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4 R x 1, x 2, x 3,... ( x n ) (n = 1, 2, 3,... ) 1.1 ( ). R ( x n ) x R ɛ N N n x n x < ɛ ( x n ) x ɛ > 0, N N, n > N : x n x < ɛ ( x n ) x n xn x lim x n = x x n x (n ) n x ( x n ) 1.1. x n = 1/n ( x n ) 0 ɛ N 1/ɛ N N n > N 0 < x n = 1 n < 1 N ɛ n > N : x n 0 < ɛ ( x n ) ( x n ) 1 x n = n n 1 n R ( x n ) ɛ > 0, N N, n > N : x n x < ɛ ɛ > 0, N N, n > N : x n x ɛ (1.1) ɛ N N n x n x ɛ ɛ = 1/2 n x n 0 = 1 0 = 1 > ɛ (1.2)

5 5 ɛ 1/2 N n (1.1) ( ) x n 0 ( ) x n 1 n ( ) x n R f : R R ( ) x n f(x1 ), f(x 2 ), f(x 3 ), f : R R f(x) = x (1.3) ( x n ) 1.1 xn = 1/n f(x n ) = x 3 n + 1 = 1 n f(x n ) 1 (n ) f(x n ) 1 x n = ( 1) n /n f(x n ) = x 3 n + 1 = ( 1)n n f(x n ) 1 (n ) ( x n ) lim n x n = 0 0 (1.3) f 0 ( x n ) f(x n ) 1 (n ) 1.2 ( ). x R f R x n x (n ) ( x n ) f(x n ) f(x) (n ) lim n f(x n) = f(x) f x f(y) f(x) (y x) lim y x f(y) = f(x)

6 6, (x n x) ( ) (f(x)) (f(x n )) lim f(x n) n lim n f(x n ) = f (lim n x n ) 1.3 f ɛ > 0, δ > 0, y R : x y < δ f(x) f(y) < ɛ R f 1 x 0 f(x) = 0 x < 0 x = 0 0 x n = 1/n x n = 1/n f n, x n = 1 n > 0 n, f(x n ) = 1 f(x n ) 1 (n ) f f(0) = 1 f(x n ) f(0) (n ) 0 1 f n, x n = 1 n < 0 n, f(x n ) = 0 f(x n ) 0 (n ) 0 f(x n ) f(0) (n ) 0 2 f

7 L R L x 1, x 2, x 3,... ( x n ) (n = 1, 2, 3,... ) Eucliden Norm 1.3 ( ). R L x = ( x 1, x 2, x 3,..., x L) x x R L x = (x 1 ) 2 + (x 2 ) 2 + (x 3 ) (x L ) 2 (1.4) L = 2, R L ( x n ) x R L lim x n x = 0 x n x 0 (n ) n ( x n ) x x ( x n ) lim x n = x x n x (n ) (1.5) n ( x n ) x ( xn ) x x n = ( x 1 n, x 2 n, x 3 n,..., x L n) 1.4 l = 1, 2, 3,..., L : x l n x l (n ) (1.6) (1.6) 1.5. R 2 0 = (0, 0) ( ) 1 x n = n, 0 x n = x n = x n = ( 0, 1 ) n ( 1 n, 1 ) n ( 1 n cos n, 1 n sin n ) (1.7) (1.8) (1.9) (1.10) xn 0 (n ) 0 R 2 0

8 L F : R L R M 1 R L R M F : R L R M F (x) = F (x 1, x 2,..., x L ) x R L F 1 (x 1, x 2,..., x L ) F 2 (x 1, x 2,..., x L ) F (x) =. F M (x 1, x 2,..., x L ) 1.5. x R L F R L x n x (n ) ( x n ) F (x n ) F (x) (n ) x n x 0 (n ) ( x n ) F (x n ) F (x) 0 (n ) F x lim F (y) = F (x) F (y) F (x) (y x) y x F 0 F (0) 1 ɛ - δ 1.5 ɛ > 0, δ > 0, x R L : x u < δ f(x) f(u) < ɛ (x 1, x 2,..., x L ) F

9 L = 2 M = 1 F (x 1, x 2 ) = F (x, y) 1.6 (1 ). 1 f : R R x R f(x + h) f(x) lim h 0 h f x f(x + h) f(x) lim = df h 0 h dx (x, y) = f (x) f x f x R f R f R R x f (x) f f : R R f x f x x f R 1.5. n f : R R ( ) 1 x n sin (x 0 ) f(x) = x 0 (x = 0 ) 1. n = 0 x 0 f(x) = sin (1/x) f x = 0 2. n = 1 f x = 0 3. n = 2 f x = ( ). F : R 2 R (x, y) R 2 g(h) = F (x + h, y) g : R R g h = 0 g(h) r(0) F (x + h, y) F (x, y) lim = lim h 0 h h 0 h F (x, y) x F (x, y) x F x (x, y) R 2 F x y F (x, y + h) F (x, y) lim h 0 h F (x, y) y F (x, y) y F y (x, y) R 2 F y

10 F 1.6. (K, L) (K +dk, L+dL) Q = F (K, L) F (K, L) K K + dk Q F K (K, L)dK L L + dl Q F L (K, L)dL Q dq 4 dq = F F (K, L)dK + (K, L)dL K L F F F (x, y) 0 xy = 0 F (x, y) = x + y xy 0 2 h (x, y) (0, 0) (h, h) F F (x, y) (0, 0) F F (0 + h, 0) F (0, 0) (0, 0) = lim = 0 x h 0 h F F (0, 0 + h) F (0, 0) (0, 0) = lim = 0 y h 0 h F df df = F F (0, 0)h + (0, 0)h = 0 (1.11) x y F (1.11) df = F (0 + h, 0 + h) F (0, 0) = h 0 = h F (0 + h, 0 + h) F (0, 0) lim = 1 h 0 h F 4 Q dq = F (K + dk, L + dl) F (K, L)

11 F 1.8 ( ). F : R 2 R (x, y) R 2 (u, v) R 2 g : R R g(ɛ) = F (x + ɛu, y + ɛv) g ɛ = 0 g(ɛ) g(0) F (x + ɛu, y + ɛv) F (x, y) lim = lim ɛ 0 ɛ ɛ 0 ɛ (1.12) F (x, y) (u, v) (1.12) F (x, y) 2 (u, v) F 1.8 (u, v) R 2 (u, v) = (1, 0) (u, v) = (0, 1) x y 1.7 F 2 (0, 0) (ɛ, ɛ) F (u, v) = (1, 1) F (0 + ɛ, 0 + ɛ) F (0, 0) ɛ = ɛ 0 ɛ (0, 0) (u, v) = (1, 1) F 1 = (1.10) (u, v) f : R R x R f(x + h) f(x) lim h 0 h f x f(x + h) f(x) lim = df h 0 h dx (x, y) = f (x) f x R f

12 f x 2. c R f(x + h) ( f(x) + ch ) lim = 0 (1.13) h 0 h (1.13) c c = f (x) f (1.13) 1 (1.13) 1.9 ( ). F : R 2 R (x, y) R 2 (c, d) R 2 F (x + h, y + k) ( F (x, y) + ch + dk ) lim = 0 (1.14) (h,k) (0,0) (h, k) F (x, y) F (x, y) (c, d) ( ) F F (c, d) = (x, y), (x, y) x y F (x, y) F (x, y) F F (1.14) lim (h n, k n ) = (0, 0) n {(h n, k n )} 1.5 (1.7) (1.8) (1.9) (1.10) (1.14) F : R 2 R F (x, y) = x y. 1. (x, y) = (0, 0) F 2. (x, y) = (0, 0) F 3. (x, y) = (0, 0) F

13 (c, d) (x, y) (grdient vector) ( ) F F F (x, y) = (x, y), (x, y) x y mx F (x + h, y + k) F (x, y) (h,k) subject to (h, k) ɛ (x, y) (x + h, y + k) F (x, y) (x, y) ɛ F F 0 ɛ (h ɛ, k ɛ ) = F (x + h ɛ, y + k ɛ ) F (x + h ɛ, y + k ɛ ) (1.15) lim h ɛ = 0, lim k ɛ = 0 ɛ 0 ɛ 0 (1.15) ɛ ɛ 0 F 1 lim ɛ 0 ɛ (h 1 ɛ, k ɛ ) = F (x, y) (1.16) F (x, y) (1.16) (x, y) F F (x, y) (1.16) F F (x, y) F (x, y) lim (ɛh,ɛk) (0,0) F (x + ɛh, y + ɛk) ( F (x, y) + F x (ɛh, ɛk) F (x + ɛh, y + ɛk) F (x, y) lim = ɛ 0 ɛ (h, k) F x F (x, y)ɛh + y (x, y)ɛk) F (x, y)h + y (x, y)k (h, k) (h, k) ( ) 1 (h, k) = F (x, y) F (x, y) = 1 F F (x, y), (x, y) F (x, y) x y (h, k) = 1 = 0 (1.17) F (x + ɛh, y + ɛk) F (x, y) lim = F (x, y) (1.18) ɛ 0 ɛ (1.18) (h, k) (1.17) (h, k) 1 (1.18) F

14 F : R L R U R L k F U k C k 1.1. F : R 2 R R 2 F (x, y) 1 F (x, y) F : R 2 R (x, y) x (x, y) (x, y) F (x, t) x F x : R2 R (1.19) (x, y) F 2 y 2 F 2 (x, y) x 2 F (x, y) y x F y : R2 R (1.20) (x, y) (1.20) 2 F (x, y) x y 2 F 2 (x, y) y 2 F/ y x 2 F/ x y x y y x 1.2 ( ). F : R 2 R (x, y) x y F x : R2 R F y : R2 R

15 15, (, b) 1.2. h, k R {0} 2 F x y (, b) = 2 F (, b) y x = F ( + h, b + k) F ( + h, b) F (, b + k) + F (, b) φ (x) = F (x, b + k) F (x, b) = φ ( + h) φ () F (, b) x φ (x) = F F (x, b + k) (x, b) x x (, + h) φ (x) θ (0, 1) φ ( + h) φ () h = φ ( + θh) h = F F ( + θh, b + k) ( + θh, b) x x h = k F/ x F F ( + θh, b + k) = x x (, b) + F θh 2 x 2 (, b) + h 2 F (, b) + o (h) y x F F ( + θh, b) = x x (, b) + F θh 2 (, b) + o (h) x2 o ( ) lim h 0 o (h) /h = 0 h 2 = 2 F o (h) (, b) + y x h lim h 0 h 2 = 2 F (, b) y x x y lim h 0 h 2 = 2 F (, b) x y 1.3 ( ). F : R 2 R (x, y) x y 2 2 F x y : R2 R 2 F y x : R2 R

16 16 (x, y) (, b) 2 F x y (, b) = 2 F (, b) y x 1.3. h = F F ( + θh, b + k) ( + θh, b) x x 2 F/ x y y 1 θ (0, 1) 2 F/ x y (, b) hk = 2 F y x ( + θh, b + θ k) lim (h,k) (0,0) hk = 2 F (, b) y x x y lim (h,k) (0,0) hk = 2 F (, b) x y 2 F x y (, b) = 2 F (, b) y x (, b) 2 (x, y) (, b) ( 1). f : R R g : R R f g : R R F 1 2 F/ t 1, F/ t 2 f g f g df ( g(x) ) dx = df(y) dg(x) dy dx

17 ( 2). F : R 2 R G : R R 2 F G : R R F G F G df ( G 1 (x), G 2 (x) ) = F (G 1 (x), G 2 (x)) dg 1 (x) + F (G 1 (x), G 2 (x)) dg 2 (x) dx t 1 dx t 2 dx 1.6 ( 3). F : R 2 R G : R 2 R 2 F G : R 2 R F G F G F ( G 1 (x 1, x 2 ), G 2 (x 1, x 2 ) ) = F (G 1(x 1, x 2 )) G 1 (x 1, x 2 ), + F (G 1(x 1, x 2 )) G 2 (x 1, x 2 ) x 1 t 1 x 1 t 2 x 1 F ( G 1 (x 1, x 2 ), G 2 (x 1, x 2 ) ) = F (G 1(x 1, x 2 )) G 1 (x 1, x 2 ), + F (G 1(x 1, x 2 )) G 2 (x 1, x 2 ) x 2 t 1 x 2 t 2 x ( ). F : R 2 R c R (, b) R 2 F (, b) = c F (, b) 0 y I R b J R x I y J : F (x, y) = c y = g(x) g : I J x I y J : ( ) F dg dx (x) = x x, g(x) ( ) x, g(x) F y F (x, y) = c (x, y) (, b) x 1.8. F (K, L) c K L F (K, L) = Q (1.21) (K, L) Q (K, L) (1.21) F (K, L) = Q L = g(k) (1.22) L g (1.22)

18 18 g g ( ) F dg dk (K) = K K, g(k) ( ) K, g(k) F L (K, L) F Q = F (K, L) = K 2 + K 5 + L 7 + L 9 K L Q 1. F (1, 1) 2. 1 g(k) F (K, g(k)) = F (1, 1) g (1) 3. 1 h(l) F (h(l), L) = 38 h (1) ( ). F : R L ++ R x R L ++, t > 0 : F (tx) = t m F (x) F m U(x, y) = x 1/2 y 1/2 t > 0 U(tx, ty) = t 1/2 x 1/2 t 1/2 y 1/2 = tx 1/2 y 1/2 = tu(x, y) U V V (x, y) = log U(x, y) = 1 2 log x log y V U V (tx, ty) = 1 2 log tx + 1 log ty 2 = 1 2 log t log x log t log y = log t + V (x, y) m t m V (x, y) x = y = 1 V 1 5 x t x R++ n t > 0 F (tx) = tm F (x) F m

19 α β x y 2 F 1. F (x, y) = x α y β. 2. F (x, y) = x α + y β. 3. G(x, y) F (x, y) = (G(x, y)) α. 1.8 ( ). F : R++ n R m x R n ++ : x F (x) = mf (x) x = x 1 x 2. Rn ++ : x 1 F x 1 (x) + x 2 F x 2 (x) + + x n F x n (x) = mf (x 1, x 2,..., x n ) x n F m t > 0 t t = 1 F (tx) = t m F (x) x 1 F x 1 (tx) + x 2 F x 2 (tx) + + x n F x n (tx) = mt m 1 F (x) x 1 F x 1 (x) + x 2 F x 2 (x) + + x n F x n (x) = mf (x 1, x 2,..., x n ) Q = F (K, L) F 1 K F (K, L) + L F (K, L) = Q (1.23) K L (K, L) (K, L ) (Q ) (1.23) p 6 K F K (K, L ) + L F L (K, L ) = Q K p F K (K, L ) + L p F L (K, L ) = pq (1.24) F m x R n ++ : x F (x) = mf (x)

20 20 1 p F K (K, L ) p F L (K, L ) (1.24) f : R R f(x) x x + h f f(x + h) 1.9 (( )). f : R R x f k + 1 θ [0, 1] f(x + h) = f(x) + f (x)h + 1 2! f k! f (k) (x)h k + R k+1 (h; x) (1.25) R k (h; x) = 1 (k + 1)! f (k+1) (x + θh)h k+1 f x k f (k) (x) f k x R k+1 (h; x) lim h 0 h k = 0 h x + h x f(x + h) f(x) + f (x)h + f f (k) (x)h k f(x + h) x f h f(x + h) 1 k = 1 f(x + h) f(x) + f (x)h 2 k = 2 f(x + h) f(x) + f (x)h f 2 k = 0 f (x + h) = f (x) + f (x + θh) h f (x + h) f (x) h = f (x + θh)

21 n F : R n R F (x) x x + h F F (x + h) x F k + 1 F (x + h) = F (x) + n i=1 F (x) x i h i + 1 2! + 1 k! n n n i=1 j=1 n i 1 =1 i 2 =1 2 F (x) x i x j h i h j +... n i k =1 k F (x) x i1 x i2... x ik h i1 h i2... h ik + R k+1 (h; x) 1 θ [0, 1] R k (h; x) = 1 (k + 1)! n n i 1 =1 i 2 =1 n i k+1 =1 R k+1 (h; x) lim h 0 h k = 0 k+1 F (x + θh) x i1 x i2... x ik+1 h i1 h i2... h ik+1 F (x + h) 1 F (x + h) F (x) + F (x) h 2 F (x + h) F (x) + F (x) h h 2 F (x)h n n F 2 2 F (x, y) x 2, 2 F (x, y) x y, 2 F (x, y) y x, 2 F (x, y) y 2 (x, y) R 2 c. h 1/3 k 1/3. F (x, y) 1 F (x+h, y +k) c

