N a.. n o s.. c a l e.. S.. y.. s t e.. m.. s.. M.. M.. T.. A bullet.. I SSN : hyphen 3290 \ centerline

Size: px
Start display at page:

Download "N a.. n o s.. c a l e.. S.. y.. s t e.. m.. s.. M.. M.. T.. A bullet.. I SSN : hyphen 3290 \ centerline"

Transcription

1 N S M M SSN : S N D O : S 8 N M N SSN : S M M SSN : V 4 S D O : / 8 N M M V M S 4 D O : / M 8 N M M V W - 4 F X - * C D J 3 S S M - W F X À V C D J 3 S - H 8 D 93 B - C F - U - B H 8 D 93B í F C B - - D M C W F S ˆ V D M C ˆ C D S z 3 S ˆ S - z U 85 P z B U 85 P B z C M R C M R WW 3 3 C C C C B 8 z z C C S S ˆ ˆ P C C - ˆ P C C ˆ ˆ H 8 ˆ D 93 R O B ˆ R O K C ˆ F ˆ U ˆ K P P N N P - P S B R 5D R 5 6 D 6 D N N PCS ˆ PCS : 44 : MSC D M C S MSC : 74: V V - z z ˆ ˆ ˆ z B k x x N N z z x---- U k-- P ˆ B z - ˆ x k C M R / k B z W 3 B z C 4 7 B 8 z z z x z z x z- --k-----x k P C k- C ˆ k 34 z k k H R O - k K H P - N ˆ R 5 D ˆ - ˆ ˆ N x PCS : x k G k - G MSC K- : K x τ - κ -x 7 8 k - τ x k - k V z - M κ x k Z - : x B k k x N : k k - - Z z k - * N MM : Vx z : x - : -j S : - - N MM V 4 B

2 M k k z τ /3 G K G z K k O x - τ x M τ τ τ τ τ τ x k - τ x - τ x - O - κ z τ τ κ M τn τ/5 τ N x τ z z F x τ τ τ τ τ k κ κ G K x z x z F κ x z x τ z z M x C V τ κ 3 k z B z z / - H x k x z k k x x k k z k k xk k

3 M k M - -k z- 3 k G k K - x ˆ 34 ˆ / 3 G G ˆ K ˆ z- K k- O - k z x K x M : κ x 5 z k x M ˆ ˆ x ˆ ˆ x O x k K K K / k x z x z z k O k k M z N 5 N x κ 6 ˆ k M N MM V z-z ˆ - - N / 5 - N k F x zz - ˆ k k F k G - - x - K x ˆ ˆ ˆ k z x k G K x z F - : k x z z x x z z M C ˆ V z k : 3 - k k- k k z B z z- x z z H x k x - z ---- ˆ -z--- M --k-- x - - C ˆ -k V ˆ -x--x k z k- - - k k x- -k-- - -k - - k

4 À F X z V C D J S N - S k 7 η η x x xx W N - - k x x x - K k F H x / k z R k F x - z z! N - S k - - : ξ F - 8 ξ ξ S - 8 k ξ M k C k z B z

5 F X z V C D J S -x F X z V N C S k D J S - xx W x ˆ M x x k k : 7 k - / k - 5 /9 -x N ˆ MM V x x--x-- x x ˆ W ˆ N x x k x x x K k F H N x k x x x F k H x k z ˆ ˆ R - - k- -F x z R k z k F x k! N S k 34 ] : 3 x x 5 ] x x! 3 x F 8 x x S 34 8 ] k M k- -C k ˆ

6 M D - - zxzzx J - k C ξ 9 C ξ α ξ x k M C αx z zzx k R - x - x : K x K x x k k : x k K O K x K x k x x k L x R kk k x k x kk k - x U x x k k - k M - k x C α z k B z z z k k k k R 7 z k - M x F - x x x x Mx - C xx k τ C ξ α ξ z τ x - S k M τ τ τ τ k τ τ τ k 9

7 M D M D k -- -z----x- z-z-x τ - - J k k zx S zzx C x 9 D k F ˆ k C x 9 C x x M C -- --x---z- τ D D -z-z-x-- D -k C - - x R N MM V x ˆ x x : x K k M C -x - x z K zzx k x x R k k : ˆ x x x k K x x ˆ : K k O K K O x K K x x k K x k x x- - --k -L - -x - - R k k x k k x L x R k k k ˆ k x --x --k k k k x k k k x k k x U - -x - ˆ U x- - k- x k k k M k - x x-- -- k k C k M ˆ x - -z - - k B C k -- z-z-z -k-- k-k ˆ z k - -k B R k zzz k k k ˆ k 7 z k- 7 k z M k x F M x - x x x Mx ˆ C

8 F X z V C D J S D / D D / D k k D / D R / ˆ 34 D R : F D k D k W D / D D O F k x F D W x x k x x : k ˆ k ˆ x j Z Z z k Z ˆ k k S k k k S ˆ Z k Z O x j S G P x H N MM V 4 6 F X z V C D J S D D C D C D D C k k k C D D R C D R : F D C D k D k W C D D C D F O- - - F k F D C D W x- - k-x x z R k k x : k k - : P x j Sk Z Z - -x z- ---k--- k : Z k k S k k k S k Z - - -k k Z O x j x S G P x B z k H k F X À z V C D J S D / D / D k k k / D τ R / R τ : F D k D / D O F k x F D W x x k x x z R k - x : k k : P x j Sk Z Z x z k Z σ κ κ S k σ κ κ S Z k Z O x j x S G P x B z H k z k N MM V 4 6

9 M x : x F z x x k k x τ κ 4 τ κ Πσ Πσ 4 τ σ 4 τ ; τ ; τ ; x 5 x z x Π P z z S P 956 ku k k C k S N S L D L D R xk S z k k S S xz k - κ φ k k κ φ κ 5 φ D z z - z F L Q

10 M M x- : - -x x ˆ x ˆ κ : 3 Q L 6 - ˆ ˆ - Sk 3 ˆ ˆ L F z ˆ x- -- ˆ 3 -x - k- k x ˆ : 4 ˆ - N MM V 4 7 k F ˆ - z k x C P ˆ ˆ x k k C P 4 - ˆ ˆ ˆ k ˆ ˆ ˆ ˆ x k 5 -x-z --x ˆ C P ˆ P P ˆ ˆ ˆ z P - 4 z ˆ S ˆ ˆ 4 P k-u k - ; - - -k - ; x x - - z k C - x k P S ˆ N S L D L D z -R-- S x k S P k k U ˆ k k k - k k S C k S x--- N --z--k S L - -- R x -- - k ˆ k S z k k k k- S -k : 5 k x zk ˆ D z k -

11 À F X z V C D J S z z S K z x x z F S S z C α S B z S - x S - W x S z M k G O - k z x k S z - - F z - x - L k z F x k k k - κ L 7

12 F X z V C D J S F X z V C D J S x / z / z z z κ C S S K L 8 z N MM V 4 8 z x z x z S F - -S - - S - - z S z - - C C - -S - - S B B - -z z ˆ ˆ ˆ ˆ S S x -S - - ˆ W x x S W x - -S z S M z M --k k ˆ G O k G z O k x k x S z k F x F ˆ z k z F x L x

13 M F M M -x - -k k F F k k L x k /; /] x-z - /; /] L k / ; / ] x --C -x-z -- --k -- - R k k - - Rk - - κ K κ K k K K C / k K ] ; 9 / K / ] K W K k C K k κ K κ 9 K 6 C K xz K K W K K x C x z k x ˆ k k W x k K k K k x x C k z K k K K K K x- C K ] x----k 9 K / K W K ˆ x x k k - - -x -- -x- - - k K k K ˆ -z x R K x x z k -- K -- ˆ x R x-- --z K x ˆ -- W O x ˆ O k x x x ˆ z κ x K κ K 9 π γ 5 5 π γ ] κ K 8 π γ 3 π 3 γ ] C k γ K / Uk - K x γ k 5 K 5 κ K κ K 63γ 6 C K k 7γ] k K N k 3 κ k 3 z k C γ U γ x x k N MM x V K 4 z 9 k K k 63 6 C K 7 k N k -- -z O N MM V 4 9 x k K k K ˆ 9 ˆ 5 ˆ ] k K 8 ˆ 3 ˆ ] C / U k K k K ˆ 63 6 C K 7 ] N k ˆ z

14 F X z V C D J S F X z V C K D J S F X À z V C D J S K K - - N P N P S P G N S O O S P G N k N P S O O L W O k O j x k L k F W F 3 N L z j x j x k -F - - z kx F - W L W F 3 N L z - z x M F k 3 N K k L z x - x K / W L W - z x k k x z x M k W K L k W z x K x; κ x z x M Φ 3 L k Φ / x K - - / k- k x k : 3 x ˆ k C P k x C x x κ C D L Φ Φ ] 4 L C P z - x - - ; - k - - ˆ : 4 P ˆ x x ˆ ˆ k D C L C P 3 k - x κ K κ K K C Φ P CS φ] 5 C D L S - -z Φ - P k k / x ˆ : 5 k K k k K ˆ C P CS k Φ S SΦ C P Φ - - k- x x k k ] 6 C F : x 6 ˆ S ˆ P C ˆ P ] k L C K Pκ 884W K C Φ P / F N MM V F z K k 884 W K C P F k K k K K P N ˆ MM V CS ] 4 5 S P k x S P 6976 P ˆ 95 P ˆ ] 6 F K k 884 W ˆ K ˆ 557 P /

