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! " " # $ % & &% &' %& % % %" % % % ()* +, -./ 01 1 2 3! 45 +, 01 4./ 61 178 44 9!4 44 :; < 78!:=:5 +, 0:=>4 :>/ ? ==./ % $ % @1=1! / A7:!:> 7 :; 7:!4 174B C D > / 617:== =/ ?8 !4:!/ E !/ 6 7 : 7:!:> 7 <8 : 5/ < := / <8 <8 : 5 :; 9!24 1= F 1! G14 :!" H$ % % % %I H& - J H " K / % & L / $ %/ % % %M %I % $ % 6 = :> 1!2 N:O 4 P M M " Q%/ / 12 = 74 :! %% RI $R/ %% R $ & $R/ >: 1= 81G1 2 S %% R % R R$T% %R/ R R/ !7 !4 7:>D14 O = 45 7:!4 1 !4 S R& R R $R/ P =14 :! D !7 D= 7:>> 4> !4" ½
! " t # # θ θc(t) $c(0) = c(0) = 0% c(t), c(t) > 0& ' θ > 0 θ > θ ( θ π (t, w)% t ' % w ' ) 0 $*& + "% , θ - $.& + " % θ - / 0" % , 1 2 2 P ≥ 0% B ≥ 0% % 3 ' $θ& $θ > θ& +
D > 0 ( 4 5 π # 6 u(x) = −e−ax - 0 $*& + "% - " $.& + " % 7 8 / 2
% $*& " 1
% x 8 s(x) % s(x) − c(e) $e ' & 2 / −x − s(x) + "% x 6 f (0, e) = 1 − p(e)% f (x, e) = p(e)g(x)% x > 0 7 p(e) ' % e% p ' , # $*& $08+&1 $.& - - 9 a% n " q1 < q2 < · · · < qn % # qi πi (a) -
I1% % In 8" %
" % u% , # u(I, a) = −e−r(I−a) .
$*& + " % u $.& + " % $:;<=>? .;:@& u $<& + " % " Iˆ1 +k% % Iˆn +k% k , # % u $Ii u & + , # V (I)−a% V > 0 V < 0 A
$*&'$<& 1 +
B %
% N 9 i xi % / " ai x2i /2 ai ≥ 0 $i = 1% % n& 8 N / x = i=1 xi $*& - " $.& + "% % " x + " " % % N N i fi(x) = αi x + βi% i=1 αi = 1 i=1 βi = 0 - αi βi CDEFG + "% ,% θH θL ( θL λ H θ% x # T % u(x, T ) = θv(x) − T %
v(x) =
1 − (1 − x)2 . 2
B " # # c > 0 $*& # + " % " %
θL λ 4 5 $.& % " % " " # pi " θi - # $<& + " + % " % , (F, p)% F ' ,% % p ' # # 8I + " 1
$?& - , 2 % $*&'$<&
CDEFG 4JKL MN*>OLKP*5 % " Q - 2 " 6 R % 9 " % ' "
H "% # P W ( 6 ( 6
v − θB P − W, v − θT P − W,
0 < θB < θT $7 % , W % & ( " λ ! " " c $*& - " (P, W ) - 2, # # CH 6 !
# # % # # % G $.& + " % % $<& + " %
% % , $ & $?& 8 # % " % " " A $λ% θB % θT % c&1 $;& + " 1
B 6 G B π > 0% ' 0 9 " p ∈ [0, 1] ! " C(p)% C(0) = 0% C(1) = ∞% C (0) = 0% C ' , # - $G B &% % /
(WG , WB ) ) 0 ! $*& $:;<=>?.;:@& $.& - 2 $*&
$<& + "% % Wi " # - $*& $.&
CS=PT:@LU=TG ' " B % x [0, 2]
, # FH (x) = x/2 B % x [0, 1]
, # FL (x) = x ! " c% % # " $U(w, c) = u(w) − c& + 6 u(w) = 1 − e−w ,
V $*& - % ,# $.& % $*&
' 9 % x% x1 > x0 = 0 -
x1 W pH pL % 0 < pL < pH < 1 B % " c% [0, x1 ] , #
F (c) ! " 9 % " # $w ≥ 0& $*& + "% c " p - $.& + c p% + , / % " % , c 2 % c [0, x1 ] $<& + "% $.& " % " % c + , - % - % x% , # % 6 x = θa - θ ∈ [0, 1]% a ≥ 0% 9 % [0, 1] ! " " c(a) = a2 /2
$*& - $ & " $.& - w(x) ' , # 7 # A $ & " $<& 2, # X $.& 1 ( " % /
-
√ , # I − a% I ' % a ' 9 " $a = 2& $a = 1& B " 6 q1 = 40 q2 = 100% % q2 3/4% % % q2 1/2 ) 4 $*& 2, % %
$.& 2, % "
% $:KO>*PPK>O& + " % / 6 $*& % θ e% θ " e
e $.& % θ e% θ " e