22 m n A R m n n A =..... m mn 1j 2j A = ( n ), j =. Rm, j = 1, 2, 3,..., n m1 A n x = x 1 x 2. Rn, x n 11 x x x n x n 21 x x x n x n Ax =. = x x x n n R m m1 x 1 + m2 x 2 + m3 x mn x n R n x R m Ax x Ax 2.1 ( ). F : R n R m x R n, y R n : F (x + y) = F (x) + F (y) x R n, t R : F (tx) = tf (x) F 2.1. A R m n F : R n R m F (x) = Ax F

23 23 F : R n R m x R n, F (x) = Ax A R m n 2.1. j 1 0 R n e j (j = 1, 2, 3,..., n) F x R n x = x 1 x 2. x n = x x x 0 n. = x 1 e 1 + x 2 e x n e n 0 e 1, e 2, e 3,..., e n e 1, e 2, e 3,..., e n F F F (e j ) = F (x) = F (x 1 e 1 + x 2 e x n e n ) 1j 2j. m1 = x 1 F (e 1 ) + x 2 F (e 2 ) + + x n F (e n ) 0 = j j = 1, 2, 3,..., n, A = ( n ) 1 F (x) = x x x n n = ( n ) x 1 x 2. = Ax x n F A R m n F (x) = Ax 2.1. A F F (x) = Bx B R m n A A = B

24 x x x n x n = b 1 21 x x x n x n = b 2. m1 x 1 + m2 x 2 + m3 x mn x n = b m n A =..... m mn Ax = b x = x 1 x 2. b 1 b 2 Rn, b =. Rm x n b m, x x x n n = b. b A Ax = b x 2. b A (1 ). n m 1, 2,..., n x = x 1 x 2. x n Rn : x x x n n = 0 x = 0 1, 2,..., n 1, 1,, x = x 1 x 2. x n Rn \ {0} : x x x n n = 0 1, 2,..., n 1

25 25 1, 2,..., n x R n, k, x k 0 : x x x k k +... x n n = 0 k = x 1 x k 1 x 2 x k 2 x k 1 x k k 1 x k+1 x k k+1 x n x k n , 2,..., n 1 j : j 0 j, k : j k j k 1 j : j = 0 j, k, j k : j = k 1, 2,..., n , 2,..., n 1 i1, i2,..., ik (k n) , 2,..., n 1 k 1, 2,..., n, n+1,..., n+k n α F : R n ++ R F (x) = (min{x 1, x 2,..., x n }) α 1. F 2. x R++ n x x 3. x R n ++ x 1 = x 2 = = x n 2.3. n m n > m n x 1,..., x n m 1 11 x n x n = 0,. m1 x mn x n = 0 x j 0 j (x 1,..., x n ) m

26 , 2,..., n R m 1 n m 2.3. n > m. 1,..., n 1,, α 1 = = α n = 0., α α n n = 0 11 α n α n = 0,. m1 α mn α n = 0., 1,..., n n., 1,..., n 1., n m ( ). R m V v V, w V : v V, t R : v + w V tv V V R m : R 2 R n m 1, 2,..., n 1 1, 2,..., n R m 1, 2,..., n R m 2.4 ( ). R m V, V n 1, 2,..., n 1. 1, 2,..., n V 1, 2,..., n = V 2. 1, 2,..., n 1 1, 2,..., n V V = R 2 V ( ) ( ) =, 2 = 0 1

27 27 1, 2 1 ( ) x = x 1 x 2 V : x = x x 2 2 1, 2 = V 1, 2 V ( ) ( ) =, 2 = 0 1 1, 2 V 1, 2 1 ( ) x = x 1 x 2 1, 2 = V V : x = (x 1 x 2 ) 1 + x ,..., k b 1,..., b l V k = l 2.5. k > l. 1,..., k b 1,..., b l V, 1,..., k b 1,..., b l., 1 = β1b βl 1b l,. k = β1 k b βl kb l. 1,..., k 1, ( ) α α k k = 0 ( ), α 1 = = α k = 0. ( ) ( ), b 1,..., b l 1, β 1α β1 k α k = 0,. βl 1α βl kα k = 0. ( ) k > l, ( )., 1,..., k 1., k l., k l., k = l. 2.5 ( ). V V dimv 2.6. F : R n R m A = ( n ) R m n F (x) = Ax 1. F 1, 2,..., n = R n 2. F 1, 2,..., n 1 3. F 1, 2,..., n R n

28 ( ). m n A = ( n ), j = 1j 2j. Rm, j = 1, 2, 3,..., n A R m A ColA 2.7 ( ). m n 1 2 A =. m m1 ColA = 1, 2,..., n, i = ( i1 i2... in ) R n, i = 1, 2, 3,..., m A A RowA RowA = 1, 2,..., m 2.8 ( ). m n 1 Ax = 0 x R n A KerA KerA R n. KerA = { x R n Ax = 0 } 2.7. m n A R m n dim (ColA) + dim (KerA) = n 2.7. v 1, v 2,..., v l KerA dim (KerA) = l, KerA R n v 1, v 2,..., v l = R n l = n x R n Ax = 0 A = 0 dim (ColA) = 0 dim (ColA) + dim (KerA) = 0 + n = n v 1, v 2,..., v l R n v 1, v 2,..., v l. v l+1, v 1,..., v l, v l+1 1

29 29. n l, (n l) v l+1,..., v n R n v1, v 2,..., v l, v l+1, v l+2,..., v n = R n Av l+1,..., Av n ColA. y ColA y A 1, 2,..., n x R n : y = Ax v 1, v 2,..., v l, v l+1, v l+2,..., v n R n x z 1 z n z 2 z =. Rn : x = z 1 v 1 + z 2 v z l v l + z l+1 v l z n v n y = A (z 1 v 1 + z 2 v z l v l + z l+1 v l z n v n ) = z 1 Av 1 + z 2 Av z l Av l + z l+1 Av l z n Av n v 1, v 2,..., v l KerA Av 1 = Av 2 = = Av l = 0 y = z l+1 Av l z n Av n y (n l) Av l+1, Av l+2,..., Av n y ColA y Av l+1, Av l+2,..., Av n ColA Av l+1, Av l+2,, Av n y Av l+1, Av l+2,..., Av n z = c l+1 c l+2. c n Rn l : y = c l+1 Av l+1 + c l+2 Av l c n Av n y = A(c l+1 v l+1 + c l+2 v l c n v n ). y Av l+1, Av l+2,..., Av n y ColA, Avl+1, Av l+2,, Av n ColA.. Avl+1, Av l+2,..., Av n = ColA

30 30 z l+1 Av l+1 + z l+2 Av l z n Av n = 0 A (z l+1 v l+1 + z l+2 v l z n v n ) = 0 z l+1 v l+1 + z l+2 v l z n v n KerA z 1 z l z 2 z =. Rl : z l+1 v l+1 + z l+2 v l z n v n = z 1 v 1 + z 2 v z l v l z 1 v 1 + z 2 v z l v l z l+1 v l+1 z l+2 v l+2 z n v n = 0 v 1, v 2,..., v l, v l+1, v l+2,..., v n R n 1 z 1 = z 2 = = z l = z l+1 = z l+2 = = z n = 0 z l+1 Av l+1 + z l+2 Av l z n Av n = 0 z l+1 = z l+2 = = z n = 0 Av l+1, Av l+2,..., Av n 1 (n l) Av l+1, Av l+2,..., Av n ColA dim (ColA) = n l dim (KerA) = l dim (ColA) + dim (KerA) = n l + l = n 2.9 ( ). R n V V n V = { x R n x z = 0, z V } V R n R n V dimv + dimv = n

31 , V v 1,..., v r, i j, e i e i = 1, e i e j = 0 (i j) e 1,..., e r. (. 7 ) R n x, r x 1 = (x e i )e i i=1, x 1 V. x 2 = x x 1, x 2 e 1,..., e r, x 2 V., R n V V., V V = {0}, R n V V., dimr n = dimv + dimv m n A R m n dim (ColA) = dim (RowA) dim (ColA) = dim (ColA ), dim (RowA) = dim (RowA ) 2.9. x Ax = 0 x KerA 1 x = 2 x = = m x = 0 z RowA : z x = 0 x (RowA) KerA = (RowA) dim (KerA) = dim ( (RowA) ) dim (KerA) = n dim (ColA) dim ( (RowA) ) = n dim (RowA) dim (ColA) = dim (RowA) 2.7, m n A R m n dim (KerA) = n dim (RowA) (2.1) 1 Ax = 0 dim (KerA) n dim (RowA) 8 (2.1) ( ) = ( ) ( ) 2.10 (rnk A). A dim (ColA) = dim (RowA) = rnka 7,,,,, p , m = c c m 1 m 1 1 x = = m 1 x = 0, m x = 0.

32 , ( ). A A 2.12 ( ). A R n n B R n n AB = I BA = I B A A R n n 1. A, A A B n AB = I BA = I, B A, AB = I BA = I A BA = [ ] 2 3 B B 2. AB = B B A [ ] 1. BA = B B 2. AB = [ ] 3 2 B B

33 A R n n F : R n R n 3 1. F 2. F 3. F : F, 1,..., n = R n., dimr n = dim(cola) = n, dim(kera) = 0., 0 = α α n n, α 1 = = α n = 0., 1,..., n 1., F. 2 1: dimr n = n., F, 1,..., n n 1., R n x 1,..., n., 1,..., n = R n., F., 1,..., n R n, F : R n R n F 1 : R n R n F F ( ). A R n n A, F : R n R n F (x) = Ax, F F A R n n F : R n R n A A R n n F : R n R n F (x) = Ax F, F 1 : R n R n, 2.13, F 1 F 1 n B F 1 (x) = Bx F F 1 F F 1 : R n R n (F F 1 )(x) = F (Bx) = ABx (2.2) F F 1 F F 1 : R n R n (F F 1 )(x) = x (2.3) (2.2) (2.3) AB I A B n 2.11 BA I B A B = A 1

34 m n m n A R n n det : R n n R deta deta A (2 ). 2 ( ) A = R det : R 2 2 R deta = A 2.14 deta A A ( ) 2 ( ) ( ) 1 0 e 1 =, e 2 = 0 1 A A e 1 Ae 1, e 2 Ae 2 Ae 1 Ae 2 e 1, e 2 A Ae 1, Ae 2 Ae 1 Ae 2 ( Ae 1 = ), Ae 2 = A e 1 e 2 ( )

35 35 e 1 e 2 e 1 e e 1 e ( ). 1. ( ) ( ) ( ) det = det + det t R ( ) ( ) 11 t det = tdet 21 t ( ) ( ) t det = tdet t ( ). ( ) ( ) det = det det : R 2 2 R

36 F : R 2 2 R n F : R n n R, n ,, ( ). n 1, 2,..., n M = { 1, 2, 3,..., n } π : M M π M 2.18 ( ). 2 k = 1, 2, 3,..., n i k = j σ(k) = j k = i k k i, j σ : M M

37 π π = τ 1 τ 2 τ r π = π 1 π 2 π s 2 r s r s ( ). sgn(π) 1 π sgn(π) = 1 π M = { 1, 2, 3,..., n } Π n n A = R. n n.... n nn (sgnπ) π(1)1 π(2)2... π(n)n = deta π Π (,, deti=1, ) n A A deta = deta deta = π Π(sgnπ) π(1)1 π(2)2... π(n)n. π Π, π π 1 Π, deta = π Π(sgnπ 1 ) π 1 (1)1 π 1 (2)2... π 1 (n)n. π 1 (1), π 1 (2),..., π 1 (n), π 1 (i) = k i = π(k), deta = π Π(sgnπ) 1π(1) 2π(2)... nπ(n)., deta.

38 ( ). n n A =..... R n n n nn i j n j 1 1 j n A ij = i i 1 j 1 i 1 j+1... i 1 n i i+1 j 1 i+1 j+1... R (n 1) (n 1) i+1 n n1... n j 1 n j+1... nn A ij det (A ij ) A n n A n ij ( 1) i+j deta ij = deta j=1 n ij ( 1) i+j deta ij = deta i=1 i j 2.23 ( ). A n, b R n, Ax = b x. x j = det( 1,..., j 1, b, j+1..., n ) deta ( ) ( ) ( ) ( ) A =, B =, C =, D = deta = detb = detc = detd = 16

39 ( ) ( ) 1 0 e 1 =, e 2 = 0 1 A e 1 Ae 1, e 2 Ae 2 Ae 1 Ae 2 A e 1, e 2 A Ae 1, Ae 2 A B C A B C e 1 e 2 A 4 4 B 4 4 C 8 2 ( ) ( ) 4 0 Ae 1 = = 4e 1, Ae 2 = = 4e ( ) ( ) 4 0 Be 1 = = 4e 1, Be 2 = = 4e ( ) ( ) 8 0 Ce 1 = = 8e 1, Ce 2 = = 2e D D e 1 e 2 ( ) ( ) 1 1 b 1 =, b 2 = 1 1 D Db 1 Db 2 ( ) ( ) 8 2 Db 1 = = 8b 1, Db 2 = = 2b e 1 e 2 C 8 2 D b 1 b D

40 ( ). n A Ax = λx, x R n \ {0}, λ R λ A x λ 2.2 A 4 e 1 e 2 B 4 e 1 e 2 C 8 2 e 1 e 2 D 8 2 b 1 b ( ) 0 1 A = 1 0 Ax = λx, x R 2 \ {0}, λ R (2.4) λ x (2.4) λ x (A λi) x = 0 (2.5) x x = 0 (A λ) (A λ) 1 (2.5) (A λi) 1 (A λi) x = 0 x = 0 (2.5) (2.5) (A λ) (A λ) det (A λi) = 0 A λ 1 det (A λi) = 1 λ = λ2 + 1 = 0 (2.6) (2.6) λ A ( ) 1 1 A = 0 1 Ax = λx, x R 2 \ {0}, λ R (2.7)

41 41 x 2.3 λ det (A λi) = 0 1 λ λ = (λ 1)2 = 0 λ = 1 (2.7) ( ) ( ) 0 1 x 1 Ax = x (A I) x = = x 2 ( ) x = t 1 0, t R \ {0} (2.8) A n A Ax = λx (A λi) det (A λi) = 0 (2.9) λ (2.9) (2.9) λ det (A λi) = 0 λ det (A λi) = 0 λ n A n n n. λ Ax = λx Ax = λx (A λi) x = 0 x λ

42 ( ). n A n P Λ A = P ΛP 1 A, P, P 1 AP., AP = P Λ. λ 1 0 P = (v 1 v n ) Λ =... 0 λ n Av i = λ i v i i = 1,... n ( ). n A R n,, A i = 1, 2,..., n : Av i = λ i v i (2.10) v 1, v 2,..., v n R n c 1 c 2 x R n, c =. Rn : x = c 1 v 1 + c 2 v c n v n (2.11) c n P P = (v 1 v 2... v n ) P n (2.11) x = P c c = P 1 x (2.12) A (2.11) (2.10) Ax = c 1 Av 1 + c 2 Av c n Av n = c 1 λ 1 v 1 + c 2 λ 2 v c n λ n v n λ λ 2 Λ = λ n

43 43 c 1 λ 1 λ c 2 λ 2. Ax = P. = P 0 λ 2 c Ax = P Λc (2.13) c n λ n λ n (2.12) (2.13) c Ax = P ΛP 1 x A = P ΛP P 1 AP,, P R n : A n v 1,..., v n. P P = (v 1,..., v n ), P R n e 1,..., e n v 1,..., v n., P R n. : P i A λ i v i. (i = 1,..., n). P 1 AP i b i,., P 1 AP. b i = P 1 Av i = P 1 λ i v i = λ i e i 2.5. ( ) 5 3 D = P = 3 5 ( ) 2.24 A R n A n A l (l n) l v 1, v 2,..., v l 1, n λ 1,..., λ l A, v 1,..., v l. v 1,..., v l 1, v 1,..., v k 1 1, v 1,..., v k 1, v k 1 k., v k = α 1 v α k 1 v k 1 ( )

44 44. A ( ),, ( ) λ k, Av k = α 1 Av α k 1 Av k 1 λ k v k = α 1 λ 1 v α k 1 λ k 1 v k 1. λ k v k = α 1 λ k v α k 1 λ k v k 1. ( ) ( ) ( ), α 1 (λ 1 λ k )v α k 1 (λ k 1 λ k )v k 1 = 0. v 1,..., v k 1 1, α 1 (λ 1 λ k ) = = α k 1 (λ k 1 λ k ) = 0. λ 1 λ k = = λ k 1 λ k 0, v k = 0., v k A., A l 1. 1 n R n 2.26 n A n R n A n 1 n n 1, I n = ( 1 ). 2 A λ R A λ A λi = 0 ( ) x = x 1 x 2 : Ax = λx (2.14) R 2 λ R A A λi 0 dim {Ker (A λi)} = 1 1 R

45 45 v ( v / Ker (A λi)) v 1 = (A λi)v (2.15) v 2 = v (2.16) v 1 v (1) (A λi) 2. (2) v 1, v 2 R (A λi)v 1 = (A λi) 2 v = 0 v 1 λ Av 1 = λv 1 (2.17) (2.15) (2.16) (2.17) Av 1 = λv 1 Av 2 = v 1 + λv 2 ( ) λ 1 A(v 1 v 2 ) = (v 1 v 2 ) 0 λ P = (v 1 v 2 ) R 2 2 (2.18) P P (2.18) ( ) P 1 λ 1 AP = 0 λ 2.7 ( 0 ). 0 det (A λi) = 0 λ 1 = µ + iν (2.19) λ 2 = µ iν i (i 2 = 1), µ ν ν 0 Ax = λ 1 x x C 2 x = u + iw (2.20)

46 46 Ax = λ 2 x x C 2 x = u iw u w R 2 w 0 (2.19) (2.20) Ax = λ 1 x A(u + iw) = (µ + iν)(u + iw) Au + iaw = µu νw + i(µw + νu) Au = µu νw (2.21) Aw = µw + νu (2.22). λ 2 = µ iν, x = u iw Ax = λ 2 x,. P = (u w) R 2 2 P (2.21) (2.22) ( ) P 1 µ ν AP = ν µ (2.23) (2.23) ν < 0 λ 1 λ 2 ρ = µ 2 + ν 2 ( ) 2 ( ) 2 µ ν + = 1 ρ ρ θ (0, π) (2.24) (2.25) (2.23) ( ) ( ) µ ν ρ cos θ ρ sin θ = ν µ ρ sin θ ρ cos θ ( ) cos θ sin θ = ρ sin θ cos θ µ = ρ cos θ (2.24) ν = ρ sin θ (2.25) θ ρ m n A F (x) = Ax F (y, x) = y Ax

47 m n A R m n F : R m R n R F (y, x) = y Ax, y R m, x R n F ( 1 ) F : R m R n R A R m n F (y, x) = y Ax, y R m, x R n m = n y = x (2 ). n x R n n n Q(x) = ij x i x j (2.26) j=1 i=1 Q : R n R Q : R n R 2.27 n n A =..... R n n n nn (2.26) n n ij x i x j = x Ax j=1 i= ( ). A, A = A, A. Q A., ( ) Q(x) = x Ax = x x 2 2, ( ) 1 1., Q 1 1 Q(x) = x Ax = x x 1 x 2 x 1 x 2 + x 2 2 = x x 2 2 x Ax = x ( 1 2 A A ) x Q(x) = x Ax A. A.