15 M F 4 M P-- - M -- 5 F 4 F 4 5 K K P P 5 3 K 5 K S 5 5 K K K 3 K 5 K C P K S K / 5 K Φ C P K ˆ 3 K 5 K S k Φ κ F R L F k K / P R - R L - - k K K F - - F - - F 5 F 5 C C R - R L L P z z k x K x K F x x N MM L V k 4 N MM V ˆ 4 K F F 5 C R K x N MM V 4

16 À F X z V C D J S W - R L M κ z 7 z z x F x x z B R z k κ J / ] J R/ L 8 z / L x J z z - B k - κ C J ] R/ J R/ J / L 9 J z - B B J z/z J z S R W R z κ K κ K J/K J /K ] J /K J /K C ] 3 K K / R K x x k κ K κ 8K 4 C K 3 K K z 3 R k F W F - D z k z k R C α x x x x x x C α D α C D 3 C α D / D k

17 F X z V C D J S W F X z V C D J S x α / D O W D R L M C x NL MM M V : 7 ˆ k z z z - -x k --F - -- z - ˆ x ˆ 7 x z B R z z k z : x 8 k J F J x R k C D L z C D L x z z x -B - - z k B R z : 9 k C k J R J R k k J J C J D L R / ] D L J z B J z z J z S z R D / L J z z - B - - W R z- - k k C J R k / J ˆ K k R / ˆ ] K k J J / K D J L K 9 k k J K J C k 3 K J z K R K J z x / z J z x S - R -- --k k k K W k 8 K 3 K K - z- - - ˆ 3 R ˆ k - F W- -- z C D D - z- - k K k k k K - -z k-- / K J / K - ] - ˆ - - R J C / K - - J / K - C ] - : 3 x-x-x--x-x-x- C D K D K C / R C D x D C x C D D C D D C D k x k C D D O D

18 M H C α / D R C α z k B x R K R κ κ 8K 4CK 4αK 33 κ K ; D κ 8K 4C K κ K D C α K 34 C α R k S x K κ 884W K S 557 C 46 α 8 C α z k R k x k R k z α x C x G - K C K R K α C - x k R 4 C G - K K! R ] κ W K ] κ W K ] κ W K! - 3 D 6 k k R k k K 3 K k 3 D 6 C x x

19 M H M C- C D D - H R- - k C C D / D! C D R C ˆ R ˆ ] κ W K ] κ W K ] - -z - -k- D k B-x -- z ˆ - k - -! R K R k B x : 33 k k 8 K K K 4 K R zk R k V V k K xc D D k 8 K 4 C K k K C D k C k - 8 K ˆ K 4 C K 4 33 K x C R - - -k - - G - K S x KN MM V 4 3 k k 884 W K ; K D D S k 557 K --- ˆ C ˆ K k K C C ˆ 46 ˆ K 34 8 C -z - -k R - - C k k k-- -x k S - R k x k W ˆ - K ˆ S C D z x C ˆ x 8 G K C ˆ z k R C k - x ˆ K R k R k K C x k R k - 4 C ˆ G K K D x C! x G K R k k k K k k k W K k k K k! K R K D 3 D 6 C k - - k R-- x -- ˆ - -k k- K k 3 R 4 K k C ˆ D 3 D 6 C k x - -- x - G K - - R K k -! 34 ] R k k k k W k k k W K k

20 À F X z V C D J S k F F R k ±/ κ K K ] L 35 O W k κ κ α K C K ] 36 - G - K 4 F - k x z z k R - x k : ω ω; δ 37 ; ω; R 4 ω 37 ω x k k - P x zk k k F K z R k ; ω; κ ω R 4 ωτr ]δ L ω 38 k - k - ω; ω 39

21 F X z V C D J S - - F X - - z V C D J S z z - --k τ C R - α - - k ˆ R -F N MM k V 4 4 k F x F : 35 k k - - -K / K- k C D O k - ˆ - - ˆ ˆ K -W ˆ K - - L - -k k k k K C K O k 36 G W K F k k K K ] 36 k x z - -z - -k R -- k ˆ 4 F x z ˆ k ˆ k : : 37 C D - x k : D ; ˆ x k k -P-x- z k-- ; ; k -- -k- R ˆ -- ˆ 4 - ˆ - ˆ F K - z- - R - - -k ˆ ˆ : 38 ˆ k k R 4 R k ˆ - L - - x k k k k Px z k k k ˆ : 39 F - -- z-z

22 M F 6 M B k M Z F 6 FB6 k B - k z z x k - Z Z z - z x k x-x P z - - F z F P F x- x- -k z x x k z R k x x R P k C ˆ - / D k τ k R R / 8 43 F 3 F F R 97 R R x x k R τ 5 F F-- k k R k --k k z - - -z D D κ z k R / ˆ 34 κ ω; k K R - Q L z- πr - k δ K ω τ κ ω τ ωτ 6K 4ω ] k K R Q K 43 R L ˆ F K R 97 k R k C 6 K τ K κ ω τ α 4 ω τ ω R τ κ ω ττ K ω τ ] R 4 τ K K k : 4 5 ˆ C4 K k F - ωτ - k k K z R R K ωτ k z κ K τ C τ - - k τ τ K α - ; K τ K ] α κ τ K 4 R τ Q K ˆ ˆ ˆ k : ˆ 6 K 4 ˆ 4 x k 4 K ˆ 4 K R ] C R 97 F F R K K - k ωτ W k K z 3 D ˆ - - -x - C - - K k K 5 K F ˆ - - ˆ ˆ K 5 K F R 97 ˆ ˆ - F- -- F- k D x - K ˆ - ˆ ˆ π 4 /5 / B K 3 G S W /M S S 33 ] z 3 K C 4 D 6 3 B 645 K D G 83 J K M 3 K - S 8 -F - 3 K - K M x τ 5 K - F τ - 4 N MM V B 3 G S M S S 33 3 K 3 B 645 K D G 8 3 J K M S 8 3 K k M K x - - ˆ K ˆ ] N MM V k K ˆ ˆ R 97 F F W D 3 D 6

23 F X z V C D J S F -- F --- X F z - V - - C D - - J - -F S - - F X À z V C D J S F F F F F - F k ω F k - - F α k B B - F F z F - F- -z - - B z z - - M F F k z F F 7 B - k ωτ z k F 7 B ˆ 3 K 5 K M K - k k 3 K 5 K K 686 K 689 R 97 3 K D F 67 B z K 68 6 K 68 9 R 97 κ 3 D 6 z 5 S z - - k 3 K 5 S z 5 K z K K S 97 D Z 3 k D z- z Z k S Z - k - k- - - Z k k Z 5 S z k - x x z 3 Z Z k- - - k k- z Z x- --x-z S - - k 3 Z k Z z 3 k S Z 3 Z k F x z Z k S κ k xz xz F ˆ F x ˆ x σ Z 3 z Z 3 N MM V 4 6 Z - z F F z- x k k N MM V 4 6 Z F x ˆ z ˆ N MM V 4 6

24 M F 8 M B M -- - k F 8 FB 8 B - K 5 K ωτ k K K 5 7 K 9 k R 97 C K 5 K K D 6 z k K 5 K 57 K 9 R x- - -k - D 6 K 5 z z 7 k K 9 97 D 3 κ D 6 W z x k z R k k W k x R x R < ˆ k : 43 k k x k W k R < k C P k C P k k κ ˆ ˆ 43 κ k ˆ - z Πσ Πσ ] 43 : σ z k C D L R 4 ˆ ˆ : ˆ 44 k P P ] C D 43 k k C ˆ P C D ˆ L C P 44 R 43 4 z C D κ C D L R 44 ˆ 4 k 44 D L R ˆ ˆ σˆ L - ˆ - R 44 4 D ˆ k R P D L P R ] ˆ 4 R ˆ R R 44 N N MM V 4 7 MM V 4 7 ˆ D ˆ D L ˆ 4 ˆ 44 ˆ R R R N MM V 4 7

25 F X z V C D J S F 9 F B X z V C D J S F X À z V C D J S - k F 9 F B 9 B - 3 K 5 K ωτ k R 97 3 K 5 K z k R 97 -N z- k-- k 3 K 5 K 97 z κ : 45 Nz k z : k N z k : x : : ˆ 46 x ˆ : ˆ x-x-x 45 ˆ ˆ x-x-x R ˆ C C x-x-x R 46 x-x-x R : C C x-x-x 46 R x-x-x R C R 46 C - - ˆ C - C R ˆ C C k ˆ C z M ˆ - - R ˆ 46 C M zk R ˆ R k 46 : ˆ ˆ C - ˆ -z k-r- ˆ - σ - 4C -k R ˆ - ˆ Z 47 - : κ ˆ 4C ˆ C ˆ k ˆ R κ 4C : 47 ˆ Z R ˆ ˆ ˆ 4 R ˆ ˆ k 4 C N MM V 4 8 ˆ R ˆ k ˆ 4 C M R N MM V 4 8 z k R k ˆ : Z ˆ 4 C ˆ 3 k 4 C ˆ 34 ˆ k ˆ ˆ 4 47 N MM V 4 8