e ' % " a1 % a2 ! " C = 12 (c11 a21 + c22 a22 + 2c12 a2 a2 ) - " % 6 x1 = a1 + γa2 + ε1 x2 = a2 + ε2 7 ε1 ε2
σ12 σ22 # YJZJ , # % CE = µ − 2r σ2 % µ σ2 '
" H # 6 CE ≥ u¯ x1 + x2 - $*& - $.& 8
$<& + "% c12 = γ = 0 - $?& + "% ε1 $σ12 = ∞& - A c12 *>? γ 1 $;& + " % ε2 $σ22 = ∞& - A c12 *>? γ 1 $[& ! /
# % # θa θb $J \&% % % [0, 1] ! " # % 6 " " V% ] ^ # $ ^% ] V #& $*& 8 % /,, ;_ `=:@ $ " # , # # & $.& % %
#
- R -/ # bi = αi + βiθi % i = a, b " $2 % , #& $<& A / 1 X R 1 X /,, ;_ `=:@1 X # 1 A 0 '2 / 1 ' % 9 % " x% 6 x1 > x0 = 0 - x1 " p pH pL % 0 < pL < pH < 1 ! " % pL % [0, x1] , # F (c)% pH ) 7 " "
6 w ≥ 0 $*& + "% c % "
p 8 $.& 3 "% " c % p% + , % % " c 0" / 1
$<& + " % $.& " % , % c + ,
0 a b , # 6 U(a, b) = θa a + θb b H " % % a b 8# θa θb ,# [0, 1] × [0, 1] ! " # " c ) $*& - $.& 8 , # % / " + a , H a
#, % " " , A c " 8, d " $ , " d/2& B , % a y1 y2 -
a w < c + d b % a Y 6 a
" P1 P2% Y , , - % w " + " % # # + $c& x , , Q $M& A 6 % d W % % d % " % A Q + %
+ / % A M% c 2" c , e d1 + "% M " d $ % & $ " / &% d " R M /1 +" / , 1 0" , 1 A 1 + \ # # M 8 #, # ! # 6 " " % 1.9 a_ `=:@ # v " c 6 β=0 β = 1.9 σ=0 v = 6, c = 10 v = 9, c = 7 σ = 1.9 v = 9, c = 7 v = 10, c = 6
$*& - # 0 % β σ " 1 B v c " 1 $.& + "% ,# ' \ % 0" / 1 - (p0 , p1)
" \ # M + t = 0 \ M + t = 1/2 # σ ∈ [1, Σ] + t = 1 M \ q # # 6 V (q) = q ! " C(q) = q2 /(2σθ)%
θ ' % [0, 1] , # F (θ) = θ2 ! " # ασ% α < 1/3 $*& -
# $.& b % $ " % f -/ &1 $<& + "% 6 4M \ q¯ # p¯ 5 X / 1 -
# " % $ & - q¯%
$?& + "% %
q0 # 6 " % q0 # % + t = 1 # q ≥ q0 - # "
q0 - q0 %
$;& 2 $?& $<& ( α Σ $?& % $<&1 9 # # + #
# n + % # % 8# θi [0, 1] $*& 8 R -/ $.& - " # $<& 2 $*& $.& # # A $?& B $*&'$<& " " % n = 3
# % # θa θb $J \&% % % [0, 1] ! " # % 6 " " V% ] ^ # $ ^% ] V #& $*& 8 % /,, ;_ `=:@ $ " # , # # & $.& % %
#
- R -/ # bi = αi + βiθi % i = a, b " $2 % , #& $<& A / 1 X R 1 X /,, ;_ `=:@1 X # 1 A 0 '2 / 1 - % - % x% , # % 6 x = θa - θ ∈ [0, 1]% a ≥ 0% 9 % [0, 1] ! " c(a) = a2 /2 $*& - $ & " $.& - w(x) ' , # 7 # A $ & " $<& 2, # X $.& 1 ( " % / " 6 S g # S θH u(x) − r% x ' % r ' + g θH u(x) − r %
θL < θH H S uH % % g uL < uH ! " # # c% g π $*& 8
$.& 8 % , "
R # " "
, # r / # , # ( R ;_ `=:@% ;_ *>@; [R¯ −θ, R¯ +θ] H θ ,# , + , C % % % # $H 6 , max{R − r, −C}% min{r, R+ C}& 3 , θ θ1 π θ2 1 −π ! , ¯ % C % π / R $*& - % # % " , % θ - $.& ( r % , $<& - # % ,# $H 6 # % # % # % ,% % ,% # % #
B ' # $]& $^& ! #
# % # θ1 θ2 a_ *>@; θ1 [0, 1]% % θ2 [a, 1 + a]% a ∈ (0, 1) $*& + % # % "
$.& 3 "% # ,# - % % " # 2 $*& + a H a #, % " " , A c " 8, d " $ , " d/3& B , % a y1 y2 - a w < c + d b % a Y 6 a " P1 P2 % Y ,
, Y " # r - % w "
+ \ # # M 8 #, # ! # 6 " " % ! " 6 1, c= 3,
σ% 1 − σ%