48 (n ). n A x R n \ {0} : x Ax > 0, A postive definite) x R n : x Ax 0, A (positive semidefinite) x R n \ {0} : x Ax < 0 A (negtive definite) x R n : x Ax 0 A (negtive semidefinite) 2.8. I 2.25 A = I x = x 1 x 2. x n Rn \ {0} : x Ax = x Ix = x x x 2 n > 0 A = I R n n x = x 1 x 2. x n Rn \ {0} : x Ax = x Ix = x 2 1 x 2 2 x 2 n < ( ). λ λ. 2 A = R. n n λ n 1. A 2. i : λ i > 0 i : λ i 0 2., A i λ i < 0 (λ i 0).

49 ( ). R n v 1, v 2,..., v n 0 (i j ) v i v j = 1 (i = j ) (2.27) v 1, v 2,..., v n R n 2.27 ( ). v 1, v 2,..., v n. P, P = (v 1 v 2... v n ) R n n P P = P P = I P = P 1, P., P, P = P ( ). n A 1. A 2. A P 2 :,., A : A.,., n A P., α 1 0 P 1 AP =... 0 α n., α 1,..., α n A. n = 1,, n. α 1 A, v 1. v 1 W 1 W 1, W 1 A-. e 1, e 2,..., e n e 1 = v 1, e 2,..., e n = W1 A A ( ) A α 1 0 = 0 A 1,. {e i } {e i } P 1, P 1 A., A 1,, A 1., n 1 P P 1 A 1P.

50 50,, ( ) 1 0 P = P 1 0 P ( ) 1 ( ) ( ) P α AP = 0 P 0 A 1 0 P ( ) α 1 0 = 0 P 1 A 1P. 2 1: n A, P 1 AP n P A 2.9. A n v A x R n v x = 0 v (Ax) = A P Λ, Λ = P 1 AP P. x R n : (P 1 x) Λ(P 1 x) = x Ax (P 1 x) Λ(P 1 x) = x (P 1 ) Λ(P 1 x) = x P ΛP 1 x = x Ax P (P 1 ) = P, 2.30., P = (v 1,..., v n ), z R n,,. λ 1 0 Λ =..., 0 λ n x = z 1 v z n v n x Ax = λ 1 z λ n z , ( ) ( ) 2.28

51 ( ). n A ( ) A ( )., n A ( ) A ( ) ( ). A n i 1, i 2,..., i m 1 i 1 < i 2 < < i m n m B b 11 b b 1m i1i 1 i1i 2... i1i m. b i2i B M = = b m b mm im i im i m B M detb M A m principl minor i 1 = 1, i 2 = 2,..., i m = m A m leding principl minor m,., A n 2 n 1 n A = A ( ) ( ) ( ) ( ) ( ) ( ) det 2 4 5, det, det, det, det 1, det 4, det det 2 4 5, ( ) 1 2 ( ) det, det n A k detb k A k detbk 1. A 2. A m ( ) m ( )

52 52 3. A 4. A m ( ), m ( ) :. : 1, 2 A. : 3, 4 : A = ( , detb 1 = 0, detb 2 = 0 A. (, ( ) 1 x = 1, x Ax = 1 < 0.) : :, ) ( ) 1 1 A = 5 1 Q(x) = x Ax = x 2 1 4x 1 x 2 + x 2 2, ( ) 1 2 B = 2 1 Q(x) = x Ax = x 2 1 4x 1 x 2 + x 2 2., A det(1) > 0, deta > 0, B det(1) > 0, detb = 1 4 = 3 < 0.

53 ( ). R n C x C, ɛ > 0, y R n : x y < ɛ y C C 3.2 ( ). R n C C C c = { x R n x / C } C 3.1 ( ). C x C x (x n ) N n > N : x n C. 3.2 ( ). C x R n x (x n ) n : x n C x C ,. 3.3 ( ). C x, y C, t [0, 1] : tx + (1 t)y C C., x, y C, [x, y] C ( ). C C F : C R x, y C, t [0, 1] : F (tx + (1 t)y) tf (x) + (1 t)f (y) F

54 ( ). C C F : C R, F, x, y C, t [0, 1] : F (tx + (1 t)y) tf (x) + (1 t)f (y) F C C F : C R F x C, h R n : F (x + h) F (x) + F (x) h (3.1) ( :.) 3.3. F : R L R, F x R L y R L F (y) F (x) + F (x)(y x) 3.3 (3.1) F (x + h) { F (x) + F (x) h } 0 F (x + h) = F (x) + F (x) h + h 2 F (x)h + R 2 (h; x) lim h 0 R 2 (h; x) h 2 = 0 (3.1) h 2 F (x)h + R 2 (h; x) 0 (3.2) 2 ( ) ( ) 1 1 h h 2 F (x) h h + R 2(h; x) h 2 0., h = tv, v = 1, t 0, v 2 F (x)v 0. h 0 h 2 F (x)h R 2 (h; x) (3.2) 2 v 2 F (x)v C C 2 F : C R 1. F 2. x C F 2 F (x)

55 F : R L R, F 2 x R L 2 F (x) C C F : C R F x C, h R n : F (x + h) F (x) + F (x) h 3.6. C C 2 F : C R 1. F 2. x C F 2 F (x) ( ). C C F : C R x, y C, t [0, 1] : F ( tx + (1 t)y ) mx {F (x), F (y)} F (qusi-convex function) 3.5. F,. 3.7 ( ). C C F : C R, F, x, y C, t [0, 1] : F ( tx + (1 t)y ) min {F (x), F (y)} F (qusi-concve function) 3.7. C C F : C R 1. F 2. z R : { x C F (x) z } 3.8. C C F : C R 1. F 2. z R : { x C F (x) z }

56 C C F : C R R G : R R G F : C R 3.6. F (x 1, x 2 ) = x α 1 1 xα 2 2, α 1, α 2 > 0 α 1 + α 2 1 C C C F : C R F y R n : F (x) y = 0 y 2 F (x)y 0 (3.3) F (x) F (x) 3.10 (3.3) 2 F (x) F (x) C C F : C R 1. F 2. y R n : F (x) y = 0 y 2 F (x)y 0, C. C 1 F : C R F x y : F (y) F (x) F (x) (y x).,. x y: (i) F (x) (y x) > 0 F (y) > F (x). (I.e., F (y) F (x) F (x) (y x) 0.) (ii) F (x) (y x) 0 F (y) F (x). (I.e., F (y) < F (x) F (x) (y x) < 0.) (i)., (ii). 3.8 ( ). C C F : C R x, y C : F (x) (y x) 0 F (y) F (x) F (pseudo-convex function) 3.7. (i). 9 F 2 F (x) R n.

57 (i) (ii)., F F. 2. x F (x) 0, (i) (ii)., (i) (ii), F (x) 0. C = R, F (x) = x (i) : x C, y C, F (x) (y x) > 0, ε > 0 z = (1 ε)x + εy = x + ε(y x) F (z) > F (x) (3.4) F (z) (y z) > 0. (3.5), (3.5) (ii), F (y) F (z)., (3.4) F (y) > F (x). (i). (ii) : x C, y C, F (x) (y x) 0, ε > 0 z = y + ε F (x), C z C, F (x) 0 F (x) (z x) > 0., (i) F (z) > F (x). ε 0, z y F (y) F (x)., (ii). F : C R (i) F F F (ii) x C : F (x) 0, F F ( ). J N A R J N n b R N 1. z R+ J : b = z A 2. x R N : Ax R+ J b x < 0 2

58 58 1 A z = b z z z 1 z 2 z =. RJ + A = 2. RJ N, 1 j = ( j1 j2... jn ), j = 1, 2,..., J z J J z A b = z A b = z A = z z z J J (z j 0, j = 1, 2,..., J) 1 b A (cone) b 2 Ax R J + 2 Ax R J + j x 0, j = 1, 2,..., J 1 x 0, 2 x 0,..., J x 0 b x < 0 x R N A 90 b 90 x b 3.9 ( ). R N b R N b 0 x R N = { x R N x = 0 } x = v + λb v λ R v x b 3.8. x b v v = x x b b z R J + x RN 1 z b = z A x b x = z Ax (3.6)

59 59 z 2 z Ax = z 1 1 x + z 2 2 x + + z J J x 0 b x < 0 (3.6) 1 2 J J = 1 J 2 J A = 1 2. J 1 J R J N, A = 1 2. J 1 R(J 1) N A A A 1 A 1 z R J 1 + b = z A (3.7) A x R J 1 + b x < 0 x R N J x 0 x Ax R J + b x < 0 2 J x < 0 1, 2,..., J 1 b x J â 1, â 2,..., â J 1 b   = â 1 â 2. â J 1 R(J 1) N b  b = w  (3.8) w R+ J 1 w RJ 1 + w = w 1 w 2. w J 1 RJ 1 + b = w  = J 1 w j â j j=1

60 60 â j j x J 3.8 â j = j x j x J J, j = 1, 2,..., J 1 b = = = J 1 w j â j j=1 J 1 ( ) w j j x j x J J j=1 J 1 w j j j=1 ( J 1 j=1 w j x j x J ) J (3.9) b b x J (3.9) b = b x b x J J ( ) J 1 b x J 1 b x j = w j j + x w j j=1 J x J j=1 J ( ) = w A x J 1 b x j + x w j J x J (3.10) J J x < 0 w j 0, j = 1, 2,... J 1 j=1 A x R J 1 + b x < 0 (3.10) x J 1 b x j x w j J x 0 J j=1 w J = x J 1 b x j x w j J x J j=1 z = w 1 w 2. w J 1 w J z z b = w A + w J J = z A

61 61 A 1 b  b = w  w R+ J 1  1  2  x R+ J 1 (3.11) x R N x J x x b x < 0 (3.12) Ax R J + b x < 0 A 2 â j, j = 1, 2,... J 1 b λ R : â j = j + λ J (3.13) â j x = 0 (3.14) λ R : b = b + λ J (3.15) b x = 0 (3.16) x λ R : x = x + λx (3.17) (3.13) (3.18) J x = 0 (3.18) j x = â j x (3.19) (3.14) (3.17) (3.11) â j x = â j x 0 (3.20) (3.19) (3.20) (3.18) (3.21) j x 0 j = 1, 2,... J 1 (3.21) Ax R+ J (3.15) (3.18) b x = b x (3.22) (3.16) (3.17) (3.12) b x = b x < 0 (3.23) (3.22) (3.23) b x < 0

62 =, b j, j = 1, 2,..., J b 3.13 ( ). R N C R N C b R N \ C c C : x c d > x b x R N d R (d = 0.) A R J N 1. z R+ J \ {0} : z A = 0 2. x R N : Ax R J R J e à e =. RJ 1 à = (A, e) R J (N+1) z à = (0, 0,..., 0, 1) }{{} N z R J + 1 x R N+1 à x R J + (3.24) (0, 0,..., 0, 1) x < 0 (3.25) x = ( x x N+1 ) R n

63 63 x R N Ã x = (A, e) ( ) x x N+1 = Ax + x N+1 e (3.26) (3.25) x N+1 = (0, 0,..., 0, 1) x < 0 (3.26) Ã x < Ax (3.24) Ax R++ J A z = 0 z z, A, x > 0, 2 x > 0,..., J x > 0 x R N A 90 x R N,, A,,. 3.3 L M N R L X N f n : X R (n = 1, 2,..., N) M g m : X R (m = 1, 2,..., M) (X, f 1, f 2,..., f N, g 1, g 2,..., g M ), mx (f 1(x), f 2 (x),..., f N (x)) x X subject to g 1 (x) 0, g 2 (x) 0,. g M (x) 0. (3.27) x X m : g m (x) 0, n : f n (x) f n (x ), n : f n (x) > f n (x )

64 64 x X x X f n g m ( ). x X (3.27) N + M (µ 1, µ 2,..., µ N, λ 1, λ 2,..., λ M ) R N+M + 1. µ 1,..., µ N, λ 1,..., λ M 1 ; 2. m : g m (x ) > 0 λ m = 0; 3. N M µ n f n (x ) + λ m g m (x ) = 0. n=1 m= ( ). M (g m (x ) = 0) (g m (x ) > 0) K ( M) n : f n (x ) v > 0 m K : g m (x ) v > 0 v R L f 1 (x ). f N (x ) g 1 (x ) v R++ N+K. g K (x v R L 3.14 f 1 (x ). (µ 1,..., µ N, λ 1,..., λ K ) f N (x ) g 1 (x ) = 0. N µ n f n (x ) + n=1 g K (x K λ m g m (x ) = 0 m=1 (µ 1, µ 2,..., µ N, λ 1, λ 2,..., λ K ) R N+K + \ {0}

65 65 M K λ m = 0, m > K N + M (µ 1,..., µ N, λ 1,... λ K, λ K+1,..., λ M ) R N+K + N + M µ λ 0 (µ 1,..., µ N, λ 1,..., λ M ) 3.15 t > 0 (tµ 1,..., tµ N, tλ 1,..., tλ M ) (µ 1,..., µ N, λ 1,..., λ M ) 0 1 µ 1 = 1 µ 1 0 µ 1 = x L µ N λ K 0 L + N + K 1 L N µ n f n (x ) + n=1 K λ m g m (x ) = 0 m=1 g 1 (x ) = 0, g 2 (x ) = 0,. g K (x ) = 0. K L + K N ( ). (3.27) X f n g m x X (µ 1,..., µ N, λ 1,..., λ M ) R N+M + 1. m : g m (x ) 0; 2. (µ 1,..., µ N, λ 1,..., λ M ) R N ++; 3. m : g m (x ) > 0 λ m = 0; 4. N M µ n f n (x ) + λ m g m (x ) = 0. n=1 m=1 x (3.27) 3.16 ( ). M (g m (x ) = 0) (g m (x ) > 0)

66 66 K ( M) x n : f n (x ) v 0 n : f n (x ) v > 0 m K : g m (x ) v 0 v R L N µ n f n (x ) v + n=1 ( N µ n f n (x ) + n=1 M λ m g m (x ) v > 0 m=1 ) M λ m g m (x ) v m=1 4 x x u : R 2 + R u(x 1, x 2 ) = (x 1 + 1) (x 2 + 1) p 1, p 2 w mx x R 2 + (x 1 + 1) (x 2 + 1) subject to w p 1 x 1 p 2 x 2 0. (3.28) R 2 + X = { x R 2 x 1 > 1, x 2 > 1 } x 1 0 x 2 0 mx x X (x 1 + 1) (x 2 + 1) subject to w p 1 x 1 p 2 x 2 0, x 1 0, x 2 0. (3.29) (3.29) (3.28) 3.1