26 M F M B - - M -- k F F B B - K 5 K ωτ k R 97 k K 5 K z k R 97 z κ K F R ˆ F97 R : 48 z Z k k R C R C σ Z 48 F R C κ κ Z C C C Z C C C Z ˆ C C - k C k C x x 48 / - - x -x H H Z Z Z C C /C C C C F Z C C B K C F Z /C B 3 3 K C k R x k / k-- B R x - B 3 4 V B K k k 4 V ] C B / 3 C 3 K - - k F Z κ W K ]κ W k K B ] ] k W ] 3 3 KJ 3 K C / k C k k W K k k k! k k J 3 K k k C O k R k V C k k! B F V 3 3 K kc C Z ] z R C Z 34 < R ] < z R x R x N MM V 4 9 k K N ˆ MM ] V k W K 4 ] 9 ] ] J 3 K ] O ˆ ] V ˆ C ˆ ] 3 x x 5 ] x! 3 x 34 ] ˆ ˆ F C C Z z R

27 F X z V C D J S F F B X Z z V - C 8 D J S F X À z V C D J S 3 3 K C C R F F B B Z Z 8 B K B C /C 3 R < 3 K k x / C R < k x x x - x k - k x x x x x ˆ L -- -z- --x k x L z x k k 49 z κ x k k 49 C C P 49 ˆ ˆ k ˆ κ 49 Πσ Πσ ] 49 σ : : ˆ 5 k k P C P k P C P 49 : C P C P κ Πσ κ Πσ 5 5 k k R 8 Πσ : Πσ 5 5 R : κ κ 8 5 O σ Z : 5 Z σ : 5 R σ σ 5 k k 8 N MM V 4 3 k O P - - Z : k P 5 P P 5 k k Z σ R R ˆ 8 ˆ 5 5 κ 5 κ 8 5 N MM V 4 3 O Z ˆ R ˆ 5 k ˆ 34 8 ˆ N MM V 4 3

28 M x Z F / R z κ Z k σ /κ - - R B 3 x z x x κ C R 5 κ κ R 8 4C R 53 Z z k Z σ C κ κ R R F K / R Z Z σ /κ F Z K C B 3 3 K F C k - C Z x C F L K R 6 P - z x S z k z S k z k R k η D - D η x R R /ν ν k S k D η 6πη 55 D η R H W

29 M x M Z- F - x k - -R x z Z - -z k k / ˆ R C k z ˆ k Z - R k Γ / k D η / 6πη k /Γ x : D η k R B 3 - -z - x k x : k 5 z x x-x-x D η 6πη /Γ Γ Γ N MM V x-x-x 4 3 R x k x ˆ : 53 ˆ k k R 8 ˆ 4 C R z x Z - -z - -k k : 54 Z k k ˆ R C R C R R ˆ R F 5 K R Z Z k - -- x k F Z K C k k B 3 R 3 ˆ K F- 8 - ˆ C - ˆ 34 - ˆ R 53 C k k C Z x - - C - -L Z K z k R Z 6 P z ˆ k k x S R ˆ z k-z k - - K / R z - -- Z - -k Z / k R k- Z K C D D x B 3 F R ˆ k S k C k : ˆ C 55 D 6 Z x D R F L K H W R x k 6 P z z x S k ˆ - S C - k - k : 56 C G 57 4

30 F X z V C D J S F F B X Z z B V C 3 D J 3 K S F X À z V C D J S : - C F F B B Z Z B 3 3 K B K 3 : - 4 C - K : C R D C C k 4C < R < D C k x Z R Z R 4 C C < R < D C k - Γ C x Z : 57 Γ 76 C 3 G SCD SCD C Γ G B CCD 3 57 B CCD C G SCD S- --C D C C D C- C- RD D Γ G B k x RD SCD 57 C G B k x L - - k L C G 3 Γ 3 RD RD G -- -R -k k S B C R - x k - k - S - z x k F- - ˆ S ˆ k C D - F ˆ S C C D S - RD G B k x - W G W 3 RD : ] 6π 6 59 C G Γ G k Γ κ C Γ G C G --- -x Γ --- x R k S z x φ - - N ˆ MM k N MM V 4 3 V 4 3 F S W 6 ˆ G G k 59 G G x ˆ N MM V 3

31 M F 3 M Sk S - - M F 3 F 3 Sk Sk S S RD x x RD x x SC D B CC D F k - SC D B CC D RD F k x ˆ B CC D F k z z κ F ˆ z k - F - : 6 V k V k V F κ φ k 6 N- - - V - k - N N S V k V N S - - V : 6 k 6 N C G C G V V k k k N x-s πN V - - x N S Γ N Γ N φ ] V 6 κ V V V V x S x N φ N V / φ V - -k- k V 4/3π 3 : 6 k Q k 6 N ˆ k G 9 G K C V G C ] G k V 6 Q x κ Q κ x S x φ 9 φ 6 K Γ Γ K x- - x - K-- ˆ L k N Q x N V / V x : K /x K V L k 4 / 3 K ˆ K 3 : z k φ K K F K -x-- --k-- z F - K xk k k x Q - - k k 9 K ˆ G G ˆ x 6 x 3 φ φ x - - x - S - - S - Q x κ κ k φ k 3 x K / x 3 K L k x x : -x -x K K N MM V F K x k ˆ - k N MM V 4 33 x ˆ ˆ 3 x S k k ˆ ˆ 3 x x ˆ

32 F X À z V C D J S 5x κ S φ 3 φ 3 K S φ%] ] κ W K ] x κ W K ]! 6 S x D SCD BCCD RD φ z 3 K 4 κ 48W K ] κ W K ] x SCD κ W x K BCCD ] κ W K! F κ κ φ 3 φ 6 K SCD BCCD RD F k φ φ 3 kz 4 ] H L x x x z F K S κ B x F B C C D S Γ - xxk x k C - W z k F

33 F X z V C D J S 5 F x X z V C D J S 5 x x ˆ k k k S ˆ k k ˆ S ˆ ˆ ˆ C ˆ 3 ˆ α k ˆ ˆ R 3 K Γ W L L x x : S k k k k k 34 W K k R x k 34 k ] W C K k L L S Γ! 3C % ] ] W ˆ K ˆ ] x 63 3α π L L k 6 ˆ C W ˆ α ˆ K ˆ S ] x- D SCD BCCD RD xzk 34 S k z 34 3 K ] x - S N MM 4 V k 48 4 W 34 K L L S k k k k k W K k R -x 3 x x SCD k 5 ] x! k W -x 3 x K BCCD k C k k W K k C RD L L 34! 34 ] L L - F - - k k 6 S x D 3 6 SCD K BCCD RD SCD BCCD RD z F k - 48 W ˆ -k z-- 4 k H L 34 x- - -x ] -x z - - S % - -- ] F ] K W ˆ K ˆ S ] & x & SCD k W x ˆ ˆ k K ˆ ] & k W ˆ K ˆ ] & x RD B--x F S ] C G --- -x--x --k x k C 3 x k x 5 ] x! 3 x W - --z -- -k F x k k 34 ] C - k R k F C G W k k k - x- - 3 x : 6 C : 63 C G 3 C 3 F C

34 M F 4 M B - - k- k K 3 K M 4 8 W K 4 9 F 4 F 4 B B k κ k K 3 K - -x- κ 48W K K K k x 4 8 W ˆ 4 ˆ 9 φ φ 8 M M SCD BCCD φ SCD BCCD R- -D R D - F k F - 5 D F k F 5 D - 8 F 5 P M F 5 SCD P BCCD k k D W k k D W R k D k O ˆ Γ C G H 5 S D x H k x S x k 7 O -x F F 5 P 7 O F D k x D W k k - - H x G S x z x k N MM V H O F z C k N MM V 4 35 x H D k N MM V 4 35

35 F X À z V C D J S D - λ x k α λ D x C α k k D F κ C α x C α - κ O S x - k - k x - x - C α kf k O k - - x S z x τ x τ k x k z x x zk k x x k x z S k k J x 5] G - K k z k τ - τ τ k k M F O z - x x z xk F O k

36 F X z V C D J S F X D z V C D J S x C D F z D - x k zk D D x C H x ˆ z x D x C ˆ x k - k - k ˆ - D F k C k k D F - k C C ˆ ˆ x C ˆ ˆ S k k O S N MM O V 4 36 S k k x k x x- - x C C k F k-- --F k O - -O k - - k ˆ - x x S- - S z z x x x k x z k x x x k z z k k ˆ -x- --z k-- x - - k x - k x- z x - -k x - S z k S ˆ k J x 5 ] k G - - K k - k J x- k 5 ˆ z G k K k -- --z k

37 M z k x τ L θ L κ MH θ 64 τ k x k τ κ ρ γ γ G ü Θ x M ] L L λρ/γ θ M H ρ/ θ γ θ x k M - x κ MH x x W x MH x z k x L x x x x L x z j zzk x R x z M zk x x k x k x H R x x x x H x z x x z x R z τ µ µ κ β 65 β µ µ x W x x z S zk

38 M - - M z-- -- k- x k x - L - L k z R k 64 x x x k x L k k M ˆ H G C -x - k x k L k k k k x k ˆ M H x - z k x G D J F X À z k Dó G ó - M ] -- S M-- - -k- S - M- F - S C C Pj N CS D Dó G k R G -x - - C SGR k L / N MM V M H 4 37 x x- ˆ M - H / W x x M k H x- - M z - k --x L x - x x k x x - -L x z W z-z- -- -k- R k x z M ˆ H x z k - -z k- - x k L x x k x x x k x ˆ ˆ L -x - k x x H R -- k x x x j zz k x H x x z x x z - -z z x- - - k R- - k z - - x k x k 65 -x W x-- x --- k x

