Báo cáo nghiên cứu khoa học: "Biểu diễn Doob - Mayer đối với Martingale trên thang thời gian" docx

Introduction [8] D [10] [5] stochastic calculus on the time scale 1. Preliminaries on time scales [5] R T T forward jump operator backward jump operator σ, ρ : T → T σ(t) = inf{s ∈ T : s > t} inf ∅ = sup T ρ(t) = sup{s ∈ T : s < t} sup ∅ = inf T graininess µ : T → R + ∪ {0} µ(t) = σ(t) − t t ∈ T right- dense σ(t) = t right-scattered σ(t) > t left-dense ρ(t) = t left-scattered ρ(t) < t 1 isolated t a, b ∈ T [a, b] {t ∈ T : a  t  b} T k T T T f T R m f delta differentiable differentiable t ∈ T k f ∆ (t) ∈ R m f  > 0 t f(σ(t)) − f(s) − f ∆ (t)(σ(t) − s)  |σ(t) − s| s ∈ f t ∈ T k f differentiable on T T = R f  (t) T = Z ∆f f T rd−continuous rd− T X C rd (T, X) f T R m × R m regressive det(I + µ(t)f(t)) = 0 t ∈ T f : T → R f σ : T → R f σ = f ◦ σ f σ t = f(σ(t)) t ∈ T I = {t : right-scattered points of T}. 1.1. Proposition ([7]). The set I of all right-scattered points of T is at most countable. A rd− T F 1 T F 1 = {[a; b) : a, b ∈ T}. F 1 T m 1 F 1 m 1 ([a, b)) = A(b) − A(a). m 1 F 1 . µ A ∆ m 1 , F 1 ∆− A T E A ∆ − T\{max T, min T} f : T → R, A ∆ − f µ A ∆ E Lebesgue - Stieljes ∆− integral  E f(s)∆A(s). 1.2. Example. If A(t) = t for all t ∈ T we have µ A ∆ is Lebesgue ∆− measure on T and  E f(s)∆A(s) is Lesbesgue ∆− integral. 1.3. Remark. By the definition of µ A ∆ we see that (1) For each t 0 ∈ T k , the single- point set {t 0 } is ∆ A − measurable, and µ A ∆ ({t 0 }) = A(σ(t 0 )) − A(t 0 ) (1.1) (2) If a, b ∈ T and a  b, then µ A ∆ ((a, b)) = A(b) − A(σ(a)) µ A ∆ ((a, b]) = A(σ(b)) − A(σ(a)) µ A ∆ ([a, b]) = A(σ(b)) − A(a) 2. Doob - Meyer decomposition a ∈ T k T a = {x ∈ T : x  a} (Ω, F, {F t } t∈T a , P) {F t } t∈T a rd− X = {X t : t ∈ T a } (Ω, F, {F t } t∈T a , P) 2.1. Definition. A process A = {A t } t∈T a is called increasing if it is F t −adapted, A a = 0 and the almost sure sample paths of A are increasing on T a . 2.2. Proposition. If M is a right continuous bounded martingale, A is increasing then for any t ∈ T a , then EM t A t = E  [a,t) M σ s ∆A s . (2.1) Proof. Fix a t ∈ T a . For any n ∈ N, consider a partion π (n) = {a = t (n) 0 < t (n) 1 < · · · < t (n) k n = t} of [a, t]. Denote δ π (n) = max t i ∈π (n) |t i+1 − σ(t i )|. Let N π (n) s =        M σ(a) if s = a M σ(t (n) i+1 ) if s ∈ (t (n) i , t (n) i+1 ] ∀i = 0, , k n − 2 M t if s ∈ (t (n) k n −1 , t). Since M is right continuous, M σ s is also right continuous. Therefore, M σ s = lim δ π (n) →0 N π (n) s ∀s ∈ [a, t). Hence, by the b ounded convergence theorem we have E  [a,t) M σ s ∆A(s) = E  lim δ π (n) →0  [a,t) N π (n) s ∆A(s)  = lim δ π (n) →0 E   [a,t) N π (n) s ∆A(s)  = lim δ π (n) →0 E  M σ(a) A σ(a) + k n −1  i=1 M σ(t (n) i ) (A σ(t (n) i ) − A σ(t (n) i−1 ) ) + M t (A t − A σ(t (n) k n −1 ) )  = lim δ π (n) →0 E  M t A t + k n −1  i=1 A σ(t (n) i−1 ) (M σ(t (n) i ) − M σ(t (n) i−1 ) ) + A σ(t (n) k n −1 ) (M t − M σ(t (n) k n −1 ) )  = EM t A t . The proof is complete.  