67 u : R 2 + R u(x 1, x 2 ) = x 1 + x 2 p 1, p 2 w mx x R 2 + u(x 1, x 2 ) = x 1 + x 2 subject to w p 1 x 1 p 2 x 2 0. (3.30) R 2 + u (x 1, x 2 ) x 1 (x 1 0) u (x 1, x 2 ) x 2 (x 2 0) x = (x 1, x 1) x 1 > 0 x 1 > 0 x R 2 ++ R 2 ++ mx x R 2 ++ u(x 1, x 2 ) = x 1 + x 2 subject to w p 1 x 1 p 2 x 2 0. (3.31) (3.30) (3.31) w(> 0) u : R+ 2 R u(x 1, x 2 ) = x e x 2 mx x R 2 + u(x) subject to w x 1 x 2 0. (3.32) x R+ 2 (3.32) x { x R 2 x 1 > 0, x 2 0 } u X = { x R 2 x 1 > 0 }

68 68 e x 2 x 2 R X g 1 : X R g 1 (x) = w x 1 x 2 mx x X u(x) subject to g 1 (x) 0. (3.33) X (3.33) (3.33) mx x X u(x) subject to g 1 (x) 0, g 2 (x) 0. (3.34) (3.32) g 2 : X R (3.34) (3.32) K L M X R L P R K f : X P R g m : X P R (m = 1, 2,..., M) X P 2 p P mx x X subject to g 1 (x, p) 0, f(x, p) (3.35) g 2 (x, p) 0,. g M (x, p) 0. (3.35) p P p P (3.35) 3.10 (Policy Function). p P (3.35) P : P X (p) Policy Function 3.11 (Vlue Function). Policy Function b : P R b(p) = f((p), p) b(p) Vlue Function

69 p P (1, λ 1,..., λ K ) R 1+K ++ x X (3.35) = (L+M) (L+M) M 2 xf(x, p ) + λ m 2 xg m (x, p ) x g 1 (x, p )... x g M (x, p ) m=1 x g 1 (x, p ) x g M (x, p ) p Q P Policy Function (p) Vlue Function b(p) Q ( ). p P (1, λ 1,..., λ K ) R 1+K ++ x X (3.35) Policy Function (p) Vlue Function b(p) M b(p ) = p f(x, p ) + λ m p g m (x, p ) m= p P : g m ((p), p) = 0, m = 1, 2,..., M p p = p x g(x, p ) (p ) + p g m (x, p ) = 0, m = 1, 2,..., M (3.36) (3.36) λ m m M λ m x g(x, p ) (p ) + m=1 M λ m p g m (x, p ) = 0 (3.37) (1, λ 1,..., λ K ) R 1+K ++ x X (3.38) (3.37) x f(x, p ) + m=1 M λ m x g m (x, p ) = 0 (3.38) m=1 x f(x, p ) (p ) + Vlue Function M λ m p g m (x, p ) = 0 (3.39) m=1 p P : b(p) = f ((p), p) p p = p b(p ) = x f(x, p ) (x ) + p f(x, p ) (3.40) (3.40) (3.39)

70 mx x V f(x) (3.41) subject to g j (x) 0, j = 1, 2,..., m V R n f : V R g j : V R (j = 1, 2,... m) 3.19 ( ). x (3.41) m + 1 λ 0, λ 1,..., λ m m λ 0 f(x ) + λ j g j (x ) = 0 (3.42) j=1 λ j g j (x ) = 0, j = 1, 2,..., m (3.43) λ 0, λ 1,..., λ m 3.19 (3.42) 1 First Order Condition: FOC (3.43) Complementry Slckness λ 0 > 0 (3.42) (3.43) λ 0 λ 1, λ 2,..., λ m m f(x ) + λ j g j (x ) = 0 (3.44) j=1 g j (x ) = 0, j = 1, 2,..., m (3.45) (3.44) (3.45) 3.19 λ 0 = 0 Constrint Qulifiction 3.20 ( ). x (3.41) x m λ 1, λ 2,..., λ m m f(x ) + λ j g j (x ) = 0 j=1 g j (x ) = 0, j = 1, 2,..., m λ 1, λ 2,..., λ m g j 2. x 0 R n j = 1, 2,..., m g j (x 0 ) > λ 0 0 λ j > 0 g j (x ) 1.

71 Slter Condition ( ). x (3.41) 1 g j 2 m λ 1, λ 2,..., λ m m f(x ) + λ j g j (x ) = 0 j=1 g j (x ) = 0, j = 1, 2,..., m λ 1, λ 2,..., λ m 3.22 ( ). x x m λ 1, λ 2,..., λ m (3.44) (3.45) g j f 12 x (3.41) ( ). R n f (3.41) (3.44) (3.45) 14 (3.41) m L(x, λ) = f(x) + λ j g j (x) L : V R m + R x V j=1 λ R m + L(x, λ ) L(x, λ ) L(x, λ) (x, λ ) V R m + L(x, λ) sddle point 3.24 ( ). (3.41) (x, λ ) V R+ m L(x, λ) 1. x i L(x, λ ) 0, i = 1, 2,..., n 12 h φ > 0 φ : R R f(x) φ`h(x)

72 72 2. x i 3. x i L(x, λ ) = 0, λ j L(x, λ ) 0, 4. λ j λ j L(x, λ ) = 0, 3.24 x i x i L(x, λ ) = x i L(x, λ ) = x i i = 1, 2,..., n j = 1, 2,..., m j = 1, 2,..., m m f(x ) + x i j=1 ( f(x ) + x i g j (x ) 0, i = 1, 2,..., n (3.46) x i ) m λ j g j (x ) = 0, i = 1, 2,..., n (3.47) x i λ j j=1 L(x, λ ) = g j (x ) 0, j = 1, 2,..., m (3.48) λ j λ j L(x, λ ) = λ j g j (x ) = 0, j = 1, 2,..., m (3.49) λ j (3.48) (3.49) i x i > 0 (3.46) (3.47) x i f(x ) + m j=1 λ j f(x ) + g j (x ) = 0, x i m λ j g j (x ) = 0 j=1 i = 1, 2,..., n 1 (3.41) x f(x ) + m λ j g j (x ) = 0 (3.50) j=1 g j (x ) = 0, j = 1, 2,..., m (3.51) g j (x ) 0, j = 1, 2,..., m (3.52) ( ). f g j (j = 1, 2,..., m) x (3.41) λ R m + (x, λ ) V R+ m L(x, λ) 3.25 f g j (j = 1, 2,..., m) (3.41) 3.24 (3.41) 15 (3.50) (3.51) (3.52) (3.50) (3.51)

73 ( ). f g j (j = 1, 2,..., m) (x, λ ) V R+ m 3.24 (x, λ ) L(x, λ) (3.41) (3.41) 3.27 ( ). (x, λ ) V R+ m L(x, λ) x (3.41) L(x, λ) x f : R R 2 f > 0 f > 0 f 0 min x R g(x) := 1 x f(x) x > 0 g g (x) = 0 x g (x) = 0 x

74 x 2 mx (c 0,c 1 ) subject to u(c 0 ) + δu(c 1 ) (4.1) c 0 + c 1 x c 0 0 c 1 0 u c 0 c δ (4.1) (c 0, c 1) u (c 0) = λ δu (c 1) = λ λ u (c 0) + δu (c 1) = 0 c 0 + c 1 = x x T (4.1) mx (c 0,...,c T 1 ) subject to u(c 0 ) + δu(c 1 ) + δ 2 u(c 2 ) + + δ T 1 u(c T 1 ) (4.2) c 0 + c c T 1 x c 0 0 c 1 0. c T (4.2) (c 0,..., c T 1 ) u (c 0) = λ δu (c 1) = λ. δ T 1 u (c T 1 ) = λ 16 Stokey-Lucs, chpeter4

75 75 δ t 1 u (c t 1) + δ t u (c t ) = 0, t = 1, 2,..., T 1 c 0 + c c T 1 = x T T x mx {c t :t=0,1,... } subject to δ t u(c t ) (4.3) t=0 c t x t=0 c t 0, t = 0, 1,... Dynmic Progrmming: DP 4.2 X R n + x t X δ 0 < δ < 1 f : X X R DP( x 0 ) mx {x t :t=1,2,... } δ t f (x t, x t+1 ) t=0 subject to Γ (x t, x t+1 ) 0, t = 0, 1,... x 0 = x 0 x X Γ (x, y) 0 y X 4.1 ( ). x z u (z) δ {c t } (4.3) t {x t } c t = x t x t+1 x = x 0

76 76 (4.3) mx {x t:t=0,1,... } subject to δ t u (x t x t+1 ) t=0 x 0 = x x t x t+1 0, t = 0, 1,... f (x t, x t+1 ) := u (x t x t+1 ) Γ (x t, x t+1 ) := x t x t+1 f (x t, x t+1 ) Γ(x t, x t+1 ) mx {x t:t=0,1,... } subject to δ t f (x t, x t+1 ) t=0 x 0 = x Γ (x t, x t+1 ) 0, t = 0, 1,... DP 4.1 ( ). β (β > 0) t t + 1 x t+1 = β(x t c t ) DP 4.2 ( ). t x t c t s t i t t + 1 c t + s t = x t x t = (1 + i)s t δ t u (c t ) t=0 DP 4.1 ( ). {x t : t = 0, 1,... } DP( x 0 ) {x t : t = 0, 1,... } x 0 Γ (x t, x t+1 ) > 0 t X f : X X R

77 77 Euler eqution f (x t 1, x t ) + δ f (x t, x t+1 ) = 0, t = 1, 2,... (4.4) x t x t DP( x 0 ) 4.1 ( ). {x t : t = 0, 1, 2,... } DP( x 0 ) (4.4) 4.1. {x t } DP( x 0 ) δ t f (x t, x t+1 ) = f (x 0, x 1 ) + δf (x 1, x 2 ) + + δ t 1 f (x t 1, x t ) + δ t f (x t, x t+1 ) +... (4.5) t=0 t x t V (z) = f (x 0, x 1 ) + δf (x 1, x 2 ) + + δ t 1 f (x t 1, z) + δ t f (z, x t+1 ) +... (4.6) V (z) mx z V (z) (4.7) subject to Γ (x t 1, z) 0 Γ (z, x t+1 ) 0 z = x t x t V (x t) > V (x t ) f (x 0, x 1 ) + + δ t 1 f (x t 1, x t) + δ t f (x t, x t+1 ) + > δ t f (x t, x t+1 ) {x 0, x 1,... x t,... } {x 0, x 1,... x t,... } DP( x 0 ) (4.8) z = x t z = x t (4.8) 1 d dz V (z) = 0 z=xt δ t 1 f (x t 1, x t ) + δ t f (x t, x t+1 ) = 0 x t x t δ t 1 f (x t 1, x t ) + δ f (x t, x t+1 ) = 0 (4.8) x t x t t (4.4) t=0

78 ( ). mx {x t :t=0,1,... } δ t u (x t x t+1 ) t=0 subject to x t x t+1 0, t = 0, 1,... x 0 = x u (x t 1 x t ) + δu (x t x t+1 ) = 0 c t u (c t 1 ) + δu (c t ) = 0 δu (c t ) u (c t 1 ) = 1 (4.9) t 1 δ t 1 u (c t 1 ) t δ t u (c t ) (4.9) t 1 t t 1 t 17 (4.9) DP( x 0 ) 1 lim t δt f (x t, x t+1 ) x t = 0 (4.10) x t Trnsverslity Condition: TC 4.2 ( ). f (x t, x t+1 ) X X R R x t f (x t, x t+1 ) 0 X X {x t : t = 0, 1,... } 1. (4.4) 2. (4.10) {x t : t = 0, 1,... } DP( x 0 ) 4.2. {x t : t = 0, 1,... } (4.4) (4.10) {x t : t = 0, 1,... } δ t f (x t, x t+1 ) δ t f ( x t, x ) t+1 t=0 t=0 17

79 79 δ t f (x t, x t+1 ) δ t f ( x t, x ) t+1 t=0 t=0 f (x t, x t+1 ) ( T lim T ( T T = lim δ t f (x t, x t+1 ) δ t f ( x t, x ) ) t+1 T t=0 t=0 ( T = lim δ t( f (x t, x t+1 ) f ( x t, x t+1) )) (4.11) T f ( ) x t, x t+1 f (xt, x t+1 ) Df (x t, x t+1 ) (x t x t, x ) t+1 x t+1 = f 1 (x t, x t+1 ) (x t x t ) + f 2 (x t, x t+1 ) ( x ) t+1 x t+1 f (x t, x t+1 ) f ( x t, x t+1) f1 (x t, x t+1 ) (x t x t) + f 2 (x t, x t+1 ) ( x t+1 x ) t+1 t=0 δ t( f (x t, x t+1 ) f ( x t, x ) )) t+1 lim T t=0 ( T δ t( f 1 (x t, x t+1 ) (x t x t) + f 2 (x t, x t+1 ) ( x t+1 x t+1) )) (4.12) t=0 x 0 = x 0 T t=0 δ t( f 1 (x t, x t+1 ) (x t x t) + f 2 (x t, x t+1 ) ( x t+1 x ) ) t+1 = f 1 (x 0, x 1 ) (x 0 x 0) + = f 2 (x 0, x 1 ) (x 1 x 1) + δf 1 (x 1, x 2 ) (x 1 x 1) + δf 2 (x 1, x 2 ) (x 2 x 2) + δ 2 f 1 (x 2, x 3 ) (x 2 x 2) + δ 2 f 2 (x 2, x 3 ) (x 3 x 3) + δ 3 f 1 (x 3, x 4 ) (x 3 x 3) +. δ T 1 f 2 (x T 1, x T ) (x T x T ) + δ T f 1 (x T, x T +1 ) (x T x T ) + δ T f 2 (x T, x T +1 ) ( x T +1 x ) T +1 T 1 δ t( (xt+1 f 2 (x t, x t+1 ) + δf 1 (x t+1, x t+2 )) x t+1) + δ T f 2 (x T, x T +1 ) ( x T +1 x ) T +1 t=0 (4.12) lim T ( T t=0 δ t( f 1 (x t, x t+1 ) (x t x t) + f 2 (x t, x t+1 ) ( x t+1 x ) )) t+1 = lim T ( T 1 t=0 (4.4) δ t( (xt+1 f 2 (x t, x t+1 ) + δf 1 (x t+1, x t+2 )) x ) ) t+1 + lim T δt f 2 (x T, x T +1 ) ( x T +1 x ) T +1 (4.13) f 2 (x t, x t+1 ) + δf 1 (x t+1, x t+2 ), t = 0, 1, 2,...