39 F X z V C D J S S k U B D G S M S C U B M GNFM P G V C k G GNFM k U B M P z z R ] C C k R G P ] F X z D J S P J P ] F X z D J S ˆ : P L ] F X z D J S J H ] V M M Z G P ˆ S ] M Y K L S S W K P L ] C L B R W B D R P ] J R B D F F B S H L J H M ] B K W J P ] L B C S ] G B L B R S P : M S P ] G B C P ˆ R L ] B k Y B J K J K F X z V C D J S S k U B j M D G S M S 337 C U B G F M GNFM P G V C k G Nz F GNFM k U B M P z z R k k C C k R G P k k F X z D J - S P -J P k 3 k F X z D J S - z - : - P L k 4 k F X z D J S - - J H 33 4 k 5 k V M M Z - G P S S k 6 k M Y K L S- -S W K G P P L k 7 k C L B R W B - -D R D L k M R P C P k 8 k J R B D F F B S L - - J -H M k 9 k B K W J P k k L B C - -S k k G B L B - R S - - P : M S P -x k k G B C - -P R L k 3 k B k Y B J J K Y W G J R S N k 4 k H C B k x S R k 5 k H B - - Ox U O x k 6 k B J B SM k 7 k L C S P- L k 8 k B Y C Z G F -J P k 9 k C C S z -- - S F U M F X À z V C D J S S k U B j M Dó G ó S M S F S C - U B G Nz F M - GNFM P G V C k G Nz F M - GNFM k U B M P z z R k R G P 5 ] F X À z D J S P - J P ] F X À z D J S P - z : P L ] F X À z D J S P J H ] V M M Z G P S S ] M Y K L S S W K G P - P L R W B D R D L k M R P C P ] J R B D F F B S H L J H M ] B K W J P ] L B C S ] G B L B R S P : M S P x C P R L 3 ] B k Y B J - K J - K Y W G - J R S N ] H C B k x S R ] H B Ox U P O x 6 ] B J B - SM ] L C S P L ] B - Y C Z - G - F J P ] C C S z S M F U M ] C C x C R ] C C H Bz C U B k R R S B k 9 ] C C S R F - M - M S B k ä V B 6 N MM V 4 38

40 M 3 ] G C N C P M P P Ox U P O x 4 ] J D C M K J H M ] V C D J N ] V C K F N P B ] V C K F G P B ] V C S D J N P R B ] V C S D J N P R B ] V C S D J N : P R B ] J C O R P L ] C O P R S L ] V D K M M R L ] 34 ] Y D B Y C Z Y G G J P ] S V N N S B 36 ] M S z G k H H S H 3 K C M ] J F L P S J P ] J F L P P L ] J F C B K Y H J P C C 4 ] D K F S M G k N C U P C 9 4 ] Y C F B S 4 ] M G G C P ˆ R ] G G L V D J F k R F J P D P M k 3 k G C N C P - M P P - Ox U - P O x k 4 k J D C M K - - J H M k 5 k V C D J- -N k 6 k V C K F N z P B k 7 k V C K F G x P- B k 8 k V C S - D J - - P R B k 9 k V C S - D J N - P R B k 3 k V C S - D J : P- R B k 3 k J C O R P- L k 3 k C O P- - - R S L k 33 k V D K - M M - -P -- - R L k 34 k Y D B Y C Z Y- - G - J P 6354 k 35 k S V N N - S- B - k 36 k M S z G k H H S H 3 K C M k 37 k J F L P S J- P 6435 k 38 k J F L P P L - 99 k 39 k J F C B K Y- H L P - J- --P - - C C 6 96 k 4 k D K F S M G k-- N - - C U P - C 9 k 4 k Y C F B - S V N Y k 9 k 4 k M G G C P R k 43 k G G L - -V D J k R F J P D P k 44 k G Gk C U P C 9 9 k 45 k G F S - G J M B F M 3 ] G C N C - P M P P Ox U P O x 4 ] J D C M K J H M ] V C D J N ] V C K F N z P B ] V C K F G z x P B ] V C S D J N P R B ] V C S D J N P R B ] V C S D J N : P R B ] J C O R P L ] C O P R S L ] V D ć K M M P R L ] Y D B - Y C Z Y G G J P ] S V N N S B 36 ] M S z - G k H H S H 3 K C M ] J F L P S J P ] J F L P P L ] J F C B K Y H S H L P J P C C ] D K F S M G k N C U P C 9 4 ] Y - C F B S - V N Y k 9 4 ] M G í - G í C P R ] G G L V D J F k R - F - J P D P ] G Gk C U P C ] G F S G J M B / F ] Z - Y G M M - M J G ] Z - Y G Q W H J SM ] Z - Y G B - C H Y Z Q G Z S P S ] R G K S z B z P R ] R G K P R N MM V 4 39

41 F X z V C D J S 5 ] N G H j C C 5 ] L D H k M S D R B ] L D H k M S D ] R J H D L K J C L M R J F M ] H R C R D D W L C G N ] P H k P M N L M P S P G F P N J N 57 ] Y H G D x J L SM ] M J H R H Y B W J H M ] K 6 ] G J H L Y L X W G Z D W R W G Z R F M N N L ] D J L R M C P ] D J J C V z z G L x P ] D J J C V z z G L x 64 ] D J M C S J C V z z H P ] D J G L M C S V P R ] D J S F X z H 5 67 ] D J V C S N J H M ] H K K G MG H N Y k 69 ] J Y K B J Y C M ] V K C ] G L D J J C V z z W Mk Wk J N ] G L D J J C V z z U 73 ] G L H M M G C D x P R S F X z V C D J S k 5 k N G H j C C P F k 5 k L D H k M S D P R B k 53 k L D H k M S D P R B k 54 k R J H D L K L M R J F M k 55 k H R C R D W L C G M N j Mj Y N k 56 k P H k P M N L P S P R G K C G F P N J N k 57 k Y H G D x J L SM k 58 k M J H R H Y B W J H M k 59 k K S S k 6 k G J H L Y L X G Z D W R W G D C C M Y M S D G C Z R F M N G B k N L k 6 k D J L R M : C P k 6 k D J J C Vz z G x P P k 63 k D J J C Vz z G x S B k 64 k D J M C S J Vz z H J P k 65 k D J G L M C V P R k 66 k D J S F X H P R S k 67 k D J V C S N : J H M k 68 k H K K G MG N Y k 938 k 69 k J Y K B J Y J C M k 7 k V K C P R S k 7 k G L D J J C Vz F X À z V C D J S ] N G H j C C - P F ] L D H k M S D - P R B ] L D H k M S D - P R B ] R J H D L K J C L M - R - J F M ] H R C R D D W L C G M N j Mj P Y N ] P H k P M N L M P S P R G K C N G F - P N J N ] Y H G D x - J L - SM ] M - J H R - H Y - B W - J H M ] K S - S ] G J H L Y L X W G Z D W R W G D C C M Y M S D G C Z R F - - M N - S G B k N L ] D J L R M : C P ] D J J C - Váz z G L x R P P ] D J J C - Váz z G L x S B 64 ] D J M C - S J C - Váz z H J P ] D J G L M C - S V P R ] D J S F X À z H - P R S ] D J V C S N : J H M ] H K K G MG - H N Y k ] J Y K B J Y J C M ] V K C P R S ] G L D J J C - Váz z W Mk Wk J N ] G L D J J C - Váz z U S B 8 73 ] G L H M M G C D x - P R S ] G L D J P C D B F - P L ] H L D V D Z W M S D Z F R G C S G J P ] J - H L J C G J R G G L S : M P L ] D L Y W P K L S P Y Mj P L ] R L L K P H C N N MM V 4 4

42 M J ] W L M P P L ] H L D ] R L z z R V J G P S M N F F P K D 8 ] F M k K G H J H M ] P M Z k j P U R S P R ] D M F C x M N ] R M k P R 86 ] N M V P L ] N M P L ] G M M : / / ˆ / M 5 P / / 4 D 89 ] Y M M R 5 J 9 ] D N k M M H Y Y H J P ] P C K M M B 3 S 3 J H ] B P Q H M Y L M B Y X D W M X C M S D G C Z R H N B B k S ] B Q L S X R L ˆ B : P 94 ] N R D G W k D Y L M J H M 95 ] H W R P C S ] S S S J M F ] S S ˆ P R S ] M J 99 k 79 k W L M P P L k 8 k H L D - x k 8 k R L z z R V J R P- - - S - M N - F F - P - K-- P k 8 k F M k K G H J H M k 83 k P M Z k j P U S - -P- R L k 84 k D M F C x M - N 3 5 k 85 k R M k - - P- R 9 3 k 86 k N M x - V P- L k 87 k N M - - V P L k 88 k G M M - : - M 5 P 4 D k 89 k Y M M R J SM k 9 k D N k M M H Y K - -J -- P k 9 k P C K M M - B- 3 S 3 J H M k 9 k B P Q H M Y- L - B Y X D W M - D V - M S D G C - Z R H P - N B - B k - -S k 93 k B Q L S X R L - - B- : P- - R B k 94 k N R D G W k D Y M - J H M - 9 k 95 k H W R P J - C S k 96 k S S S - - -J M F k 97 k S S - -P - - R S k 98 k S S - C J M F k 99 k C B S R- W J U - - B- P R 8 64 k S V- C - M J ] W L M P - - P L ] H L D x P ] R L z z Á R V J G R P S M N F F P K P D 8 ] F M á k K G á H J H M ] P M Z k j P U R - S P R L ] D M F C x M N ] R M k P R ] N M x - V P L ] N M - V P L ] G M M : // / M 5 P / / 4 D 89 ] Y M M R J - SM ] D N k M M H Y Y H K J P ] P C K M M B 3 S 3 J H M ] B P Q H M Y L M B Y X D W M D V X C M S D G C Z R H - P N B B k S ] B Q L S X R L B : P R B ] N R D G W k D Y L M J H M 9 95 ] H W R P J C S ] S S S J M F ] S S P R S ] S S C J M F ] C B S R W J U B P R S V C J - SM 34 4 S F X À z D J S - J P S F X À z D J J P S F X À z D J P - - J P S F X À z D J G J H M S V C D J P D S D J J B N N MM V 4 4