f : T a → R f t− f t− = lim s↑t f(s) t ∈ T \ {min T} f a− = f(a) 2.3. Definition. An increasing process A = (A t ) t∈T a is said to be natural if for every bounded cadlag martingale M = (M t ) t∈T a we have EM t A t = E  [a,t) M s− ∆A s . (2.2) 2.4. Proposition. The rd−continuous, increasing process (A t ) t∈T a is natural iff A σ t is F t −measurable for t ∈ I ∩ T a . Proof. Sufficient condition. Suppose that A σ t is F t −measurable for t ∈ I ∩ T a . Let M t be a F t − cadlag martingale and t ∈ T a arbitrary. For any n ∈ N, we consider a partition π (n) = {a = t (n) 0 < t (n) 1 < · · · < t (n) k n = t} of [a, t] such that δ π (n) = max |t (n) i+1 −σ(t (n) i )|  2 −n . Let M π (n) s =        M a if s = a M σ(t (n) i ) if s ∈ (t (n) i ; t (n) i+1 ] ∀i = 0, , k (n) n − 2. M σ(t (n) k n −1 ) if s ∈ (t (n) k n −1 ; t) Since M is a cadlag process, M s− = lim δ π (n) →0 M π (n) s ∀s ∈ [a, t). Therefore, by the b ounded convergence theorem we have E  [a,t) M s− ∆A(s) = E  lim δ π (n) →0  [a,t) M π (n) s ∆A(s)  . We have E  [a,t) (M σ s − M s− )∆A s = E lim δ π (n) →0  [a,t) (N π (n) s − M π (n) s )∆A s = lim δ π (n) →0 E  (M σ(a) − M a )(A σ(a) − A a ) + k (n) n −2  i=0 (M σ(t (n) i+1 ) − M σ(t (n) i ) (A σ(t (n) i+1 ) − A σ(t (n) i ) ) + (M t − M σ(t (n) k n −1 ) )(A t − A σ(t (n) k n −1 ) )  . Because σ(t (n) i )  t (n) i+1  σ(t (n) i+1 ) , A t is rd−continuous and M t is right continuous we see that the above limit converges to E   s∈I∩[a,t) (M σ s − M s )(A σ s − A s )  . On the other hand, A σ s is F s − measurable for s ∈ I ∩ [a, t) then E [(M σ s − M s )(A σ s − A s )] = E [E(M σ s − M s )(A σ s − A s )|F s ] = E [(A σ s − A s )E(M σ s − M s |F s )] = 0. Thus, E  [a,t) (M σ s − M s− )∆A s = 0. By using the proposition 2.2 we get E  [a,t) M σ s ∆A s = E  [a,t) M s− ∆A s = EM t tA t , i.e., (A t ) is natural increasing processes Necessary condition. Let A = (A t ) be a natural increasing process. We need drive that A σ t is F t −measurable for t ∈ I∩ T a . Let t ∈ I∩T a . It is easy to see the process  A s = A s − A t , s  t is also natural on T t . Therefore, by (2.2), for any cadlad, bounded martingale M t we have EM σ(t) (A σ(t) − A t ) = E  [t,σ(t)) M τ − ∆A τ = EM t (A σ(t) − A t ). Or, E(M σ(t) − M t )(A σ(t) − A t ) = 0 =⇒ E(M σ(t) − M t )A σ(t) = 0. Since EM t (A σ(t) − E[A σ(t) | F t ]) = 0, E(M σ(t) − M t )(A σ(t) − E[A σ(t) | F t ]) = 0. It is easy to see that M τ =  as<τ (A σ(s) − E[A σ(s) | F s ]) is a F τ −martingale. Therefore, E(M σ(t) − M t )(A σ(t) − E[A σ(t) | F t ]) = E(A σ(t) − E[A σ(t) | F t ])(A σ(t) − E[A σ(t) | F t ]) = 0, which implies that A σ(t) − E[A σ(t) | F t ] = 0 a.s. The proof is complete.  2.5. Corollary. (A t ) is increasing process on time scale T i) T = N then (A t ) is natural iff it is previsible. ii) T = R then evry increasing process (A t ) is natural if it is continuous. Proof. i) If T = N, then any point t ∈ T right- scattered and σ(t) = t + 1. Therefore, by the proposition 2.4, A t is natural if and only if A t+1 is F t −measurable, i.e., (A n ) is a previsible process. ii) When T = R we have σ(t) = t for all t ∈ T. Since A t is increaing, A t is F t −measurable.  [1] 2.6. Theorem (Dunford - Pettis [1]). If (Y n ) n∈N is uniformly integrable sequence of random variables, there exists an integrable random variable Y and a subsequence (Y n k ) k∈N such that weak-lim k→∞ Y n k = Y , i.e., for all bounded random variables ξ we have lim k→∞ E(ξY n k ) = E(ξY ) X - (D) {X τ : τ is a stopping time satisfying a  τ < ∞} - (DL) t ∈ T a {X τ : τ is a stopping time satisfying a  τ  t} 2.7. Theorem (Doob-Meyer decomposition). Let X be a right continuous submartingale of class (DL). Then, there exist a right continuous martingale and a right continuous increasing process A such that X t = M t + A t ∀t ∈ T a a.s. If A is natural then M and A are uniquely determined up to indistinguishability. If X is of class (D) then M and A are uniformly integrable. Proof. First we proof uniqueness. Suppose there exist two right continuous martingales M , M  and two right continuous natural increasing processes A, A  such that X t = M t + A t = M  t + A  t ∀t ∈ T a a.s. This relation implies that B t = A t − A  t = M  t − M t is right continuous martingale. Let ξ t be an arbitrary right continuous bounded martingale. For each partition π (n) : a = t (n) 0 < t (n) 1 < · · · < t (n) n = t of [a, t], we set ξ π (n) s =        ξ a if s = a ξ σ(t (n) i ) if s ∈ (t (n) i ; t (n) i+1 ] ∀i = 0, 1, , n − 2 ξ σ(t (n) n−1 ) if s ∈ (t (n) n−1 ; t) . we have ξ s− = lim δ π (n) →0 ξ π (n) s ∀s ∈ [a, t) By the bounded convergence theorem we have E  [a,t) ξ s− ∆A(s) = E lim δ π (n) →0  [a,t) ξ π (n) s ∆A(s) = E lim δ π (n) →0  ξ 0 (A σ(0) − A 0 ) + n−2  i=0 ξ σ(t i ) (A σ(t i+1 ) − A σ(t i ) ) + ξ σ(t n−1 ) (A t − A σ(t n−1 ) )  Therefore, Eξ t (A t − A  t ) = E  [a,t) ξ s− ∆A(s) − E  [a,t) ξ s− ∆A  (s) = E lim δ π (n) →0  ξ 0 (B σ(0) − B 0 ) + n−2  i=0 ξ σ(t i ) (B σ(t i+1 ) − B σ(t i ) ) + ξ σ(t n−1 ) (B t − B σ(t n−1 ) )  = lim δ π (n) →0 E  ξ 0 (B σ(0) − B 0 ) + n−2  i=0 ξ σ(t i ) (B σ(t i+1 ) − B σ(t i ) ) + ξ σ(t n−1 ) (B t − B σ(t n−1 ) )  Thus Eξ t (A t − A  t ) = 0. Now let X be an arbitrary bounded random variable and let us define the bounded mar- tingale ξ by taking a right continuous version of E(X|F t ) t∈T a . From the above, E(X(A t − A  t )) = E  E(X(A t − A  t )|F t )  = E  (A t − A  t )E(X|F t )  = Eξ t (A t − A  t ) = 0. Since the choice of X was arbitrary, it follows that A t = A  t almost surely for a fixed t > 0. By virtue of the right continuity of A and A  , we conclude that A and A  are indistinguishable. Hence, M = X − A and M  = X − A  are indistinguishable as well. Next, we prove the existence of the decomposition. By uniqueness, it suffices to prove the existence of the processes M and A on the interval [a; b] for fixed b ∈ T a . Without loss of generality we may assume that X a = 0. Let π (n) : a = t (n) 0 < t (n) 1 < · · · < t (n) N = b be a partition of [a, b] such that δ π (n) = max t i ∈π (n) |t i − σ(t i−1 )|  1 2 n . Consider the Doob - Meyer decomposition of the finite submartingale X (n) = (X t (n) j ) t (n) j ∈π (n) X t (n) j = M (n) t (n) j + A (n) t (n) j Thus, M (n) = {M (n) t (n) j } t (n) j ∈π (n) is a martingale satisfying M (n) a = X a and A (n) = {A (n) t (n) j } t (n) j ∈π (n) is a previsible and increasing. Therefore, M (n) t (n) j = E(M (n) b |F t (n) j ) = E(X b − A (n) b |F t (n) j ) 2.8. Lemma. {A (n) b : n = 1, 2, · · · } is uniformly integrable. Proof. Let λ > 0 be fix and define the random variable T (n) λ by T (n) λ =      min{t (n) j−1 : j = 1, 2, , N and A (n) t (n) j > λ} b if {t (n) j−1 : j = 1, 2, , N and A (n) t (n) j > λ} = ∅ . Since A (n) is increasing, {T (n) λ  t (n) j−1 } = {A (n) t (n) j > λ}, and this set belongs to F t (n) j−1 by the previsibility of A (n) . It is easy to see that T (n) λ is a stopping time. By noting that {T (n) λ < b} = {A (n) b > λ} and that A (n) T (n) λ  λ on this set, we obtain 0  1 2  A (n) b >2λ A (n) b dP   A (n) b >2λ (A (n) b − λ)dP   A (n) b >2λ (A (n) b − A (n) T (n) λ )dP   Ω (A (n) b − A (n) T (n) λ )dP =  Ω (X b − X T (n) λ )dP =  {A (n) b >λ} (X b − X T (n) λ )dP. By using Chebyshev inequality we have: P{A (n) b > λ}  EA (n) b λ = EX b λ → 0 where λ → ∞. Hence, lim λ→∞ P(A (n) > λ) = 0 uniformly. Since X is assumed to be of class (DL),  {A (n) b >λ} (X b − X T (n) λ )dP → 0 where λ → ∞. Therefore,  A (n) >2λ A (n) b dP → 0 where λ → ∞, i.e., {A (n) b } is uniform integrability.  Now we return to the proof of Theorem 2.8. By the Dunford - Pettis theorem, there is a subsequence (A (n k ) b ) k∈N converging weakly to an integrable random variable A b . We claim that for any sub σ− algebra G of F, weakly- lim k→∞ E(A (n k ) b |G) = E(A b |G). To prove this, fix an arbitrary bounded random variable η. Then, lim k→∞ E(ηE(A (n k ) b |G)) = lim k→∞ E(E(ηE(A (n k ) b |G)|G)) = lim k→∞ E(E(A (n k ) b |G)E(η|G)) = lim k→∞ E(E(A (n k ) b E(η|G)|G)) = lim k→∞ E(A (n k ) b E(η|G)) = E(A b E(η|G)) = E(ηE(A b |G)). We now define the processes M and A by M t = E(X b − A b |F t ); A t = X t − M t ; ∀t ∈ [a, b], where A b is the weak limit point of an appropriate subsequence of (A (n) b ). In the first definition we take a right continuous version of the martingale M t which implies that the process A is right continuous, A t is integrable for each t. We see that A a = X a − E(X b − A b |F a ) = X a − weak- lim k→∞ E(X b − A (n k ) b |F a ) = X a − weak- lim k→∞ E(M (n k ) b |F a ) = X a − weak- lim k→∞ M (n k ) a = weak- lim k→∞ A (n k ) a = 0. Let Π =  n∈N π (n) . If a  s  t  b with s, t ∈ Π are fixed then A t − A s = X t − X s − E(X b − A b |F t ) − E(X b − A b |F s ) = X t − X s − weak- lim k→∞  E(X b − A (n k ) b |F t ) − E(X b − A (n k ) b |F s )  = X t − X s − weak- lim k→∞  E(M (n k ) b |F t ) − E(M (n k ) b |F s )  weak- lim k→∞  EA (n k ) t − EA (n k ) s   0. Since Π is countable and A is right continuous, it follows that there is a version of A t such that A t  A s for all t > s in [a; b] almost surely. It follows that A is increasing. Next we check that A is natural. Let ξ be any right continuous bounded martingale. By the predictability of A (n) then E  ξ b A (n) b  = Eξ b (A (n) σ(a) − A (n) a ) +  t (n) k ∈π (n) Eξ a (A (n) σ(t (n) k ) − A (n) σ(t (n) k−1 ) ) = Eξ σ(a) (A (n) σ(a) − A (n) a ) +  t (n) k ∈π (n) Eξ σ(t (n) k−1 ) (A (n) σ(t (n) k ) − A (n) σ(t (n) k−1 ) ). Where n enough large the right hand of the obove relation equals to Eξ σ(a) (A σ(a) − A a ) +  t (n) k ∈π (n) Eξ σ(t (n) k−1 ) (A σ(t (n) k ) − A σ(t (n) k−1 ) ). Letting n → ∞ we obtain Eξ σ(a) (A σ(a) − A a ) +  t (n) k ∈π (n) Eξ σ(t (n) k−1 ) (A σ(t (n) k ) − A σ(t (n) k−1 ) ) → E  [a,b) ξ s− ∆A(s), and E  ξ b A (n) b  → E [ξ b A b ] . So, we have E [ξ b A b ] = E  [a,b) ξ s− ∆A(s). Replacing ξ = (ξ s ) by ξ = ξ t∧s for each t ∈ [a, b] it easy to conclude that E [ξ t A t ] = E  [a,t) ξ s− ∆A(s). Thus A = (A t ) is natural. Finally, if X is of class (D), then X is uniformly integrable and the limit X ∞ = lim t→∞ X t exists almost surely and this limit belongs to L 1 . The Doob - Meyer decom- positions of the discrete submartingales X (n) along the partitions π (n) = {t (n) j : j ∈ N} of T a , are then uniformly integrable as well, and we may define A ∞ as the weak limit of an appropriate subsequence of (A (n) ∞ ) n∈N , where A (n) ∞ := lim j→∞ A (n) t (n) j . The details carry over almost verbatim.  References [...]... York Inc, 1979 Tóm tắt Biểu diễn doob - Meyer đối với martingale dưới trên thang thời gian Mục đích của bài báo này là nghiên cứu biểu diễn Doob - Mayer đối với Martingale dưới trên thang thời gian Kết quả đạt được có thể xem là sự tổng quát hóa của biểu diễn Doob - Mayer với thời gian rời rạc và thời gian liên tục (a) Department of Mathematics,Vinh University, Nghe An, Vietnam ... 20 9-2 45 11 Alberto Cabada, Dolores R Vivero, Expression of the Lebesgue -Integral on Time Scales as a usual Lebesgue Integral; Application to the calculus of Antiderivatives, Mathematical and Computer Modelling, 43, (2006)194207 12 I.I Gihman, A.V Skorokhod, The Theory of Stochastic processes III, Springer Verlag New York Inc, 1979 Tóm tắt Biểu diễn doob - Meyer đối với martingale dưới trên thang thời. .. Math., 18 (1990), No 1-2 , 1856 7 M Bohner and A Peterson, Dynamic Equations on Time Scales, Birkhauser ă Boston, Massachusetts, 2001 8 P.A Meyer, A decomposition theorem for supermartingales, Illinois J Math., 6 (1962), 193205 9 P.A Meyer, Decomposition of supermartingales: The uniqueness theorem, Illinois J Math., 7 (1963), 117 10 Kunita H and Wantanabe S, On square integrable martingales, Nagoya Math...2 Doob J.L Stochastic processes, John Wiley and Sons, New York, 1953 3 N Ikeda and S Wantanabe, Stochastic differential equations and diffusion processes, North Holland, Amstardam, 1981 4 Peter Medvegyev, . inf T graininess µ : T → R + ∪ {0} µ(t) = σ(t) − t t ∈ T right- dense σ(t) = t right-scattered σ(t) > t left-dense ρ(t) = t left-scattered ρ(t) < t 1 isolated t a, b ∈ T [a, b] {t ∈ T :. have lim k→∞ E(ξY n k ) = E(ξY ) X - (D) {X τ : τ is a stopping time satisfying a  τ < ∞} - (DL) t ∈ T a {X τ : τ is a stopping time satisfying a  τ  t} 2.7. Theorem (Doob- Meyer decomposition). Let. the Doob - Meyer decomposition of the finite submartingale X (n) = (X t (n) j ) t (n) j ∈π (n) X t (n) j = M (n) t (n) j + A (n) t (n) j Thus, M (n) = {M (n) t (n) j } t (n) j ∈π (n) is a martingale