80 80 (4.13) 1 lim T ( T 1 δ t( (xt+1 f 2 (x t, x t+1 ) + δf 1 (x t+1, x t+2 )) x t+1) ) = 0 (4.14) t=0 2 lim T δt f 2 (x T, x T +1 ) ( x T +1 x ) T +1 = lim T δt +1 f 1 (x T +1, x T +2 ) ( x T +1 x ) T +1 (4.11) (4.12) (4.13) (4.14) (4.15) δ t f (x t, x t+1 ) δ t f ( x t, x t+1) lim T δt +1 f 1 (x T +1, x T +2 ) ( x T +1 x ) T +1 t=0 t=0 (4.16) lim T δt +1 f 1 (x T +1, x T +2 ) ( x T +1 x ) T +1 TC (4.15) (4.16) x t f (x t, x t+1 ) 0 x T +1 = lim T δt +1 f 1 (x T +1, x T +2 ) x T +1 lim T δt f 1 (x T, x T +1 ) x T = 0 + lim T δt +1 f 1 (x T +1, x T +2 ) x T +1 lim T δt +1 f 1 (x T +1, x T +2 ) x T +1 (4.17) (4.17) 0 (4.16) (4.17) δ t f (x t, x t+1 ) δ t f ( x t, x t+1) 0 (4.18) t=0 f (x t, x t+1 ) (4.18) t=0 t=0 δ t f ( x t, x t+1) < t=0 δ t f (x t, x t+1 ) δ t f ( x t, x ) t+1 {x t : t = 0, 1,... } DP( x 0 ) f {x t : t = 0, 1,... } δ t f ( x t, x t+1) t=0 f f t=0 f(x t, x t+1 ) = u (g(x t ) x t 1 ) u = ln(z) f R X x X R X [0, M]

81 ( ). u (z) = ln (z) mx {x t:t=0,1,... } δ t ln (x t x t+1 ) t=0 subject to x t x t+1 0, t = 0, 1,... x 0 = x c t = x t x t δ = 0, t = 0, 1,... x t x t+1 x t+1 x t+2 (4.19) t = 0, 1, 2,... 1 c t + δ 1 c t+1 = 0 c t+1 = δc t (4.19) c t = δ t c 0, t = 0, 1,... (4.20) (4.20) (4.20) c 0 c 0 x 0 (4.20) (4.20) ( c t = c δ + δ ) t=0 = c δ c δ = x 0 x 0 c 0 = δx 0 c 0 = x 0 x 1 c 1 = x 1 x 2 x 1 = x 0 c 0 = δx 0 x 2 = x 1 c 1 = δ (x 0 c 0 ) = δ 2 x 0

82 82 x t = δ t x 0, t = 0, 1,... (4.21) x 0 (4.21) {x t : t = 0, 1,... } f(z) = ln(z) 4.2 f x t f (x t, x t+1 ) 0 f (x t, x t+1 ) TC TC x t f (x t, x t+1 ) = 1 x t x t+1 = 1 c t = 1 δ t c 0 lim t δt f (x t, x t+1 ) x t = lim δ t 1 x t t δ t δ t x 0 c 0 = lim δ t x 0 t c 0 = lim δ t 1 t 1 δ = 0 TC (4.21) Bellmn eqution ( ). v : X R { } x X, v(x) = mx y: Γ(x,y) 0 f(x, y) + δv(y) (4.22) v (4.22) v 4.3 (Policy Function). v(x) { } φ(x) f(x, y) + δv(y) mx y: Γ(x,y) 0 φ(x) Policy Function 18

83 ( ). DP( x 0 ) x 0 X {x t : t = 0, 1,... } v ( x 0 ) = t=0 δ t f ( x t, x ) t+1 v (4.22) (4.23) (4.23) v Vlue Function x 0 = x {x t : t = 0, 1,... } DP( x 0 ) Vlue Function 4.3. x 0 X DP( x 0 ) { x 0, x 1, x 2, x 3,... } x 0 X Vlue Function v v ( x 0 ) = t=0 δ t f ( x t, x ) t+1 v ( x 0 ) = f ( x 0, x 1) + δf (x 1, x 2) + δ 2 f (x 2, x 3) +... (4.24) v ( x 0 ) < f ( x 0, ˆx 1 ) + δv (ˆx 1 ) (4.25) ˆx 1 {x X Γ( x 0, x) 0} ˆx 1 mx {x t :t=2,3,... } δ t f (x t+1, x t+2 ) t=0 subject to Γ (x t, x t+1 ) 0, t = 1, 2,... x 1 = ˆx 1 {ˆx 1, ˆx 2, ˆx 3,... } Vlue Function v (ˆx 1 ) = f (ˆx 1, ˆx 2 ) + δf (ˆx 2, ˆx 3 ) + δ 2 f (ˆx 3, ˆx 4 ) +... (4.26) (4.25) (4.24) (4.26) f ( x 0, x 1) + δf (x 1, x 2) +... < f ( x 0, ˆx 1 ) + δv (ˆx 1 ) = f ( x 0, ˆx 1 ) + δf (ˆx 1, ˆx 2 ) + δ 2 f (ˆx 2, ˆx 3 ) +... { x 0, ˆx 1, ˆx 2,... } { x 0, x 1, x 2,... } DP( x 0 ) (4.25) x 1 {Γ( x 0, x 1 ) 0}, v ( x 0 ) f ( x 0, x 1 ) + δv (x 1 ) (4.27) x 0 X { } x 0 X, v ( x 0 ) = mx x 1 : Γ( x 0,x 1 ) 0 f( x 0, x 1 ) + δv(x 1 ) v (4.22)

84 ( ). 4.3 u (z) = ln (z) Vlue Function x t = δ t x 0 c t = (1 δ)δ t x 0 v (x) = δ t ln ( (1 δ)δ t x ) t=0 { x X, v (x) = mx ln(x y) + δ y: Γ(x,y) 0 mx y ln(x y) + δ δ t ln ( (1 δ)δ t y )} (4.28) t=0 δ t ln ( (1 δ)δ t y ) t=0 subject to Γ(x, y) 0 (4.29) y 1 1 y ln ( (1 δ)δ t y ) = lny + ln(1 δ)δ t 1 x y + δ δ t 1 y = 0 t=0 y = δx (4.29) { mx ln(x y) + δ y: Γ(x,y) 0 δ t ln ( (1 δ)δ t y )} = ln(x δx) + δ δ t ln ( (1 δ)δ t δx ) t=0 = ln ( (1 δ)x ) + = t=0 δ t+1 ln ( (1 δ)δ t+1 x ) t=0 δ t ln ( (1 δ)δ t x ) t=0 = v (x) (4.28) 4.3. u (z) = ln (z) Vlue Function

85 {x t : t = 0, 1,... } DP( x 0 ) Vlue Function v DP( x 0 ) Vlue Function 4.4 ( ). v 1. v (4.22) 2. {x t : t = 0, 1,... } lim t δ t v(x t ) = 0 19 φ { x 0, φ( x 0 ), φ 2 ( x 0 ),... } DP( x 0 ) 4.4. v { } x X, v(x) = mx y: Γ(x,y) 0 f(x, y) + δv(y) (4.30) mx y f(x, y) + δv(y) subject to Γ(x, y) 0 φ(x) (4.30) φ x X, v(x) = f ( x, φ(x) ) + δv ( φ(x) ) (4.31) φ(x) X (4.31) x = φ(x) x X, v ( φ(x) ) = f ( φ(x), φ φ(x) ) + δv ( φ φ(x) ) x X, v(x) = f ( x, φ(x) ) + δv ( φ(x) ) = f ( x, φ(x) ) + δf ( φ(x), φ 2 (x) ) + δ 2 v ( φ 2 (x) ). = δ t f ( φ t (x), φ t+1 (x) ) t=0 φ 0 (x) = x φ t (x) = φ φ φ(x), }{{} t 1 t x x = x 0 v( x 0 ) = δ t f ( φ t ( x 0 ), φ t+1 ( x 0 ) ) (4.32) t=0 (4.32) { x 0, φ( x 0 ), φ 2 ( x 0 ),... } 19

86 86 v Γ( x 0, x 1 ) 0 x 1 v( x 0 ) f( x 0, x 1 ) + δv(x 1 ) (4.33) x 1 Γ(x 1, x 2 ) 0 x 2 (4.33) v(x 1 ) f(x 1, x 2 ) + δv(x 2 ) v( x 0 ) f( x 0, x 1 ) + δv(x 1 ) f( x 0, x 1 ) + δf(x 1, x 2 ) + δ 2 v(x 2 ) {x t : t = 0, 1,... } v( x 0 ) f( x 0, x 1 ) + δv(x 1 ) f( x 0, x 1 ) + δf(x 1, x 2 ) + δ 2 v(x 2 ) f( x 0, x 1 ) + δf(x 1, x 2 ) + δ 2 f(x 2, x 3 ) + δ 3 v(x 3 ).. T δ t f(x t, x t+1 ) + δ T v(x T ) t=0 δ t f(x t, x t+1 ) (4.34) t=0 ( T ) δ t f(x t, x t+1 ) + δ T v(x T ) lim T t=0 = lim = T T t=0 δ t f(x t, x t+1 ) t=0 δ t f(x t, x t+1 ) + lim T δt v(x T ) (4.32) (4.34) {x t : t = 0, 1,... } δ t f ( φ t ( x 0 ), φ t+1 ( x 0 ) ) δ t f(x t, x t+1 ) t=0 t=0 φ { x 0, φ( x 0 ), φ 2 ( x 0 ),... } DP( x 0 ) Policy Function Vlue Function v v

87 87 v : X R T v(x) T v(x) := mx { } f(x, y) + δv(y) y:γ(x,y) 0 mx y f(x, y) + δv(y) subject to Γ(x, y) 0 (4.35) x X, T v(x) = v(x) (4.36) x X, v(x) = mx { } f(x, y) + δv(y) y:γ(x,y) 0 v v (4.36) (4.35) T v v T 2 v(x) := mx { } f(x, y) + δt v(y) y:γ(x,y) 0 T 2 v(x) (4.37) x X, T 2 v(x) = T v(x) (4.38) x X, T v(x) = mx { } f(x, y) + δt v(y) y:γ(x,y) 0 T v (4.38) (4.37) T 2 v T n v v v v 4.5. DP( x 0 ) f X x X T n v(x) v (x) := lim n T n v(x) x X, v (x) = mx { f(x, y) + δv (y) } y:γ(x,y) 0 v v 4.5 v T n v v v v

88 v : X R v { } x X, v(x) = mx y: Γ(x,y) 0 f(x, y) + δv(y) v mx y f(x, y) + δv(y) subject to Γ(x, y) 0 Policy Function φ(x) 1 x X, f 2 (x, φ(x)) + δv (φ(x)) = 0 (4.39) x d dx v(x) = { } f(x, y) + δv(y) x y=φ(x) v (x) = f 1 (x, φ(x)) (4.39) x X, f 2 (x, φ(x)) + δf 1 (φ(x), φ 2 (x)) = 0 (4.40) {x 0, x 1, x 2,... } Policy Function φ(x t 1 ) = x t, φ 2 (x t 1 ) = x t+1, t = 1, 2,... (4.40) x = x t 1 X f 2 (x t 1, x t ) + δf 1 (x t, x t+1 ) = 0, t = 1, 2, x x 1 1 p > 0 c > 0 y + cy δ DP t x t X X X = {0, 1, 2,... } x t 1 t 1 t x t 1 t + 1 x t+1 t

89 89 x t+1 (x t 1) t + c(x t+1 x t + 1) f 1 π(x t ) ( + c(x t+1 x t + 1) ) x t+1 x t + 1 > 0, f(x t, x t+1 ) = π(x t ) x t+1 x t p x t > 0, π(x t ) = 0 x t 0. Γ(x t, x t+1 ) = x t+1 x t f, π, Γ DP Vlue Function n n r n := p(δ + δ δ n ) ( + cn) = p 1 δn 1 δ ( + cn) n ( r n 1 + δ n + ( δ n) ) = r n 1 δ n = p 1 1 δ + cn 1 δ n (4.41) 1 pδ + c pδ > + c n = 1 (4.41) (4.41) n n 4.5. n n (4.41) pδ > + c (4.41) Π Π := p 1 1 δ + cn 1 δ n

90 90 n (4.41) DP Vlue Function v x 1 v(1) = p + Π v(2) = p + δv(1) = (p + δp) + δπ v(3) = p + δv(2) = (p + δp + δ 2 p) + δ 2 Π. v(x) = p + δv(x 1) = (p + δp +... δ x 1 p) + δ x 1 Π (4.42) v(x) = p 1 δx 1 δ + δx 1 Π = 1 1 δ p v(0) (p + cn 1 δ n ) δ x 1 (4.43) lim δ t v(x t ) = 0 x ( ) t p +cn δ x 1 v(x 1 δ n t ) (4.43) v x 1 y 1 p ( + cy) δv(x 1 + y) p δv(x 1) x 1 { ( v(x) = mx p + δv(x 1), mx p ( + cy) + δv(x 1 + y)) } (4.44) y y 1 p ( + cy) + δv(x 1 + y) y (4.44) 2 y = 1 x 1 p + δv(x 1) p ( + c) + δv(x) (4.43) Π +cn 1 δ n +cn 1 δ n = 1 n (4.44) x 1 v(x) = p + δv(x 1) (4.42) 4.9. x = 0

91 91 5 Policy Function Policy Function 20 Policy Function n n h k : R n R (k = 1, 2,..., n) h 1, h 2,..., h n h : R n R n x R n, h(x) = ( h 1 (x), h 2 (x),..., h n (x) ) h x t+1 = h(x t ), t = 0, 1, 2,... (5.1) n x 0 (5.1) x 0 { x0, h(x 0 ), h 2 (x 0 ), h 3 (x 0 ),... } (5.1) {x t : t = 0, 1, 2,... } 5.1 ( ). (5.1) x R n x = h( x) x sttionry point 5.2 ( ). x R n (5.1) x 0 x 0 (5.1) {x t : t = 0, 1, 2,... } lim x t = x t x globlly stble x x 0 x 0 (5.1) {x t : t = 0, 1, 2,... } lim x t = x t x loclly stble 20 Stokey-Lucs, chpter 6

92 h 1 h(x) = βx, β β 1, 0 x = h(x) x = βx (1 β)x = 0 x = 0 (β 1 ) x = 0 x {x t : t = 0, 1, 2,... } x 0 x t+1 = βx t x t = β t x 0, t = 0, 1, 2,... x 0 lim x 0 β < 1, t = t β > 1. β 1 x c 0 h(x) = βx + c h(x) = x {z t } x = c 1 β z t = x t x, t = 0, 1, 2,... z t+1 = x t+1 x = βx t + c c 1 β = βx t β c 1 β = β(x t x) = βz t

93 93 lim z 0 β < 1, t = t β > 1. lim z t = 0 lim (x t x) = 0 t t lim x t = x t β 1 x 1 β n n A h(x) h(x) = Ax I A 21 {x t : t = 0, 1, 2,... } x t+1 = Ax t, t = 0, 1, 2,... x = Ax (I A)x = 0 I A (I A) 1 (I A)x = 0 (I A) 1 (I A)x = 0 x = 0 x = 0 x lim t x t = 0 lim t At x 0 = 0 h(x) = Ax + 0 {x t : t = 0, 1, 2,... } x t+1 = Ax t + 0, t = 0, 1, 2,... x = Ax + 0 I A x = (I A) (1 β) 0

94 94 1 {z t } z t+1 = x t+1 x z t = x t x, t = 0, 1, 2,... (5.2) = Ax t + 0 (I A) 1 0 = Ax t + (I A)(I A) 1 0 (I A) 1 0 = Ax t A(I A) 1 0 = A(x t x) = Az t lim z t = 0 lim (x t x) = 0 t t lim x t = x t x lim t z t = ( ). lim t At z 0 = 0 x t+1 = Ax t + 0, t = 0, 1, 2,... (5.3) x 1. (5.3) {x t : t = 0, 1, 2,... } lim t x t = x 2. det(a λi) = A 23 A H H AH = Λ λ i (i = 1, 2,..., n) A λ λ 2 Λ = λ n 22 A A A + bi 2 + b 2 23

95 95 (5.2) {z t } z t+1 = Az t (5.4) H H = H 1 (5.4) H H z t+1 = H Az t = H AHH z t = ΛH z t (5.5) y t = H z t, t = 0, 1, 2,... (5.6) {y t } (5.5) y t+1 = Λy t y 1 t+1 y 2 t+1. = λ λ y 1 t y 2 t. = λ 1 y 1 t λ 2 y 2 t. y n t λ n y n t λ n y n t y i t (i = 1, 2,..., n) y i t = (λ i ) t y i 0, t = 0, 1, 2,... 1 λ i 1 lim {y t : t = 0, 1, 2,... } t y i t = 0 λ i < 1, i = 1, 2,..., n lim yt i = 0, t i = 1, 2,..., n (5.7) lim y t = 0 t (5.8) H y t z t (5.8)(5.9)(5.10) lim y t = 0 lim H z t = 0 t t H lim t z t = 0 HH lim z t = 0 t lim z t = 0 (5.9) t lim z t = 0 lim (x t x) = 0 t t lim x t = x (5.10) t λ i < 1, i = 1, 2,..., n lim t x t = x

96 (5.3) 5.1. n = 1 x t+1 = x t 5.2. n = 2 ( x t+1 y t+2 ) ( ) x t = by t A n m m < n 1 n m {y t } y 1 t+1 y 2 t+1. y n t+1 λ = 0 λ λ n λ i λ 1, λ 2,..., λ m 1 λ m+1, λ m+2,..., λ n 1 m y 1 t, y 2 t,..., y m t y 1 0, y 2 0,..., y m 0 y 1 t y 2 t.. y n t lim t yi t = 0, i = 1, 2,..., m n m yt m+1, yt m+2,..., yt n lim t yi t = 0, ym+1 0, y0 m+2,..., y0 n i = m + 1, m + 2,..., n y y 0 = 0.. y0 m y m+1 y m+1 0, y m+2 0,..., y n 0 x 0 y n 0 lim y t = 0 lim x t = x t t m m m

97 97 x 0 stble mnifold ψ : R n R n m ( ) 0 0 ψ(x) := H (x x) 0 I n m y t = H (x t x) ( ) 0 0 ψ(x t ) = H (x t x) 0 I n m ( ) 0 0 = y t 0 I n m = y m+1 t.. y n t ψ(x 0 ) = 0 y m = 0 y n 0 x 0 ψ(x 0 ) = 0 {x t : t = 0, 1, 2,... } x 0 { x R n ψ(x) = 0 } lim t x t = x {x R n ψ(x) = 0} 5.3. n = 2 ( x t+1 y t+2 ) ( ) 2x t 1 = 1 2 y t 5.2 h h h : R n R n x R n, h(x) = ( h 1 (x), h 2 (x),..., h n (x) ) h x R n Dh(x) Dh(x) = h 1 (x) x 1 h 2 (x) h 1 (x) x 2... h 1 (x) x n 1 x h n (x) x 1... h n (x) x n