43 F X z V C D J S F P X R S 468 z V 7 C D J S F X À z V C D J S k 7 k S D J V C P R S P 468 R S ] S7 D J V C z 7 ] S D J - z V C M 435 M 435 M 8 ] S 435 V C D J k 8 k S V C D 8 ] S z V C - - D J J z H M ] G J J S H M S U J H M ] G J S P R L S 9 U ] D S G C k 9 k G J S S U ] D S G C P L ] P R L N Sj D H P S M L 84 J M 687 B 4 M H z k k D S G C ] N S j D H S : M J M W B - F z P L M H z P R B : ] B S H S W B k k N Sj D H S M F z ] B 3 ] H S S M H S B : x M B M H z k 3 ] H -S M M M S N Y k 5 4 ] J S : W F z P : P G M W M S k N Y k 5 B ] J S P G Z P B : C W M ] P J k k B S H S B M P W B Z L P F B Q: C M ] P S J 97 9 M P 6 ] W Z š ć M V B J ć L F S Mk Q Q k 3 k H S M S 97 9 P R L ] D Y z x k M 6 ] Z M - M M S V J N F Y k S N Mk Y k Q 8986 ] D Y z N J H M k 4 k J S P G W 7 ] D 9 ] D Y Y z k Z M - Y G N J S 8 49 ] D Y 33 z Z N P ] U B : C M J H B 9 S ] D k Y J z Z 956 k Y P G] F J Vk N O R M Q P K W L F ] U : Q P P S N Y k 5 ] Y W Y S 97 ] F Y k 9 Vk Y O R F Q K : N Y k k 5 J 6 S : S - S S ] M k Z M V J W Mk ] Y B - Y W C Q Z - Y G G Y k Y P R F L 57 J F 986 S H : SM S S 34 4 ] M W N Y35 Z Y G3 ] M k N W - F B 7 Y C k D Z Y Y G z G J P M 643 H M 5 ] L W 34 F N k Y k 997 P 4 ] L M k W N 3 Y 8 k 6 ] Z M D Z Z Y N G / z M N N F MG - H N Y J k 7 7 ] V M Z V R N - H M ] L k W k 9 k D Y P U z 4 Z Y 8 ] J M G Z k N 6 D ] Z O M K Z S S N / J M C G S G Z 49 J MG 33 B H D V N Y k 7 ] V S k M k Z k V R U N x U B 4 S P L ] J M Z J P 8 ] J k Ox M ZU P D k Ox O F K Vk S S O R C Q G K N MM : V P 4 4 P S x 87 N Y k ] J k M Z k Y P W Ox U Y P Ox Y k F J N S MM : S S S V k 3 k M W B Y C Z G F H M 34 k 4 k M W N Y Z Y N F J P k 5 k L W P L k 6 k Z M Z N M M H N Y k 7 k 7 k V M Z V R k P U k 8 k J M Z D O K S C G G Z J B D V S k x P L 87 5 k 9 k J M Z P Ox U Ox N MM V 4 4

Attention, Coordination, and Bounded Recall

Attention, Coordination, and Bounded Recall Attention, Coordination, and Bounded Recall Alessandro Pavan Northwestern University Chicago FED, February 2016 Motivation Many socioeconomic environments - large group of agents - actions under dispersed

More information

Online Appendices to Financing Asset Sales and Business Cycles

Online Appendices to Financing Asset Sales and Business Cycles Online Appendices to Financing Asset Sales usiness Cycles Marc Arnold Dirk Hackbarth Tatjana Xenia Puhan August 22, 2017 University of St. allen, Unterer raben 21, 9000 St. allen, Switzerl. Telephone:

More information

Household Saving, Financial Constraints, and the Current Account Balance in China

Household Saving, Financial Constraints, and the Current Account Balance in China Household Saving, Financial Constraints, and the Current Account Balance in China Ayşe İmrohoroğlu USC Marshall Kai Zhao Univ. of Connecticut Facing Demographic Change in a Challenging Economic Environment-

More information

Extended Libor Models and Their Calibration

Extended Libor Models and Their Calibration Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Vienna, 16 November 2007 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November

More information

Utility Maximization in an Illiquid Market

Utility Maximization in an Illiquid Market Utility Maximization in an Illiquid Market H. Mete Soner M. Vukelja April 1, 013 Abstract We consider a stochastic optimization problem of maximizing the expected utility from terminal wealth in an illiquid

More information

Particle methods and the pricing of American options

Particle methods and the pricing of American options Particle methods and the pricing of American options Peng HU Oxford-Man Institute April 29, 2013 Joint works with P. Del Moral, N. Oudjane & B. Rémillard P. HU (OMI) University of Oxford 1 / 46 Outline

More information

, c G G(c) = c σ σ p 1 F ( p) F p q(p) q(p) = G(p) q (p)p/q(p) q (p)p/q(p) = σ c c b p 1 F ( p) p p α 1 α α α α α = 0 α = 1 α (0, 1) α W S (c, p, b) = ((1 b) p c)(1 F ( p)) b p W S dw S /d p = 0 Φ( p)

More information

Extended Libor Models and Their Calibration

Extended Libor Models and Their Calibration Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Haindorf, 7 Februar 2008 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar

More information

Macroeconomics Qualifying Examination

Macroeconomics Qualifying Examination Macroeconomics Qualifying Examination January 211 Department of Economics UNC Chapel Hill Instructions: This examination consists of three questions. Answer all questions. Answering only two questions

More information

Incorporating Managerial Cash-Flow Estimates and Risk Aversion to Value Real Options Projects. The Fields Institute for Mathematical Sciences

Incorporating Managerial Cash-Flow Estimates and Risk Aversion to Value Real Options Projects. The Fields Institute for Mathematical Sciences Incorporating Managerial Cash-Flow Estimates and Risk Aversion to Value Real Options Projects The Fields Institute for Mathematical Sciences Sebastian Jaimungal sebastian.jaimungal@utoronto.ca Yuri Lawryshyn

More information

Reserve Accumulation, Macroeconomic Stabilization and Sovereign Risk

Reserve Accumulation, Macroeconomic Stabilization and Sovereign Risk Reserve Accumulation, Macroeconomic Stabilization and Sovereign Risk Javier Bianchi 1 César Sosa-Padilla 2 2018 SED Annual Meeting 1 Minneapolis Fed & NBER 2 University of Notre Dame Motivation EMEs with

More information

X t. ( t. x 1 (t) x 2 (t) T T. t 2 (u)du. t 1 (u)du T. 0 Xd s ds X

X t. ( t. x 1 (t) x 2 (t) T T. t 2 (u)du. t 1 (u)du T. 0 Xd s ds X r X X sds, X E Q e X sds X r X Xd s ds, X d, X e, ˆX f E d e Xd s ds X d, X e, ˆX f ν ν h ν H x 1 x x 1 x x 1 udu x udu r X Q dr b κr d + σ dw, κ b σ W r e κ r +e κ eκs b s ds+e κ eκs σ s dw s θ e κ r

More information

Heterogeneous Firm, Financial Market Integration and International Risk Sharing

Heterogeneous Firm, Financial Market Integration and International Risk Sharing Heterogeneous Firm, Financial Market Integration and International Risk Sharing Ming-Jen Chang, Shikuan Chen and Yen-Chen Wu National DongHwa University Thursday 22 nd November 2018 Department of Economics,

More information

The Global Rise of Corporate Saving

The Global Rise of Corporate Saving The Global Rise of Corporate Saving Peter Chen Loukas Karabarbounis Brent Neiman University of Chicago University of Minnesota University of Chicago January 2017 This paper 1 Global rise of corporate saving

More information

Risk-Adjusted Capital Allocation and Misallocation

Risk-Adjusted Capital Allocation and Misallocation Risk-Adjusted Capital Allocation and Misallocation Joel M. David Lukas Schmid David Zeke USC Duke & CEPR USC Summer 2018 1 / 18 Introduction In an ideal world, all capital should be deployed to its most

More information

Brownian Motion, the Gaussian Lévy Process

Brownian Motion, the Gaussian Lévy Process Brownian Motion, the Gaussian Lévy Process Deconstructing Brownian Motion: My construction of Brownian motion is based on an idea of Lévy s; and in order to exlain Lévy s idea, I will begin with the following

More information

The Normal Distribution

The Normal Distribution The Normal Distribution The normal distribution plays a central role in probability theory and in statistics. It is often used as a model for the distribution of continuous random variables. Like all models,

More information

Solving the Black-Scholes Equation

Solving the Black-Scholes Equation Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Initial Value Problem for the European Call The main objective of this lesson is solving

