingale. Therefore [ n∑ ] n∑ Var τ (Y ) = E |E[E[A ti |F ti ] − E[A ti +1|F ti +1]|F ti ]| = E [|E[A ti − A ti +1|F ti ]|] (59) i=1 i=1 For arbitrary σ-algebra F, G such that F ⊂ G and X ∈ L 1 , E (|E[X|F]|) = E (|E (E[X|G]|F) |) ≤ E (E (|E[X|G]||F)) = E (|E[X|G]|) (60) Thus for every τ and t i , E [|E[A ti − A ti +1|F ti ]|] ≤ E [|E[A ti − A ti +1|G ti ]|]. Therefore Var(X) < ∞ (w.r.t. {G t })implies Var(Y) < ∞ (w.r.t {F t }) and Y is ({F t }, P)-quasi-martingale. 23. We introduce a following standard result without proof. Lemma. Let N be a local martingale. If E([N, N] 1/2 ∞ ) < ∞ or alternatively, N ∈ H 1 then N is uniformly integrable martingale. This is a direct consequence of Fefferman’s inequality. (Theorem 52 in chapter 4. See chapter 4 section 4 for the definition of H 1 and related topics.) We also take a liberty to assume Burkholder- Davis-Gundy inequality (theorem 48 in chapter 4) in the following discussion. Once we accept this lemma, it suffices to show that a local martingale [A, M] ∈ H 1 . By Kunita- Watanabe inequality, [A, M] 1/2 ∞ ≤ [A, A] 1/4 ∞ [M, M] 1/4 ∞ . Then by Hölder inequality, ( ) E [A, M] 1/2 ∞ ( ≤ E [A, A] 1/2 ∞ ) 1 ( 2 E [M, M] 1/2 ∞ 23 ) . (61)
Page 1 and 2: Solution to selected problems. Chap
Page 3 and 4: 10. (a) Let N t = ∑ i 1 i (N i t
Page 5 and 6: where Z t ′ = ∫ R xN t(·, dx),
Page 7 and 8: 21. a) let a = (1 − t) −1 ( ∫
Page 9 and 10: For this, let t 0 = 0 and t n+1 =
Page 11 and 12: So that Y R ◦ θ R (ω) = { 1 R
Page 13 and 14: 6. Pick arbitrary t 0 > 1. Let Xt n
Page 15 and 16: 15. If M 0 = 0 then by Theorem 42,
Page 17 and 18: 22. Claim that for all integer k
Page 19 and 20: Chapter 3. Semimartingales and Deco
Page 21: lemma Let X t be a càdlàg adapted
Page 25: 31. Let T be an arbitrary F µ stop

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