98 ( ). x I Dh(x) 3 1. det(dh( x) λi) = 0 1 x 2. 1 x 3. 1 m x ψ : R n R n m (5.1) ψ(x 0 ) = 0 lim t x t = x 5.2 ψ ψ ψ(x) = 0 R n m {x R n ψ(x) = 0} n = 1 h(x) = x h(x) = x 3 +x x = 0 h (0) = 1 h (0) λ = n = 1 h h(x) = x(x 1)(x 2) + x x t+1 = h(x t )

99 ( ). I R f : I R I F x I, F (x) = f(x) F I f f 6.1. F f : I R 2 1. C ˆF (x) = F (x) + C ˆF f 2. G f C G(x) = F (x) + C 6.1. F f x I, F (x) = f(x) x I, ˆF (x) = F (x) = f(x) ˆF f 1 2 x I, G (x) = f(x) h(x) := G(x) F (x) h x I, h (x) = G (x) F (x) = f(x) f(x) = 0 (6.1), b I : h(b) h() b = h (c) c b (6.1), b I, h() = h(b)

100 100 h I C x I, C = h(x) = G(x) F (x) x I, G(x) = F (x) + C 6.1 F f F F (x) + C f 6.2 ( ). I R f f(x)dx 24 F f C f(x)dx = F (x) + C (6.2) (6.2) f f f C ( ). f g I α, β {αf(x) + βg(x) } dx = α f(x)dx + β g(x)dx 6.2. F G f g c 1, c 2 f(x)dx = F (x) + c 1 g(x)dx = G(x) + c 2 f(x)dx + g(x)dx = F (x) + G(x) + c 1 + c 2 (6.3) d { } F (x) + G(x) dx = F (x) + G (x) = f(x) + g(x) 24

101 101 F (x) + G(x) f(x) + g(x) c 3 {f(x) + g(x) } dx = F (x) + G(x) + c3 (6.4) c 1 c 2 c 3 c 1 + c 2 = c 3 (6.3) (6.4) {f(x) + g(x) } dx = α α 0 f(x)dx + g(x)dx (6.5) d { } αf (x) dx = αf (x) = αf(x) αf (x) αf(x) c 4 αf(x)dx = αf (x) + c 4 = { α F (x) + c } 4 α = α f(x)dx βg(x)dx = β g(x)dx (6.5) {αf(x) } + βg(x) dx = αf(x)dx + βg(x)dx = α f(x)dx + β g(x)dx 6.3 ( ). f g I f g f (x)g(x)dx = f(x)g(x) 6.3. f(x)g (x)dx d { } f(x)g(x) = f (x)g(x) + f(x)g (x) dx f(x)g(x) f (x)g(x) + f(x)g (x) c {f (x)g(x) + f(x)g (x) } dx = f(x)g(x) + c 6.2 f (x)g(x)dx + f(x)g (x)dx = f(x)g(x) + c c 0

102 ( ). f I x t x = ϕ(t) ϕ f (x)dx = f(ϕ(t))ϕ (t)dt 6.4. f F c 1 f(x)dx = F (x) + c 1 = F ( ϕ(t) ) + c 1 (6.6) d dt F ( ϕ(t) ) = F ( ϕ(t) ) ϕ (t) = f ( ϕ(t) ) ϕ (t) F ( ϕ(t) ) f ( ϕ(t) ) ϕ (t) c 2 f ( ϕ(t) ) ϕ (t)dt = F ( ϕ(t) ) + c 2 (6.7) c 1 c 2 (6.6) (6.7) R [.b] [, b] n + 1 x 0, x 1, x 2,..., x n = x 0 < x 1 < < x n = b {x 0, x 1,..., x n } [, b] = {x 0, x 1,..., x n } [, b] D { D := {x 0, x 1,..., x n } } = x0 < x 1 < < x n = b, n N D [, b] n [x 0, x 1 ], [x 1, x 2 ],..., [x n 1, x n ] ξ i [x i 1, x i ] S(, ξ) = n f(ξ i )(x i x i 1 ) i=1 f(x)

103 103 ξ = (ξ 1, ξ 2,..., ξ n ) f(x) [, b] n δ( ) := mx 1 i n (x i x i 1 ) δ( ) δ( ) 0 S(, ξ) f(x) f(x) [, b] 6.3 ( ). f [, b] [, b] δ( ) s R ɛ > 0, δ(ɛ) : δ( ) < δ(ɛ) S(, ξ) s < ɛ f [, b] s = lim S(, ξ) δ( ) 0 s [, b] f(x) s = f(x)dx b b f(x)dx := f(x)dx 6.5. f [, b] f [, b] ( ). R [, b] f [, b] 6.5 f [, b] n f 6.6 f [x i 1, x i ] i = 1, 2,..., n M i m i [x i 1, x i ] f(x) x x i x x i M i m i f( x i ) = M i, S( ) := f(x i ) = m i n M i (x i x i 1 ) i=1

104 104 n s( ) := m i (x i x i 1 ) i=1 2 S( ) s( ) 6.7. s( ) s( ) S( ) S( ) 6.8., D s( ) S( ) 6.8 s( ) {s( ) D} {s( ) D} s D, s( ) s 6.8 S( ) {s( ) D} D, s( ) s S( ) 6.4 ( ). R I f ɛ, δ(ɛ), x, y I : x y < δ(ɛ) f(x) f(y) < ɛ f I f f I f I 6.9. f I f I [, b] 6.5 f [, b] M i m i m i f(ξ i ) M i, i = 1, 2,..., n m i (x i x i 1 ) f(ξ i )(x i x i 1 ) M i (x i x i 1 ), i = 1, 2,..., n

105 105 n n n m i (x i x i 1 ) f(ξ i )(x i x i 1 ) M i (x i x i 1 ) i=1 i=1 i=1 i = 1, 2,..., n ξ 6.8 s {s( ) D} s( ) S(, ξ) S( ) (6.8) D, s( ) s S( ) (6.8) ξ D, S( ) s( ) > S(, ξ) s (6.9) f [, b] ɛ x, x [, b] : x x < δ(ɛ) f(x) f(x ) < ɛ (6.10) δ(ɛ) δ( ) < δ(ɛ) x i x i 1 < δ(ɛ), i = 1, 2,..., n x i x i [x i 1, x i ] x i x i < δ(ɛ), i = 1, 2,..., n (6.10) f( x i ) f(x i ) < ɛ M i m i < ɛ n s( ) S( ) = (M i m i )(x i x i 1 ) < ɛ i=1 n (x i x i 1 ) i=1 = ɛ(b ) (6.9) ɛ > 0 δ(ɛ) > 0 δ( ) < δ(ɛ) S(, ξ) s < S( ) s( ) < ɛ(b ) (6.11) (b ) > 0 ɛ (6.11) f [, b]

106 ( ). f g [, b] α, β { } b αf(x) + βg(x) dx = α f(x)dx + β g(x)dx [, b] D αf(x)+βg(x) S(, ξ) = = α n { αf(ξi ) + βg(ξ i ) } (x i x i 1 ) i=1 n f(ξ i )(x i x i 1 ) + β f g [, b] lim i=1 δ( ) 0 i=1 lim δ( ) 0 i=1 lim S(, ξ) = α lim δ( ) 0 = α δ( ) 0 i=1 n f(ξ i )(x i x i 1 ) = n g(ξ i )(x i x i 1 ) = n g(ξ i )(x i x i 1 ) i=1 n f(ξ i )(x i x i 1 ) + β f(x)dx + β g(x)dx f(x)dx g(x)dx lim δ( ) 0 i=1 n f(ξ i )(x i x i 1 ) 6.11 ( ). f [, b] < c < b c 25 f(x)dx = c f(x)dx [, b] = {x 0, x 1,..., x n } c f(x)dx = x 0 < < x k = c < x k+1 < < x n = b, c n f(ξ i )(x i x i 1 ) = i=1 k f(ξ i )(x i x i 1 ) + i=1 n i=k+1 f(ξ i )(x i x i 1 ) f [, b] [, c] [c, b] c δ( ) 0 lim δ( ) 0 i=1 n f(ξ i )(x i x i 1 ) = f(x)dx 25 f [, c] [c, b] [, b]

107 107 lim δ( ) 0 i=1 k f(ξ i )(x i x i 1 ) + lim n δ( ) 0 i=k+1 f(ξ i )(x i x i 1 ) = c f(x)dx + c f(x)dx 6.12 ( ). f g [, b] [, b] x [, b], f(x) g(x) f(x)dx g(x)dx f(x)dx = g(x)dx x [, b], f(x) = g(x) [, b] f(x) g(x) [, b] = {x 0, x 1,..., x n } ξ i [x i 1, x i ] f(ξ i )(x i x i 1 ) g(ξ i )(x i x i 1 ), i = 1, 2,..., n n n f(ξ i )(x i x i 1 ) g(ξ i )(x i x i 1 ) i=1 i=1 δ( ) 0 f(x)dx f(x)dx = g(x)dx x [, b], f(x) = g(x) g(x)dx (6.12) f(c) > g(c) c [, b] f g c f(x) > g(x) x [α, β], f(x) 1 2 f(c) > g(x) 1 2 g(c) c [α, β] [, b] [α, β] = {x 0, x 1,..., x n } ξ i [x i 1, x i ] f(ξ i )(x i x i 1 ) 1 2 f(c)(x i x i 1 ) > g(ξ i )(x i x i 1 ) 1 2 g(c)(x i x i 1 ), i = 1, 2,..., n n f(ξ i )(x i x i 1 ) 1 n 2 f(c) n (x i x i 1 ) > g(ξ i )(x i x i 1 ) 1 n 2 g(c) (x i x i 1 ) i=1 i=1 i=1 i=1

108 108 n i=1 f(ξ i )(x i x i 1 ) 1 n f(c)(β α) > 2 i=1 δ( ) 0 β β α α f(x)dx 1 2 f(c)(β α) β α g(ξ i )(x i x i 1 ) 1 g(c)(β α) 2 g(x)dx 1 g(c)(β α) 2 β f(x)dx g(x)dx 1 α 2 f(c)(β α) 1 g(c)(β α) 2 > 0 (6.13) f(c) > g(c) f(x)dx = g(x)dx = α α β f(x)dx + f(x)dx + α β g(x)dx + g(x)dx + α [, α] [β, b] f(x) g(x) β β f(x)dx g(x)dx α f(x)dx α g(x)dx, f(x)dx β β g(x)dx f(x)dx g(x)dx (6.13) (6.12) f(x)dx β α β f(x)dx g(x)dx α g(x)dx > 0 x [, b], f(x) g(x), f(x)dx = g(x)dx, x [, b], f(x) = g(x) 6.1. f [, b] f(x)dx f(x) dx x [, b], f(x) 0, f(x)dx = 0, x [, b], f(x) = 0

109 ( ). f [, b] c (, b) 1 b f(x)dx = f(c) f [, b] 6.6 [, b] M m f x [, b] x [, b] f( x) = M, f(x) = m M = m f [, b] k S(, ξ) = = k n f(ξ i )(x i x i 1 ) i=1 n (x i x i 1 ) i=1 = k(b ) [, b] δ( ) 0 c [, b] 1 b M > m f(x)dx = k(b ) f(x)dx = 1 k(b ) b = k = f(c) x [, b], x [, b], m f(x) M m < f(x) < M mdx < f(x)dx < Mdx m(b ) < f(x) = m < 1 b f(x)dx < M(b ) f(x)dx < M = f( x)

110 110 x < x f [x, x] [, b] 1 b f(x)dx = f(c) c (x, x) c 6.14 ( ). f [, b] x [, b] [, x] f [, x] x x [, b] F (x) := F F [, b] x f(t)dt x [, b], F (x) = f(x) x [, b] x + h [, b] h 0 h > 0 f [x, x + h] [, b] 1 h x+h c (x, x + h) F x+h x f(t)dt = x F (x + h) F (x) = (6.14) (6.15) F (x + h) F (x) h f(t)dt = f(c) (6.14) f(t)dt + x+h x x+h x f(t)dt = f(c), x < c < x + h h +0 f f(t)dt (6.15) F (x + h) F (x) lim h +0 h = lim h +0 = f( lim h +0 = f(x) (6.16) h < 0 f [x + h, x] [, b] 1 h x x+h f(t)dt = f(c) (6.17) c (x + h, x) x f(t)dt = x+h x f(t)dt + f(t)dt x+h

111 111 F (6.17) (6.18) x F (x + h) F (x) = f(t)dt (6.18) x+h F (x + h) F (x) h = f(c), x + h < c < x h 0 f (6.16) (6.19) F (x + h) F (x) lim h 0 h F (x + h) F (x) lim = f(x) h 0 h = lim h 0 = f( lim h 0 = f(x) (6.19) F x [, b] f(x) 6.14 x f(t)dt f [, b] f F C F (x) = x f(t)dt + C f [, b] f F f(x)dx = F (b) F () (6.20) C [ F (x) ] b F (b) F () = F (x) = 6.15 (6.20) = x f(t)dt + C f(t)dt + C f(t)dt [ ] b F (x) := F (b) F () f()dt C f(x)dx = [ F (x) ] b

112 f [, b] f(x) = 1, [, b] = [0, 1] 1 + x f [0, 1] = {0, 1 n, 2 n,..., n n } ξ i = i n [ i 1 n, i n ] i = 1, 2,..., n S(, ξ) = 0 n i=1 = 1 n ( = ( ( i )( i f n n i 1 ) n n n ) n n ) 1 n n n + n n δ( ) 0 ( ) 1 1 f(x)dx = lim n n n n + n F (x) = log(1 + x) f(x) 1 0 f(x)dx = F (1) F (0) = log(1 + 1) log(1 + 0) = log(2) lim n ( ) 1 n n n + n = log(2) f 2 f(x, y) y g(x, y) = g(x, y) = x x g(x, y) x F (x, y) = y F (x, y) = y x b x b y y f(t, y)dt f(t, y)dt = f(x, y) g(x, u)du g(x, u)du = g(x, y)

113 113 F F (x, y) = y x b f(t, u)dtdu 2 F (x, y) = f(x, y) x y 6.3. f [, b] f [, b] g [, b] G(x) = x g(t)dt G [, b] 6.4. f [, b] F (x) = x f(t)dt F [, b] G [, b] [, b] g 6.1. lim n lim n G(x) = f n (x) dx = f n (x) dx = x g(t)dt { } lim f n (x) dx n { } lim f n (x) dx n { } f n (x) lim f n (x) n y f (x, y) dx = { } f (x, y) dx y y f (x, y) ( ). f g [, b] f g f (x)g(x)dx = [ ] b f(x)g(x) f(x)g (x)dx

114 x [, b], d { } f(x)g(x) = f (x)g(x) + f(x)g (x) dx f(x)g(x) f (x)g(x) + f(x)g (x) 6.15 { f (x)g(x) + f(x)g (x) } dx = f (x)g(x)dx + f(x)g (x)dx = [ ] b f(x)g(x) [ ] b f(x)g(x) 6.17 ( ). f [, b] x t x = ϕ(t) ϕ [α, β] ϕ(α) =, ϕ(β) = b ϕ [α, β] [, b] t [α, β] ϕ(t) [, b] f (x)dx = β α f(ϕ(t))ϕ (t)dt (6.21) F F (x) = x f(t)dt F [, b] x [, b], F (x) = f(x) F f 6.15 f(x)dx = F (b) F () (6.22) {ϕ(t) α t β} [, b] F x {ϕ(t) α t β} ϕ t [α, β] t [α, β], d dt F ( ϕ(t) ) = F ( ϕ(t) ) ϕ (t) = f ( ϕ(t) ) ϕ (t) F ( ϕ(t) ) [α, β] f ( ϕ(t) ) ϕ (t) 6.15 β (6.22) α f(ϕ(t))ϕ (t)dt = F ( ϕ(β) ) F ( ϕ(α) ) = F (b) F ()

115 t [α, β], ϕ(t) [, b] (6.21) f [, b] [, b] ϕ(t) [, b] t [α, β] f ϕ(t) (6.21) t [α, β], ϕ(t) [, b] (6.21) ϕ [, b] ϕ f 6.5. f(x) = x 4 [0, 1] f(x) = x 4 F (x) = 1 5 x x 4 dx = x 4 dx = F (1) F (0) = 1 5 f(x) = x x 3 = [ ] x2 x 3 x 4 dx { } 1 2 x2 x x2 3x 2 dx = x 4 dx = x 4 0 ϕ(t) = 2 t ϕ ϕ (t) = t 1 2 ϕ(0) = 0, ϕ( 1 4 ) = 1 [0, 1 4 ] ϕ t [0, 1 4 ], ϕ(t) [0, 1] 1 0 x 4 dx = = = 16 = 16 = 1 5 f ( ϕ(t) ) ϕ (t)dt ( 2 t ) 4t 1 2 dt 6.6 (, 241, 7.6). f(x) = (x 2 + 1) 1 2 f [0, 1] ϕ (x 2 + 1) 1 2 = t x x 1 4 [ t 5 2 ϕ(t) = t2 1 2t t 3 2 dt ] (6.23)