More information

Online Appendix for Mislearning from Censored Data: The Gambler s Fallacy in Optimal-Stopping Problems

Online Appendix for Mislearning from Censored Data: The Gambler s Fallacy in Optimal-Stopping Problems Online Appendix for Mislearning from Censored Data: The Gambler s Fallacy in Optimal-Stopping Problems Kevin He December 31, 018 OA 1 Proofs Omitted from the Appendix OA 1.1 Completing the Proof of of

More information

Sample Questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Sample Questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Sample Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Which reasoning process is shown in the following example? 1) We examine the social

More information

Country Spreads and Emerging Countries: Who Drives Whom? Martin Uribe and Vivian Yue (JIE, 2006)

Country Spreads and Emerging Countries: Who Drives Whom? Martin Uribe and Vivian Yue (JIE, 2006) Country Spreads and Emerging Countries: Who Drives Whom? Martin Uribe and Vivian Yue (JIE, 26) Country Interest Rates and Output in Seven Emerging Countries Argentina Brazil.5.5...5.5.5. 94 95 96 97 98

More information

Hod up to AD R + Θ is measurable

Hod up to AD R + Θ is measurable Hod up to AD R + Θ is measurable Rachid Atmai Department of Mathematics University of North Texas General Academics Building 435 1155 Union Circle #311430 Denton, TX 76203-5017 atmai.rachid@gmail.com Grigor

More information

Growth and Distributional Effects of Inflation with Progressive Taxation

Growth and Distributional Effects of Inflation with Progressive Taxation MPRA Munich Personal RePEc Archive Growth and Distributional Effects of Inflation with Progressive Taxation Fujisaki Seiya and Mino Kazuo Institute of Economic Research, Kyoto University 20. October 2010

More information

Solving the Black-Scholes Equation

Solving the Black-Scholes Equation Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Initial Value Problem for the European Call rf = F t + rsf S + 1 2 σ2 S 2 F SS for (S,

More information

Error Bounds for P-matrix Linear Complementarity Problems and Their Applications

Error Bounds for P-matrix Linear Complementarity Problems and Their Applications Error Bounds for P-matrix Linear Complementarity Problems and Their Applications Xiaojun Chen The Hong Kong Polytechnic University NPA2008, Beijing, 7-9 April Outline Computational Global Error Bounds

More information

The Macroeconomics of Universal Health Insurance Vouchers

The Macroeconomics of Universal Health Insurance Vouchers The Macroeconomics of Universal Health Insurance Vouchers Juergen Jung Towson University Chung Tran University of New South Wales Jul-Aug 2009 Jung and Tran (TU and UNSW) Health Vouchers 2009 1 / 29 Dysfunctional

More information

Growth model with Epstein-Zin preferences and stochastic volatility

Growth model with Epstein-Zin preferences and stochastic volatility Growth model with Epstein-Zin preferences and stochastic volatility Håkon Tretvoll July 8, 2011 1 Introduction This document goes through a method of solving a growth model with Epstein-Zin preferences

More information

The Valuation of Bermudan Guaranteed Return Contracts

The Valuation of Bermudan Guaranteed Return Contracts The Valuation of Bermudan Guaranteed Return Contracts Steven Simon 1 November 2003 1 K.U.Leuven and Ente Luigi Einaudi Abstract A guaranteed or minimum return can be found in different financial products,

More information

Uninsured Unemployment Risk and Optimal Monetary Policy

Uninsured Unemployment Risk and Optimal Monetary Policy Uninsured Unemployment Risk and Optimal Monetary Policy Edouard Challe CREST & Ecole Polytechnique ASSA 2018 Strong precautionary motive Low consumption Bad aggregate shock High unemployment Low output

More information

1980 2007 2007 20 30 25 8% 2% 5% 1% 30 50 3% 10 5 8 4 6 3 4 2 2 1 0 20 30 40 50 60 0 20 30 40 50 60 / + 2 3% 11 50 55 +/ 2.5 1.7 5 3 15 10 6 8 5 10 6 4 5 mean tenure median tenure 4 2 3 2 1 0 20 30 40

More information

The CAPM Strikes Back? An Investment Model with Disasters

The CAPM Strikes Back? An Investment Model with Disasters The CAPM Strikes Back? An Investment Model with Disasters Hang Bai 1 Kewei Hou 1 Howard Kung 2 Lu Zhang 3 1 The Ohio State University 2 London Business School 3 The Ohio State University and NBER Federal

More information

The (λ, κ)-fn and the order theory of bases in boolean algebras

The (λ, κ)-fn and the order theory of bases in boolean algebras The (λ, κ)-fn and the order theory of bases in boolean algebras David Milovich Texas A&M International University david.milovich@tamiu.edu http://www.tamiu.edu/ dmilovich/ June 2, 2010 BLAST 1 / 22 The

More information

Financial Structure, Informality and Development

Financial Structure, Informality and Development Financial Structure, Informality and Development Pablo N D Erasmo University of Maryland Hernan J Moscoso Boedo University of Virginia September 2, 2009 Preliminary Abstract This is a theory of total factor

More information

arxiv:math/ v1 [math.lo] 9 Dec 2006

arxiv:math/ v1 [math.lo] 9 Dec 2006 arxiv:math/0612246v1 [math.lo] 9 Dec 2006 THE NONSTATIONARY IDEAL ON P κ (λ) FOR λ SINGULAR Pierre MATET and Saharon SHELAH Abstract Let κ be a regular uncountable cardinal and λ > κ a singular strong

More information

Gender Gaps and the Rise of the Service Economy

Gender Gaps and the Rise of the Service Economy Gender Gaps and the Rise of the Service Economy L. Rachel Ngai & Barbara Petrongolo American Economic Journal: Macroeconomics 2017 Presented by Francisco Javier Rodríguez for the Macro Reading Group Universidad

More information

Extended DSGE Model of the Czech Economy

Extended DSGE Model of the Czech Economy Zbyněk Štork Božena Bobková Ilkin Aliyev Moderní nástroje pro finanční analýzu a modelování 5. 6. 214 Outline 1 Extended DSGE model 2 3 Simulation 4 Outline 1 Extended DSGE model 2 3 Simulation 4 Outline

More information

Online Appendix to Financing Asset Sales and Business Cycles

Online Appendix to Financing Asset Sales and Business Cycles Online Appendix to Financing Asset Sales usiness Cycles Marc Arnold Dirk Hackbarth Tatjana Xenia Puhan August 31, 2015 University of St. allen, Rosenbergstrasse 52, 9000 St. allen, Switzerl. Telephone:

More information

Generalized Taylor Rule and Determinacy of Growth Equilibrium. Abstract

Generalized Taylor Rule and Determinacy of Growth Equilibrium. Abstract Generalized Taylor Rule and Determinacy of Growth Equilibrium Seiya Fujisaki Graduate School of Economics Kazuo Mino Graduate School of Economics Abstract This paper re-examines equilibrium determinacy

More information

Appendix: Numerical Model

Appendix: Numerical Model Appendix to: Costs of Alternative Environmental Policy Instruments in the Presence of Industry Compensation Requirements A. Lans Bovenberg Lawrence H. Goulder Mark R. Jacobsen Appendix: Numerical Model

More information

Analysis of the Bitcoin Exchange Using Particle MCMC Methods

Analysis of the Bitcoin Exchange Using Particle MCMC Methods Analysis of the Bitcoin Exchange Using Particle MCMC Methods by Michael Johnson M.Sc., University of British Columbia, 2013 B.Sc., University of Winnipeg, 2011 Project Submitted in Partial Fulfillment

More information

Notional Defined Contribution Pension, Fertility, and Efficiency. Wages in an Overlapping Generations Economy. Leran Wang

Notional Defined Contribution Pension, Fertility, and Efficiency. Wages in an Overlapping Generations Economy. Leran Wang Notional Defined Contribution Pension, Fertility, and Efficiency Wages in an Overlapping Generations Economy Leran Wang Abstract This study analyzes how fertility is affected by social security systems

More information

Financial Development and the Effects of Trade Liberalizations

Financial Development and the Effects of Trade Liberalizations Financial Development and the Effects of Trade Liberalizations David Kohn Pontificia Universidad Católica de Chile Fernando Leibovici Federal Reserve Bank of St. Louis Michal Szkup University of British

More information

Silver type theorems for collapses.

Silver type theorems for collapses. Silver type theorems for collapses. Moti Gitik May 19, 2014 The classical theorem of Silver states that GCH cannot break for the first time over a singular cardinal of uncountable cofinality. On the other

More information

Quantifying the Impact of Financial Development on Economic Development

Quantifying the Impact of Financial Development on Economic Development Quantifying the Impact of Financial Development on Economic Development Jeremy Greenwood, Juan M. Sanchez, Cheng Wang (RED 2013) Presented by Beatriz González Macroeconomics Reading Group - UC3M January

More information

Why are Banks Exposed to Monetary Policy?