116 116 ( (ϕ(t) ) ) 1 2 = t ϕ(t) = t t ϕ ϕ(1) = 0, ϕ(1 + 2) = 1 t [1, ], ϕ (t) = t t (x 2 + 1) 1 2 dx = = = = f ( ϕ(t) ) ϕ (t)dt ( ) (ϕ(t) ) t t 2 dt 2t t t dt = [ log t ] = log(1 + 2) t t 2 dt 6.6 ϕ(t) = 0 t = 1 t = 1 (6.23) ϕ α = 1, β = ϕ(α) =, ϕ(β) = b ϕ [ 1, 1 + 2] t = 0 α = 1, β = ϕ(t) = 1 t = t = 1 2 α, β α = 1, β = 1 2 α, β f [, b] f [, ) 6.5 ( ). f (, b] (, b] 0 < ɛ < b ɛ f [ + ɛ, b] I(ɛ) = +ɛ f(x)dx I(ɛ) ɛ +0 f (, b] f(x)dx = lim f(x)dx ɛ +0 +ɛ

117 117 f [, b) 0 < ɛ < b ɛ I(ɛ) = ɛ f(x)dx I(ɛ) ɛ +0 f [, b) f(x)dx = lim ɛ +0 ɛ f(x)dx f (, b) I(ɛ 1, ɛ 2 ) = ɛ2 +ɛ 1 f(x)dx I(ɛ 1, ɛ 2 ) ɛ 1 +0, ɛ 2 +0 f (, b) f(x)dx = ɛ2 lim f(x)dx ɛ ɛ ɛ f [, b) f F I(ɛ) = ɛ f(x)dx = F (b ɛ) F () f(x)dx = lim F (b ɛ) F () ɛ +0 f f [, b] [, b] f (, b], (, b], (, b) f [, b] ɛ 0 < ɛ < b [, b ɛ], [b ɛ, b] f f [, b] f(x)dx = ɛ f(x)dx + b ɛ f(x)dx ɛ +0 2 f(x)dx = lim ɛ +0 ɛ f(x)dx f [, b) [, b] (, b] (, b) 6.3. f [, b) c b c [, c) [c, b) f f(x)dx = c f(x)dx + c f(x)dx

118 118 f (, b) c 1 < c 2 < < c n (, b) (, c 1 ), (c 1, c 2 ),... (c n, b) f f f(x)dx = c1 c2 f(x)dx + c 1 f(x)dx + + f(x)dx c n f (, b) 6.7. f 0 x 0, f(x) = x 1 2 x > 0. x +0 f(x) f [ 1, 1] f [ 1, 0) (0, 1] [ 1, 1] 1 1 f(x)dx = 0 1 f(x)dx f(x)dx 6.4. < c 1 < c 2 < b f (, c 1 ) (, c 2 ) (c 1, b) (c 2, b) c1 f(x)dx + c 1 f(x)dx = c2 f(x)dx + f(x)dx c 2 [, t] I(t) = t f(x)dx t + I(t) [, ) f(x)dx = (, b] I(t) = I(t) t f(x)dx = (, ) I(s, t) = t lim t + t f(x)dx lim t t t s f(x)dx f(x)dx f(x)dx I(s, t) s, t f(x)dx = t lim s t s f(x)dx

119 f(x, y) F (y) = f(x, y)dx y F y F (y) f F (y) = d dy = f(x, y)dx d f(x, y)dx dy F (y) f x y y x 6.18 ( ). 2 f(x, y) [, b] [c, d] 1. x 2. y 3. f y (x, y) x y d dy f(x, y)dx = d f(x, y)dx dy F (y) = = lim F (y) = d dy = d { dy = lim = f(x, y)dx δ( ) 0 i=1 n f(ξ i, y)(x i x i 1 ) f(x, y)dx lim n δ( ) 0 i=1 n δ( ) 0 i=1 d f(x, y)dx dy } f(ξ i, y)(x i x i 1 ) { d dy f(ξ i, y) } (x i x i 1 )

120 lim n f n (x) dx = { } lim f n (x) dx n 6.19 ( ). 2 f(x, y) [, b) [c, d] x 2. y 3. f y (x, y) f(x, y)dx = lim ɛ +0 ɛ f(x, y)dx y x y d dy f(x, y)dx = 6.19 d f(x, y)dx dy h ɛ > 0, h : h < h < b f(x, y)dx y [c, d] f(x, y)dx < ɛ 6.4 mx f F G ( f(t) ) dt subject to Γ ( f(t) ) dt 0 (6.24) G Γ F [, b] [, b] = [0, 1] [0, 1] n [0, 1 n ) [ 1 n, 2 n 1 n )... [ n, 1] (x 1,..., x n ) k x k (x 1,..., x n ) R n F n := R n F n 27 b = + mx f F n G ( f(t) ) dt subject to Γ ( f(t) ) dt 0 (6.25)

121 G ( f(t) ) dt = (6.25) 1 n 0 2 n G(x 1 )dt + 1 n ( ) 1 = G(x 1 ) n 0 = 1 n G(x k ) n k=1 1 G(x 2 )dt + + n 1 n ( ) 2 + G(x 2 ) n 1 n G(x n )dt ( + + G(x n ) 1 n 1 n ) mx (x 1,x 2,...,x n ) R n n G(x k ) k=1 subject to n Γ(x k ) 0 (6.26) k=1 G, Γ = (6.26) λ 0 G (x k ) + λγ (x k ) = 0, k = 1, 2,..., n (6.27) n Γ(x k ) 0 λ k=1 n Γ(x k ) = 0 k=1 (6.25) (6.26) (6.24) f(t) = (6.24) G ( f(t) ) + λγ ( f(t) ) = 0, t [0, 1] (6.28) λ Γ ( f(t) ) dt 0 Γ ( f(t) ) dt = 0 f f (6.24) f (6.24) G ( f (t 1 ) ) ( Γ ( f (t 1 ) ) > G f (t 2 ) ) Γ ( f (t 2 ) ) (6.29) 2 t 1, t 2 G Γ t 1 t 2 f t 1 t 2 f (6.24) (6.29) t 1, t 2 t [0, 1] λ G ( f (t) ) + λγ ( f (t) ) = 0, t [0, 1] λ 0 (6.28)

122 122 f (6.28) f G ( f (t) ) dt G ( f(t) ) dt (6.30) t k n [ k n, k+1 n ) k [ k t n, k + 1 ) : f n n(t) = f ( t k ) n fn k f ( ) t k n f n F n t k n [ k n, k+1 n ) Γ ( fn(t) ) dt = 1 n Γ ( f ( t k )) n = 0 n k=1 f (6.28) G ( f ( t k n )) + λγ ( f ( t k )) n = 0, k = 1, 2,... n f ( ) t k n, k = 1, 2,..., n (6.27) (6.26) f t k n [ k n, k+1 n ) [ k t n, k + 1 ) : f n (t) = f ( t k ) n n f n k f ( t k n) fn F n t k n [ k n, k+1 n ) Γ ( f n (t) ) dt = 1 n n Γ ( f ( t k n)) = 0 f ( tn) ( ) k, k = 1, 2,..., n f t k n, k = 1, 2,..., n (6.26) n k=1 G ( f ( t k n k=1 )) n k=1 G ( f ( t k )) n (6.26) (6.25) f n f n (6.25) f = lim (6.30) G ( f n(t) ) dt n f n 6.5. c G ( f n (t) ) dt f = lim n f n n mx f F 1 0 f(t)dt subject to 1 0 ( f(t) ) 2dt c

123 Ω Ω F F R P : F R P 1. A F : 0 P (A) 1 2. P (Ω) = 1 ( ) 3. A k F, k l : A k Al =, P A k = P (A k ) k=1 k=1 (Ω, F, P ) A F P (A) A P (A) A (Ω, F, P ) (Rndom Vrible: RV) X : Ω R Ω X R A P (A) E = {X(ω) ω A} P (A) = P ( {ω Ω X(ω) E} ) A X E P ( {ω Ω X(ω) E} ) Pr(X E) E (, x) Pr(X x) 7.1 ( ). X : Ω R F X (x) := Pr (X x) F X : R [0, 1] Cumultive Distribution Function: CDF F X (x) X x 7.1. F X : R [0, 1] 1. x < y F X (x) F X (y) 2. x : lim h +0 F X(x + h) = F X (x) 3. lim x F X(x) = 0, lim F X(x) = 1 x 7.1 F

124 F X (x) X X [, b] R F X (b) F X () = f X (x)dx (7.1) f X : R R + f X (x) X density function f X (x) x F X (x) (7.1) b d dx F X(x) = f X (x) f X (x) f X (x) = F X (x + h) F X (x) lim h 0 h = Pr(x X x + h) lim h 0 h X [x, x + h] h h 0 [, + h] h +0 Pr( x + h) = Pr(x + h) Pr(x ) = F X ( + h) F X () Pr(x = ) = F X () F X () = 0 [, b] Pr( x b) = Pr(x b) Pr(x ) = F X (b) F X () = f X (z)dz 7.1. g 1 g f X (x) X f X (x) 1 X 7.1. F X (x) X

125 125 F X (x) = 0 x < x 0 x < x < x 1 2 x < x F X (x) F X (x) x = 0 (, 0) (0, ) x = 1 2 x = F X (x) F X (x) f X (x) = d dx F X(x) f X (x) 0 x < 0 f X (x) = 1 0 x < x < x < x f X (x) F X (b) F X () = f X (x)dx (7.2), b > 0 f X (x) X F X (x) X X p X (x) p X (x) = F X (x) lim h +0 F X(x h) p X (x) X probbility function X [ h, ] h +0 Pr( h x ) = Pr(x ) Pr(x h) = F X () F X ( h) Pr(x = ) = F X () lim h +0 F X( h) p X () X

126 x 1, x 2, x 3 p X (x 1 ), p X (x 2 ), p X (x 3 ) X p X (x i ) 0, i = 1, 2, 3, X 3 p X (x i ) = 1 i= ( ). X F X (x) [, b] 28 F X (x) x i i = 1, 2,... X h E[h(X)] = h(x i )p X (x i ) i=1 E[h(X)] h(x) exptected vlue men p X (x) p X (x) = F X (x) lim h +0 F X(x h) 7.3 ( ). X F X (x) [, b] X h E[h(X)] = h(z)f X (z)dz E[h(X)] h(x) f X (x) F X (b) F X () = f X (x)dx (7.3) h(x) = x X X F X (x) X E[X] = ( ) d x dx F X(x) dx X h(x) E[h(X)] = F X (b)h(b) F X ()h() F X (z)h (z)dz (7.4) F X () = 0, F X (b) = 1 b E[h(X)] = h(b) F X (z)h (z)dz X 28 F X () = 0, F X (b) = 1

127 x 1, x 2, x 3 p X (x 1 ), p X (x 2 ), p X (x 3 ) X p X (x i ) 0, i = 1, 2, 3, X E[X] = b 3 p X (x i ) = 1 i=1 F X (z)dz 7.2. [, ] ( ) d h(x) dx F X(z) dz = lim b = lim b ( d h(x) { h(b) ) dx F X(z) dz } F X (z)h (z)dz h(b) F X(z)h (z)dz b ( ) d b h(x) dx F X(z) dz = lim h(b) lim F X (z)h (z)dz b b ( ) d h(x) dx F X(z) dz = h( ) F X (z)h (z)dz (, ) (7.4) F X (x) h(x)df X = E[h(X)] 29 (7.4) h(z)df X = F X (b)h(b) F X ()h() F X () = 0, F X (b) = 1 h(x)df X = h(b) h(x)f X (dx) = E[h(X)] F X (z)h (z)dz F X (z)h (z)dz (7.5) (7.4) F X (x) z [, b] F X (z) = F 1 (z) + F 2 (z) (7.6) 29 Stieltjes Integrl

128 128 F X (x) F 1 (x) F 2 (x) F k (x) 0 k = 1, 2 E[h(X)] (7.4) E[h(X)] = k = 1, 2 = h(b) = h(b) = h(b) h(x)df X F X (z)h (z)dz { } F 1 (z) + F 2 (z) h (z)dz F 1 (z)h (z)dz h(z)df k (z) = F k (b)h(b) F k ()h() F 2 (z)h (z)dz (7.7) F k (z)h (z)dz F 1 (b) + F 2 (b) = F X (b) = 1 F 1 () + F 2 () = F X () = 0 h(z)df 1 (z) + (7.7) (7.8) h(z)df 2 (z) = h(b) E[h(X)] = h(z)df 1 (z) + F 1 (z)h (z)dz h(z)df 2 (z) F 2 (z)h (z)dz (7.8) (7.6) F 1 (z) F 2 (z) F X (z) E[h(X)] = h(z)df 1 (z) = h(z)df 2 (z) = h(x i )p(x i ) i=1 h(x i )p(x i ) + i=1 h(z) d dz F 2(z)dz h(z) d dz F 2(z)dz F 1 (z) F 2 (z) i=1 p(x i ) < 1 i=1 d p(x i ) + dz F 2(z)dz = 1 d dz F 2(z)dz < X F X (x) 0 x < 0, F X (x) = min { x, 1} 0 x.

129 129 F X (0) = 0 F X (1) = 1 [0, 1] F 1 (x) F 2 (x) x [0, 1] 0 x < 0, F 1 (x) = 1 0 x. 2 { 1 F 2 (x) = min } 2 x, 1 F X (x) = F 1 (x) + F 2 (x) F 1 (x) x = xdf 1 (x) = 0 p(0) = 0 F 2 (x) x [0, 1] ( d 1 dx ) 2 x = xdf 2 (x) = xdx = 1 4 X E[h(X)] = 1 xdf 1 (x) xdf 2 (x) = = (7.4) E[X] ( ). X α, β E[αX + β] = αe[x] + β

130 E[αX + β] = = αb + β ( = α b = α (αx + β)df X (x) = αe[x] + β F X (x)αdx ) F X (x)dx xdf X (x) + β 7.3 ( ). [, b] f g + β x [, b], f(x) g(x) X E[f(X)] E[g(X)] 7.3. X f X f(x) g(x) x [, b] x [, b] : (f(x) g(x)) f X (x) 0 E[f(X)] E[g(X)] = = 0 f(x)f X (x)dx (f(x) g(x)) f X (x)dx g(x)f X (x)dx X p X f(x i ) g(x i ) i = 1, 2,... (f(x i ) g(x i )) p X (x i ) 0, i = 1, 2,... E[f(X)] E[g(X)] = = 0 f(x i )p X (x i ) g(x i )p X (x i ) i=1 i=1 (f(x i ) g(x i )) p X (x i ) i=1

131 ( ). X V r[x] = E [(X E[X]) 2] V r[x] X vrince V r[x] σ 2 [X] [ E (X E[X]) 2] = E [X 2 2E[X]X + ( E[X] ) ] 2 = E [ X 2] 2E[X]E[X] + ( E[X] ) 2 = E [ X 2] ( E[X] ) 2 V r[x] = E [ X 2] ( E[X] ) 2, b X + b V r[x + b] = E [ (X + b) 2] ( E[X + b] ) ( ). X = E [ 2 X 2 + 2bX + b 2] ( E[X] + b ) 2 = 2 E[X 2 ] + 2bE[X] + b 2 2( E[X] ) 2 2bE[X] b 2 = 2( E[X 2 ] ( E[X] ) 2 ) = 2 V r[x] (7.9) σ[x] = V r[x] σ[x] X stndrd divition (7.9) X + b σ[x + b] = V r[x + b] = 2 V r[x] = σ[x] 7.3. xf(x)dx xf(x) x x 2 f(x) x f(x) x xf(x) x 2 f(x)

132 F G x, F (x) G(x) F G 31, b F () = G() = 0, F (b) = G(b) = 1 (7.10) F G [, b] u [, b] u(x) X F G u(x)df (x) = u(b) F (z)u (z)dz u(x)dg(x) = u(b) G(z)u (z)dz u(x)df (x) u(x)dg(x) = u(b) u (x) 0 = = F (z)u (z)dz u(b) + G(z)u (z)dz { } G(z) F (z) u (z)dz F (z)u (z)dz G(z)u (z)dz x [, b], F (x) G(x) u(x)dg(x) u(x)df (x) (7.11) u u u(x) X (7.11) G F F (x) G(x) 30 MWG