Why are Banks Exposed to Monetary Policy? Why are Banks Exposed to Monetary Policy? Sebastian Di Tella and Pablo Kurlat Stanford University Bank of Portugal, June 2017 Banks are exposed to monetary policy shocks Assets Loans (long term) Liabilities

More information

The Semi-Weak Square Principle

The Semi-Weak Square Principle The Semi-Weak Square Principle Maxwell Levine Universität Wien Kurt Gödel Research Center for Mathematical Logic Währinger Straße 25 1090 Wien Austria maxwell.levine@univie.ac.at Abstract Cummings, Foreman,

More information

Parametric Inference and Dynamic State Recovery from Option Panels. Nicola Fusari

Parametric Inference and Dynamic State Recovery from Option Panels. Nicola Fusari Parametric Inference and Dynamic State Recovery from Option Panels Nicola Fusari Joint work with Torben G. Andersen and Viktor Todorov July 2012 Motivation Under realistic assumptions derivatives are nonredundant

More information

Unstiffened Element with torsional Restraint - an Analytical Approach for Postbuckling Behavior using GBT

Unstiffened Element with torsional Restraint - an Analytical Approach for Postbuckling Behavior using GBT Unstiffened Element with torsional Restraint - an Analytical Approach for Postuckling Behavior using GBT Institute for material and mechanics in civil engineering Technical University of Darmstadt Lissaon

More information

Equity correlations implied by index options: estimation and model uncertainty analysis

Equity correlations implied by index options: estimation and model uncertainty analysis 1/18 : estimation and model analysis, EDHEC Business School (joint work with Rama COT) Modeling and managing financial risks Paris, 10 13 January 2011 2/18 Outline 1 2 of multi-asset models Solution to

More information

Default Risk and Aggregate Fluctuations in an Economy with Production Heterogeneity

Default Risk and Aggregate Fluctuations in an Economy with Production Heterogeneity Default Risk and Aggregate Fluctuations in an Economy with Production Heterogeneity Aubhik Khan The Ohio State University Tatsuro Senga The Ohio State University and Bank of Japan Julia K. Thomas The Ohio

More information

Optimal Taxation Under Capital-Skill Complementarity

Optimal Taxation Under Capital-Skill Complementarity Optimal Taxation Under Capital-Skill Complementarity Ctirad Slavík, CERGE-EI, Prague (with Hakki Yazici, Sabanci University and Özlem Kina, EUI) January 4, 2019 ASSA in Atlanta 1 / 31 Motivation Optimal

More information

Fiscal Multipliers in Recessions. M. Canzoneri, F. Collard, H. Dellas and B. Diba

Fiscal Multipliers in Recessions. M. Canzoneri, F. Collard, H. Dellas and B. Diba 1 / 52 Fiscal Multipliers in Recessions M. Canzoneri, F. Collard, H. Dellas and B. Diba 2 / 52 Policy Practice Motivation Standard policy practice: Fiscal expansions during recessions as a means of stimulating

More information

ECON 815. A Basic New Keynesian Model II

ECON 815. A Basic New Keynesian Model II ECON 815 A Basic New Keynesian Model II Winter 2015 Queen s University ECON 815 1 Unemployment vs. Inflation 12 10 Unemployment 8 6 4 2 0 1 1.5 2 2.5 3 3.5 4 4.5 5 Core Inflation 14 12 10 Unemployment

More information

ADDING A LOT OF COHEN REALS BY ADDING A FEW II. 1. Introduction

ADDING A LOT OF COHEN REALS BY ADDING A FEW II. 1. Introduction ADDING A LOT OF COHEN REALS BY ADDING A FEW II MOTI GITIK AND MOHAMMAD GOLSHANI Abstract. We study pairs (V, V 1 ), V V 1, of models of ZF C such that adding κ many Cohen reals over V 1 adds λ many Cohen

More information

Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models

Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models David Prager 1 1 Associate Professor of Mathematics Anderson University (SC) Based on joint work with Professor Qing Zhang,

More information

IMPA Commodities Course : Forward Price Models

IMPA Commodities Course : Forward Price Models IMPA Commodities Course : Forward Price Models Sebastian Jaimungal sebastian.jaimungal@utoronto.ca Department of Statistics and Mathematical Finance Program, University of Toronto, Toronto, Canada http://www.utstat.utoronto.ca/sjaimung

More information

Optimal asset allocation under forward performance criteria Oberwolfach, February 2007

Optimal asset allocation under forward performance criteria Oberwolfach, February 2007 Optimal asset allocation under forward performance criteria Oberwolfach, February 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 References Indifference valuation in binomial models (with

More information

Supplement to Adaptive Estimation of High Dimensional Partially Linear Model

Supplement to Adaptive Estimation of High Dimensional Partially Linear Model Supplemet to Adaptive Estimatio o High Dimesioal Partially Liear Model Fag Ha Zhao Re ad Yuxi Zhu May 6 017 This supplemetary material provides the techical proos as well as some auxiliary lemmas. For

More information

Portfolio Management and Optimal Execution via Convex Optimization

Portfolio Management and Optimal Execution via Convex Optimization Portfolio Management and Optimal Execution via Convex Optimization Enzo Busseti Stanford University April 9th, 2018 Problems portfolio management choose trades with optimization minimize risk, maximize

More information

. Social Security Actuarial Balance in General Equilibrium. S. İmrohoroğlu (USC) and S. Nishiyama (CBO)

. Social Security Actuarial Balance in General Equilibrium. S. İmrohoroğlu (USC) and S. Nishiyama (CBO) ....... Social Security Actuarial Balance in General Equilibrium S. İmrohoroğlu (USC) and S. Nishiyama (CBO) Rapid Aging and Chinese Pension Reform, June 3, 2014 SHUFE, Shanghai ..... The results in this

More information

Investment strategies and risk management for participating life insurance contracts

Investment strategies and risk management for participating life insurance contracts 1/20 Investment strategies and risk for participating life insurance contracts and Steven Haberman Cass Business School AFIR Colloquium Munich, September 2009 2/20 & Motivation Motivation New supervisory

More information

Stochastic Proximal Algorithms with Applications to Online Image Recovery

Stochastic Proximal Algorithms with Applications to Online Image Recovery 1/24 Stochastic Proximal Algorithms with Applications to Online Image Recovery Patrick Louis Combettes 1 and Jean-Christophe Pesquet 2 1 Mathematics Department, North Carolina State University, Raleigh,

More information

Characterisation of Strongly Normalising λµ-terms

Characterisation of Strongly Normalising λµ-terms Characterisation of Strongly Normalising λµ-terms Ugo de Liguoro joint work with Steffen van Bakel and Franco Barbanera ITRS - June 2012, Dubrovnik Introduction Parigot s λµ-calculus is an extension of

More information

A Full Asymptotic Series of European Call Option Prices in the SABR Model with

A Full Asymptotic Series of European Call Option Prices in the SABR Model with A Full Asymptotic Series of European Call Option Prices in the SABR Model with β = 1 Z. Guo, H. Schellhorn November 17, 2018 Stochastic Alpha Beta Rho(SABR) Model The Black-Scholes Theory Generalization

More information

How Important Are Terms of Trade Shocks?

How Important Are Terms of Trade Shocks? How Important Are Terms of Trade Shocks? Stephanie Schmitt-Grohé Martín Uribe Columbia University October 28, 215 1 Conventional View: Terms-of-trade shocks are a major source of business-cycle fluctuations

More information

A Model Calibration. 1 Earlier versions of this dataset have, for example, been used by Krohmer et al. (2009), Cumming et al.

A Model Calibration. 1 Earlier versions of this dataset have, for example, been used by Krohmer et al. (2009), Cumming et al. A Model Calibration This appendix illustrates the model calibration. Where possible, baseline parameters are chosen in the following such that they correspond to an investment in an average buyout fund.

More information

Consumption Dynamics, Housing Collateral and Stabilisation Policy

Consumption Dynamics, Housing Collateral and Stabilisation Policy Consumption Dynamics, Housing Collateral and Stabilisation Policy A Way Forward for Macro-Prudential Instruments? Effective Macroprudential Instruments - CFCM-MMF-MMPM Conference Jagjit S. Chadha University

More information

No-Arbitrage Bounds on Two One-Touch Options

No-Arbitrage Bounds on Two One-Touch Options No-Arbitrage Bounds on Two One-Touch Options Yukihiro Tsuzuki March 30, 04 Abstract This paper investigates the pricing bounds of two one-touch options with the same maturity but different barrier levels,

More information

Chapter 3: Black-Scholes Equation and Its Numerical Evaluation

Chapter 3: Black-Scholes Equation and Its Numerical Evaluation Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random

More information

The data-driven COS method

The data-driven COS method The data-driven COS method Á. Leitao, C. W. Oosterlee, L. Ortiz-Gracia and S. M. Bohte Delft University of Technology - Centrum Wiskunde & Informatica Reading group, March 13, 2017 Reading group, March

More information

A Model of Financial Intermediation

A Model of Financial Intermediation A Model of Financial Intermediation Jesús Fernández-Villaverde University of Pennsylvania December 25, 2012 Jesús Fernández-Villaverde (PENN) A Model of Financial Intermediation December 25, 2012 1 / 43

More information

Monetary Economics. Financial Markets and the Business Cycle: The Bernanke and Gertler Model. Nicola Viegi. September 2010

Monetary Economics. Financial Markets and the Business Cycle: The Bernanke and Gertler Model. Nicola Viegi. September 2010 Monetary Economics Financial Markets and the Business Cycle: The Bernanke and Gertler Model Nicola Viegi September 2010 Monetary Economics () Lecture 7 September 2010 1 / 35 Introduction Conventional Model

More information

The data-driven COS method

The data-driven COS method The data-driven COS method Á. Leitao, C. W. Oosterlee, L. Ortiz-Gracia and S. M. Bohte Delft University of Technology - Centrum Wiskunde & Informatica CMMSE 2017, July 6, 2017 Álvaro Leitao (CWI & TUDelft)