133 133 F G (7.11) 7.4. (7.10) F G 2 1. x [, b] F (x) G(x) 2. [, b] u u(x)dg(x) u(x)df (x) ˆx [, b] F (ˆx) < G(ˆx) F G ˆx [α, β] 32 x [α, β] : F (x) G(x) < 0 (7.12) x [α, β] : u (x) > 0, x [, b] \ [α, β] : u (x) = 0 (7.13) u u(x) = x u (t)dt u [, b] 2 u u(x)dg(x) u(x)df (x) 0 (7.12) (7.13) u(x)df (x) u(x)dg(x) = u(b) = = < 0 β α F (z)u (z)dz u(b) + { } G(z) F (z) u (z)dz { } G(z) F (z) u (z)dz G(z)u (z)dz x [, b] F (x) G(x) F G G F 7.6 (1 ). [, b] X, Y F X (x), F Y (x) x [.b] F X (x) F Y (x) X Y 1 X first-order stochsticlly domintes Y F X F Y 1 32

134 x [, b] F (x) G(x) F (ˆx) > G(ˆx) ˆx [, b] x [, b] u (x) > 0 G F u(x)df (x) u(x)dg(x) < 0 1 X Y 1 u E[u(X)] E[u(Y )] (7.14) u(z) = z u (7.14) E[X] E[Y ] X Y 1 X Y Jensen s Inequlity 7.5 ( ). [, b] u 1. u 2. [, b] X E[u(X)] u ( E[X] ) [, b] X E[X] = µ u s x [, b] : u(x) u(µ) s(x µ) (7.15) 33 E [ u(x) u(µ) ] E [ s(x µ) ] E [ u(x) ] u(µ) s(e[x] µ) = u s = u (µ) (7.15) s 34 X X x 1, x 2,... x n u P p(x i )u(x i ) u`p p(x i )x i 2

135 u x 1, x 2 [, b] p (0, 1) pu(x 1 ) + (1 p)u(x 2 ) > u ( px 1 + (1 p)x 2 ) p x 1 (1 p) x 2 X E[u(X)] > u ( E[X] ) X E[X] 1 X E[X] 7.5 X Y X Y F, G X, Y Y X u(x)df (x) u(x)g(x) 0 u F G X Y 7.6. F () = G() = 0 F (b) = G(b) = 1 F, G 2 1. x [, b] x F (t)dt x G(t)dt 2. [, b] u u(x)df (x) u(x)g(x) X Y E[X] = b E[Y ] = b E[X] = E[Y ] F (x)dx G(x)dx { } F (x) G(x) dx = 0 (7.16)

136 ˆF (x) 1 ˆF (x) = ( ) b F (x)dx x F (t)dt (7.17) ˆF ˆF () = 0, ˆF (b) = 1 F ˆF (x) x ˆF (x) ˆF [, b] ˆF (x) ˆF (x) d dx ˆF (x) = = x 1 d ( ) b F (x)dx dx 1 ( )F (x) F (x)dx F (t)dt f [, b] f(x)df (x) = F (b)f(b) F ()f() = f(b) f (x)f (x)dx F (x)f (x)dx f (x)d ˆF (x) = = = f (x) d dx ˆF (x)dx 1 ( ) b F (x)dx 1 ( ) b F (x)dx f (x)f (x)dx ( f(b) ) f(x)df (x) (7.18) X Y (7.17) F (x)dx = G(x)dx (7.19) (7.19) 1 1 ( ) b F (x)dx x F (t)dt 1 ( ) b G(x)dx x G(t)dt (7.18) ˆF (x), Ĝ(x) ˆF () = Ĝ() = 0 ˆF (b) = Ĝ(b) = 1 x [, b] : ˆF (x) Ĝ(x) (7.20) 35 b R b F (x)dx = R b G(x)dx b

137 ˆF (x) Ĝ(x) 1 [, b] u u [, b] 7.4 ( u (x) ) d ˆF ( (x) u (x) ) dĝ(x) u (x)d ˆF (x) (7.18) ( 1 ( ) b u(b) F (x)dx ) u(x)df (x) (7.19) 2 u(x)df (x) u (x)dĝ(x) ( 1 ( ) b u(b) G(x)dx u(x)dg(x) ) u(x)dg(x) 2 1 [, b] u u(x)df (x) u (x)d ˆF (x) u(x)g(x) 0 [, b] v v(x) = x u (x)dĝ(x) (7.21) v (t)dt v [, b] u = v (7.21) v (x)d ˆF (x) v (x)dĝ(x) ( v (x) ) d ˆF ( (x) v (x) ) dĝ(x) v v 7.4 x [, b] : ˆF (x) Ĝ(x) (7.22) ˆF, Ĝ (7.19) x [, b] x F (t)dt x G(t)dt G F F G

138 (2 ). [, b] X, Y F X (x), F Y (x) x [.b] x F X (t)dt x F Y (t)dt X Y 2 X second-order stochsticlly domintes Y F X F Y F () = G() = 0 F (b) = G(b) = 1 F, G x [, b] : ˆx [, b] : x ˆx F (t)dt F (t)dt < x ˆx G(t)dt G(t)dt x [, b] u (x) > 0 G F u(x)df (x) u(x)dg(x) > X, Y F X, F Y X 36 Z Y = X + Z X Z E[X + Z] = E[X] + E[Z] = E[X] Y X X Y Y X men-preserving spred Y X men-preserving spred Y X [, b] X, Y 2 1. X Y 2 2. X Z Y = X + Z X Y Z Y = X + Z E[Z] = 0 X Z Pr(X x, Z z) Pr(X x) Pr(Z z) X x Z z F X (x)f Z (z)

139 mx E[ u(w + xr) ] (7.23) x [0,w] w > 0 R [α, β] α β α < 0 < β u (0, ) 2 u > 0 u < 0 (7.23) 1 i 1 Y z Y [α + i, β + i] x z x (z x)(1 + i) + x(1 + Y ) = (1 + i)z + (Y i)x w := (1 + i)z, R := (Y i) x w + xr E [ u(w + xr) ] (7.23) w z R F F E [ u(w + xr) ] = β α u(w + xr)f (dr) = u(w + xβ) β α F (r) u(w + xr)dr r 1 x x u(w + xβ) = u (w xβ)β 2 x x β α F (r) r u(w + xr)dr = β α (F (r) ) x r u(w + xr) dr u 2 x x E[ u(w + xr) ] = β u(w + xβ) x = u(w + xβ) x = β α α β α u(w + xr)f (dr) x (F (r) r ) x u(w + xr) dr F (r) r ( ) u(w + xr) dr x x E[ u(w + xr) ] [ ] = E u(w + xr) = E [ u (w + xr)r ] x

140 x 2 E[ u(w + xr) ] [ ] 2 = E u(w + xr) = E [ u (w + xr)r 2] x E [ u(w + xr) ] x 2 (7.23) 1 λ, µ E [ u (w + xr)r ] = λ + µ (7.24) λx = 0 (7.25) µ(w x) = 0 (7.26) 7.7 x 1 1 λ = µ = 0 R R (7.23) x E[R] > 0 Pr(R < 0) > 0 x (0, w) R E[R] 0 x = 0 Pr(R 0) = 1 x = w E[R] 0 x = 0 E [ u(w + xr) ] x f(x) = E [ u(w + xr) ] (7.27) 1 f (x) = E [ u (w + xr)r ] f (x) x f (x) = E [ u (w + xr)r 2] < 0 (7.28) f (x) x u < 0 E[R] 0 f (0) = E [ u (w)r ] = u (w)e[r] 0 f (x) x x [0, w] f (x) 0 f(x) = E [ u(w + xr) ] [0, w] x = 0 Pr(R 0) = 1 x = w u > 0 R 0 1 r u (w + wr)r 0 E [ u (w + wr)r ] 0 (7.27) f(x) f (w) = E [ u (w + wr)r ] 0

141 141 (7.28) f (x) x x [0, w] f (x) 0 f(x) = E [ u(w + xr) ] [0, w] x = w E[R] > 0 Pr(R < 0) > 0 x (0, w) E[R] > 0 f (0) = u (w)e[r] > 0 Pr(R < 0) > 0 f (w) = E [ u (w + wr)r ] < 0 (7.28) f (x) x x (0, w) f (x ) = E [ u (w + x R)R ] = 0 (7.24) (7.25) (7.26) x = x, λ = µ = 0 1 x E[R] > 0 Pr(R < 0) > 0 E [ u (w + xr)r ] = 0 u w R x = φ(u, w, R) u w R φ(u, w, R) R ˆR E [ u (w + xr)r ] = 0 (7.29) E [ u (w + x ˆR) ˆR ] = 0 (7.30) φ(u, w, R), φ(u, w, ˆR) φ(u, w, R) φ(u, w, ˆR) ˆR R 1 ˆR R 2 ˆR 1 ˆR φ(u, w, ˆR) φ(u, w, R) ˆR 2 ˆR φ(u, w, ˆR) φ(u, w, R) x E [ u (w + xr)r ] E [ u (w + xr)r ] E [ u (w + xr)r ] 7.8. (7.29) (7.30) φ(u, w, R), φ(u, w, ˆR) 2 1. φ(u, w, R) φ(u, w, ˆR) ( 2. E [u ˆR) ] w + φ(u, w, R) ˆR 0

142 x = φ(u, w, R), ˆx = φ(u, w, ˆR) ˆx (7.30) E [ u (w + ˆx ˆR) ˆR ] = 0 (7.31) E [ u (w + x ˆR) ˆR ] x (7.31) x ˆx E [ u (w + x [ ˆR) ˆR] E u ] (w + ˆx ˆR) ˆR E [ u (w + x ˆR) ˆR] x = φ(u, w, R) ˆx = φ(u, w, ˆR) E [ u (w + x ˆR) ˆR] E [ u (w + x R)R ] R R ˆR 7.9. x = φ(u, w, R), ˆx = φ(u, w, ˆR) g g(r) = u (w + x r)r 1. g ˆR R 1 ˆx x 2. g ˆR R 2 ˆx x 7.9. g E [ u (w + x R)R ] = E[g(R)] = 0 (7.32) E [ u (w + x ˆR) ˆR] = E[g( ˆR)] (7.33) 1 g ˆR R E [ g( ˆR) ] E [ g(r) ] (7.32) (7.33) E [ u (w + x ˆR) ˆR] 0 (7.34) 7.8 ˆx x 2 g ˆR R E [ g( ˆR) ] E [ g(r) ] (7.32) (7.33) E [ u (w + x ˆR) ˆR] 0 (7.35) 7.8 ˆx x 7.4. u g g u u g u u R u

143 κ, γ, θ > 0 u u(x) = κx 1 2γ x2, u(x) = e θx g(r) = u (w + x r)r u u α < 0 < β [α, β] h g h z [α, β] h (z) 0 g z [α, 0) g(z) 0 z [0, β] g(z) G β α β α h(x)g(x)dx 0 G(z) = z α g(x)dx G(α) = 0 G(β) = 0 g(x)dx = 0 z [α, β] : G (z) = g(z) G g β [ ] β β h(x)g(x)dx = h(z)g(z) h (z)g(z)dz α α α = h(α)g(α) h(β)g(β) β α h (z)g(z)dz β = h (z)g(z)dz (7.36) α z [α, 0) G (z) 0 z [0, β] G (z) 0 G [α, 0) [0, β] z [α, 0) : G(z) G(α) = 0 z [0, β] : G(z) G(β) = 0 z [α, β] G(z) 0 z [α, β] h (z) 0 z [α, β] h (z)g(z) 0 β α h (z)g(z)dz 0 (7.36)

144 u v (0, ) 2 u > 0, v > 0 u < 0, v < 0 2 f z (0, ) : v(z) = f ( u(z) ) (7.37) R, w φ(u, w, R) φ(v, w, R) R w R f R (z) φ(v, w, R) v 1 E [ v (w + φ(v, w, R)R)R ] = 0 E [ v (w + φ(u, w, R)R)R ] 0 E [ v (w + φ(u, w, R)R)R ] E [ v (w + φ(v, w, R)R)R ] E [ v (w + xr)r ] x φ(v, w, R) φ(u, w, R) E [ v (w + φ(u, w, R)R)R ] 0 x = φ(u, w, R) 2 f v(z) = f ( u(z) ) v(w + xr) = f ( u(w + xr) ) x x = x v (w + x R)R = f ( u(w + x R) ) u (w + x R)R (7.38) E [ v (w + x R)R ] = E [ f ( u(w + x R) ) u (w + x R)R] = β α f ( u(w + x z) ) u (w + x z)zf R (z)dz (7.39) h(z) = f ( u(w + x z) ) (7.40) g(z) = u (w + x z)zf R (z) (7.41) h u > 0, v > 0 (7.38) f ( u(w + x z) ) = v (w + x R) u (w + x R) > 0 h(z) > 0 f f 0 u > 0, x (0, w) h (z) = f ( u(w + x z) ) u (w + x z)x 0 g u > 0, f R 0 z [α, 0) g(z) 0 z [0, β] g(z) 0 x u 1 E [ u (w + x R)R ] = β α u (w + x z)zf R (z)dz = 0

145 145 β α g(z)dz = β α u (w + x z)zf R (z)dx = 0 (7.40) (7.41) h(z), g(z) 7.10 β α h(z)g(z)dz = β α f ( u(w + x z) ) u (w + x z)zf R (z)dz 0 (7.39) E [ v (w + x R)R ] 0 u R u R w φ(u, w, R) r A (z) = u (z) u (z) r A (z) z 37 MWG 6.C 7.5 Job Serch w t = 0, 1, 2,... t t t + 1 [0, w] f t t w θ (1 θ) w t + 1 θ w u(w) 0 u(0) = 0 u u > 0 u < 0 δ (0, 1) δ t E[u(W t )] t=0 37 r A (z) z u decresing bsolute risk version: DARA

146 Policy Function Vlue Function y {, } w Policy Function φ(w) {, } w Vlue Function v(w) v(w) { v(w) = mx u(w) + δ { (1 θ)v(w) + θv(0) } w }, δ v(z)f(z)dz 0 (7.42) v A A = δ w 0 v(z)f(z)dz (7.43) v(w) 0 (7.42) { v(0) = mx u(0) + δ { (1 θ)v(0) + θv(0) }, δ { } = mx δv(0), A = A w 0 } v(z)f(z)dz w = 0 A u(w) (1 θ)v(w) + θv(0) θ w = 0 Vlue v(0) u(w) (1 θ)v(w) + θv(0) w w v(w) w v w w w w w v (w) = u (w) + δ {(1 θ) v (w) + θv (0)} = A (7.44) w w w w [0, w ) : w [w, w] : u (w) + δ {(1 θ) v (w) + θv (0)} < A u (w) + δ {(1 θ) v (w) + θv (0)} A w w w < w A w A v w < w, (w) = ( ) 1 u (w) + δθa w w. 1 δ(1 θ) (7.45)

147 147 v w (7.44) v (w ) = u (w ) + δ {(1 θ) v (w ) + θv (0)} = A u (w ) + δ {(1 θ) v (w ) + θv(0)} = A v (w ) = A v (0) = A (7.45) v (w ) = 1 1 δ (1 θ) = 1 1 δ (1 θ) = A u (w ) = (1 δ) A (7.46) ( ) u (w ) + δθa ( ) (1 δ) A + δθa v w = w v u (w) + δ {(1 θ) v (w) + θa} A (7.45) v (7.42) v δ t v (w t ) t 0 v Vlue Function v Policy Function φ w < w, φ (w) = w w. w reservtion price reservtion wge A w A (7.43) (7.45) w A = δ v (z) f (z) dz 0 [ w ] w = δ v (z) f (z) dz + v (z) f (z) dz 0 w [ w w u (z) + δθa = δ Af (z) dz + f (z) dz w 1 δ (1 θ) = δ = δ [ [ 0 w A 0 AF (w ) + w f (z) dz + w u (z) 1 δ (1 θ) f (z) dz + δθa 1 w u (z) f (z) dz + 1 δ (1 θ) w ] 1 δ (1 θ) w w f (z) dz δθa ( 1 F (w ) )] (7.47) 1 δ (1 θ) ] F (w) = w 0 f (z) dz ( w w ) f (z) dz = 1 F (w) (7.47) δ w u (z) f (z) dz = 1 δ (1 θ) w ( 1 δf (w ) = 1 + θδ δf (w ) (1 δ)a 1 δ(1 θ) δ 2 θ ( 1 F (w ) )) A 1 δ (1 θ)

148 148 (7.46) A δ w 1 + δθ δf (w u (z) f (z) dz = (1 δ) A ) w = u (w ) (7.48) f, F, u δ θ w 7.4. w = 1 [0, 1] z [0, 1] f (z) = 1 F (z) = z u (w) = w w w u (z) f (z) dz = = zdz w (1 (w ) 2) (7.48) δ 1 ( 1 + δθ δw 1 (w ) 2) = w 2 w = 1 δ ( θδ + ) 2θδ δ 2 + θ 2 δ