More information

Lecture 8: Two period corporate debt model

Lecture 8: Two period corporate debt model Lecture 8: Two period corporate debt model Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 213 A two-period model with investment At time 1, the firm buys capital k, using equity issuance s and

More information

Fiscal Devaluations. Emmanuel Farhi Gita Gopinath Oleg Itskhoki. Harvard Harvard Princeton 1 / 32

Fiscal Devaluations. Emmanuel Farhi Gita Gopinath Oleg Itskhoki. Harvard Harvard Princeton 1 / 32 Fiscal Devaluations Emmanuel Farhi Gita Gopinath Oleg Itskhoki Harvard Harvard Princeton 1 / 32 Motivation Currency devaluation: response to loss of competitiveness What if devaluation impossible? 1 /

More information

Physics of muscle contraction

Physics of muscle contraction Review Article Physics of muscle contraction M. Caruel matthieu.caruel@u-pec.fr L. Truskinovsky lev.truskinovsky@espci.fr Abstract. Contents . Introduction.. Background Z disk M line half-sarcomere cross-bridge

More information

2O, p. 577, sol. 4.90: Setting the partial derivative of the loglikelihood with respect to λ equal to 0: = exp[d 1 σ T ] exp[-σ 2 T/2] exp[-d 1 2 / 2]

2O, p. 577, sol. 4.90: Setting the partial derivative of the loglikelihood with respect to λ equal to 0: = exp[d 1 σ T ] exp[-σ 2 T/2] exp[-d 1 2 / 2] Errata, Mahler Study Aids for Exam 3/M, Fall 2010 HCM, 1/26/13 Page 1 2B, p. 57, 3rd line from bottom: The likelihood is 2O, p. 577, sol. 4.90: Setting the partial derivative of the loglikelihood with

More information

Credit Frictions and Optimal Monetary Policy

Credit Frictions and Optimal Monetary Policy Vasco Cúrdia FRB of New York 1 Michael Woodford Columbia University National Bank of Belgium, October 28 1 The views expressed in this paper are those of the author and do not necessarily re ect the position

More information

Household Debt, Financial Intermediation, and Monetary Policy

Household Debt, Financial Intermediation, and Monetary Policy Household Debt, Financial Intermediation, and Monetary Policy Shutao Cao 1 Yahong Zhang 2 1 Bank of Canada 2 Western University October 21, 2014 Motivation The US experience suggests that the collapse

More information

Fast Pricing and Calculation of Sensitivities of OTM European Options Under Lévy Processes

Fast Pricing and Calculation of Sensitivities of OTM European Options Under Lévy Processes Fast Pricing and Calculation of Sensitivities of OTM European Options Under Lévy Processes Sergei Levendorskĭi Jiayao Xie Department of Mathematics University of Leicester Toronto, June 24, 2010 Levendorskĭi

More information

Parametric Inference and Dynamic State Recovery from Option Panels. Torben G. Andersen

Parametric Inference and Dynamic State Recovery from Option Panels. Torben G. Andersen Parametric Inference and Dynamic State Recovery from Option Panels Torben G. Andersen Joint work with Nicola Fusari and Viktor Todorov The Third International Conference High-Frequency Data Analysis in

More information

Modeling Commodity Futures: Reduced Form vs. Structural Models

Modeling Commodity Futures: Reduced Form vs. Structural Models Modeling Commodity Futures: Reduced Form vs. Structural Models Pierre Collin-Dufresne University of California - Berkeley 1 of 44 Presentation based on the following papers: Stochastic Convenience Yield

More information

Preprint núm January Robust pricing of european options with wavelets and the characteristic function. L. Ortiz-Gracia, C. W.

Preprint núm January Robust pricing of european options with wavelets and the characteristic function. L. Ortiz-Gracia, C. W. Preprint núm. 1144 January 213 Robust pricing of european options with wavelets and the characteristic function L. Ortiz-Gracia, C. W. Oosterlee ROBUST PRICING OF EUROPEAN OPTIONS WITH WAVELETS AND THE

More information

Economia Financiera Avanzada

Economia Financiera Avanzada Model Economia Financiera Avanzada EBAPE- Fundação Getulio Vargas Universidad del Pacífico, Julio 5 21, 2011 Economia Financiera Avanzada Model Default and Bankruptcy in GE Models Economia Financiera Avanzada

More information

Non-Renewable Resources and the Sustainability of the Economic Growth under Bilateral Trade

Non-Renewable Resources and the Sustainability of the Economic Growth under Bilateral Trade Non-Renewable Resources and the Sustainability of the Economic Growth under Bilateral Trade Francisco Cabo 1 Guiomar Martín-Herrán 1 M.Pilar Martínez-García 2 1 Departamento de Economía Aplicada Universidad

More information

Income-Induced Expenditure Switching

Income-Induced Expenditure Switching Income-Induced Expenditure Switching Rudolfs Bems 1 Julian di Giovanni 2 1 Bank of Latvia Research Department, IMF 2 UPF, Barcelona GSE CREI and CEPR February 20, 2014 The views expressed in this paper

More information

Debt consolidation: Aggregate and distributional implications

Debt consolidation: Aggregate and distributional implications Debt consolidation: Aggregate and distributional implications Eleftherios Roumpanis Athens University of Economics and Business Abstract This paper builds and solves numerically, by using Eurozone data,

More information

ARONSZAJN TREES AND THE SUCCESSORS OF A SINGULAR CARDINAL. 1. Introduction

ARONSZAJN TREES AND THE SUCCESSORS OF A SINGULAR CARDINAL. 1. Introduction ARONSZAJN TREES AND THE SUCCESSORS OF A SINGULAR CARDINAL SPENCER UNGER Abstract. From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the Singular

More information

Currency Risk Factors in a Recursive Multi-Country Economy

Currency Risk Factors in a Recursive Multi-Country Economy Currency Risk Factors in a Recursive Multi-Country Economy R. Colacito M.M. Croce F. Gavazzoni R. Ready NBER SI - International Asset Pricing Boston July 8, 2015 Motivation The literature has identified

More information

Fiscal Multipliers in Recessions

Fiscal Multipliers in Recessions Fiscal Multipliers in Recessions Matthew Canzoneri Fabrice Collard Harris Dellas Behzad Diba March 10, 2015 Matthew Canzoneri Fabrice Collard Harris Dellas Fiscal Behzad Multipliers Diba (University in

More information

Corporate Strategy, Conformism, and the Stock Market

Corporate Strategy, Conformism, and the Stock Market Corporate Strategy, Conformism, and the Stock Market Thierry Foucault (HEC) Laurent Frésard (Maryland) November 20, 2015 Corporate Strategy, Conformism, and the Stock Market Thierry Foucault (HEC) Laurent

More information

Structural Change in a Multi-Sector Model of Growth

Structural Change in a Multi-Sector Model of Growth Structural Change in a Multi-Sector Model of Growth L Rachel Ngai Centre for Economic Performance London School of Economics Christopher A Pissarides Centre for Economic Performance London School of Economics,

More information

Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model

Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 1(23) Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility

More information

A Translation of Intersection and Union Types

A Translation of Intersection and Union Types A Translation of Intersection and Union Types for the λ µ-calculus Kentaro Kikuchi RIEC, Tohoku University kentaro@nue.riec.tohoku.ac.jp Takafumi Sakurai Department of Mathematics and Informatics, Chiba

More information

Oil Price Uncertainty in a Small Open Economy

Oil Price Uncertainty in a Small Open Economy Yusuf Soner Başkaya Timur Hülagü Hande Küçük 6 April 212 Oil price volatility is high and it varies over time... 15 1 5 1985 199 1995 2 25 21 (a) Mean.4.35.3.25.2.15.1.5 1985 199 1995 2 25 21 (b) Coefficient

More information

Capital Goods Trade and Economic Development

Capital Goods Trade and Economic Development Capital Goods Trade and Economic Development Piyusha Mutreja B. Ravikumar Michael Sposi Syracuse U. FRB St. Louis FRB Dallas December 2014 NYU-FRBATL Conference Disclaimer: The following views are those

More information

Optimal monetary policy when asset markets are incomplete

Optimal monetary policy when asset markets are incomplete Optimal monetary policy when asset markets are incomplete R. Anton Braun Tomoyuki Nakajima 2 University of Tokyo, and CREI 2 Kyoto University, and RIETI December 9, 28 Outline Introduction 2 Model Individuals

More information

Making the European Banking Union Macro-Economically Resilient: Cost of Non-Europe Report

Making the European Banking Union Macro-Economically Resilient: Cost of Non-Europe Report Making the European Banking Union Macro-Economically Resilient: Cost of Non-Europe Report Gaël Giraud Thore Kockerols Centre d Économie de la Sorbonne Labex Réfi January 28, 2016 Outline 1. Introduction

More information

The Extensive Margin of Trade and Monetary Policy

The Extensive Margin of Trade and Monetary Policy The Extensive Margin of Trade and Monetary Policy Yuko Imura Bank of Canada Malik Shukayev University of Alberta June 2, 216 The views expressed in this presentation are our own, and do not represent those

More information

Two Stationary Sets with Different Gaps of the Power Function

Two Stationary Sets with Different Gaps of the Power Function Two Stationary Sets with Different Gaps of the Power Function Moti Gitik School of Mathematical Sciences Tel Aviv University Tel Aviv 69978, Israel gitik@post.tau.ac.il August 14, 2014 Abstract Starting

More information