σ ' # M% " σ2 9 % # 4, β % c= 2, 1 − β ! " # \ β 2 $*& - # $.& 8 % #% # 4
5% # 4 5 2 %
/,,
" \ # M + t = 0 \ M + t = 1/2 # σ ≥ 0 + t = 1 M \ # # θσ % θ ' % [0, A] ! " c ! " # σ2 $*& -
# $.& b % , # $ \ p1% % p0 & - p0 p1 $<& 0" =`@K=> 1 8 =`@K=> ' % - E{x − w(x)}% E{w(x) − C(a)}% C(a) ' " % a ∈ [0, 1]% C (a) > 0% C (a) > 0% C (0) = 0% C (1) = ∞ ( x 6 X a 0 1−a ) $*& - a∗ % % $.& + "% a " w(x) - 2
a∗
$<& + "%
% " w(x) ≥ 0 - a∗
' % % # 9 a ∈ (−∞, +∞) - x ∈ R $*& + " f (x, a)% $08+&% , # $& $.& + " f (x, a)% 08+% % F (x, a) a
# + q # θq − t% t ' % % θ = {θ% θ} % θ = θ π θ % θ 6 θ/θ < 1/(1 − π) ! " q2 /2 ( $*& - $.& - $<& + "% I % "
$M*P*>K;% YNh& 8 $?& 8 % $ &
" 6 \ # M 8# θb 1 + α + β x α 1 − x 8# # 0 y 1 + α 1 − y + t = 0 + t = 1 " " " #6 \ x% M y ! " # x2 y 2 t = 2 # $*& 8 x y $.& + "% ;_ `=:@ f -/ - x y $<& + "% # p% " % " - x y p - p%
$?& + "% \ M % % ;_ `=:@ - x y $;& + "% M \ % % ;_ `=:@ - x y $[& ( α ∈ [0, 1] β ∈ [0, 1] $.&'$?& α β % #
" % i 0 $]jkk& 8# # v = 2+18xβ c = 20−18yσ 7 β σ ' # \ M% " t = 1/2% v c % x y % [0, 1] ! # 6 β, σ = {0, 1} ! " # β σ 3 % q = {0, 1} $*& - β % σ% x% y $.& - # $<& + "% , # p0 , p1 - , # $?& + "% ,# - =`@K=> % $+ " % " f & q 2 " 6
π 1 − π " θg C(q, σ)% % θbC(q, σ)% θb > θg > 0 7 σ ' " # # 8# V (q) + "% Vq > 0% Cq > 0% Vqq < 0% Cqq > 0% Cσ < 0% Cqσ < 0 + # γ ∈ (0, 1) $*& - " % # # σ 1 $.& 8 # $<& + "% " - $?& + "% 0 1
# ,# # p(ω)% % ω c(β, σ, ω) < p(ω) < v(β, σ, ω).
7 β, σ ' # # √ √ $
" & E{v(β, σ, ω)−p(ω)} = b +(1−γ) 2β +γ 2σ % E{p(ω)−c(β, σ, ω)} = 0 √ √ s0 + (1 − δ) 2σ + δ 2β % b0 + s0 = w0 > 0 + γ, δ ∈ [0, 1] /,, % # " # /,, H /,, x = γ+δ 2 y = δ − γ + "% y ≥ 0 - # 6 $*& H # ,# $.& ! $ # p(ω)) $<& 8 $ -/ ;_ `=:@& $?& ! # 4 5 $;& ! # 4 5 $[& H 6
π% # ' 1 − π - π%
$O& ! " % " λ% # ' µ% 1 − λ − µ # p(ω) - λ µ%
% i 0 $]jkj% ]jjk& $ i% ]jjl% l& , # + t = 0 , w R
# % " ,# % yt = 2αt kt−1 % yt ' t = 1, 2% kt ' # t + % $α1 = 1 < α2 = 3& A " V 9 # 6 ∆ = 4/9 # + ] %
% " % k1 ≥ 0 y2 $*& - # # $.& 6 B , Pˆ B , % % # # + "% " , ( w 1 - A (B, Pˆ ) w 1 B I # % [0, 1]
$*& - R -/ % # # # $.& - R -/ % # # # $<& 2 # $*& $.& " # θ1 % θ2 % % θI 2 " $*& $.&
B ' # $]& $^& ! #
# % # θ1 θ2 a_ *>@; θ1 [0, 1]% % θ2 [a, 1 + a]% a ∈ (0, 1) $*& + % # % "
$.& 3 "% # ,# - % % " # 2 $*& + # # B # c > 0 B n % # θi ,# a_ *>@; # [0, ∞) , # F (θ) + # " # p% " p% # $*& - # p∗% # $.& X # # # p∗ #1 X ;_ `=:@ /,, 1 $<& 8 " # # $?& $*& / # 7 # 2 $ 7 % + A m % n% ]h'^k 0 % ]jj^& A m 2 " % " % " A" " % m , # 6 m " " $ & 6 " % % % / a % " / b < a B % 6
c b% % " d% c < d < b
$*& 0" , , # -/ B % %
$.& 0"
-/ B % %
$<& $*&'$.& " % " , H 6 m 2 0 $]joo& 0 ' $]jkk&% % 7 " " , #
$8 9 % ]jjk& % % # √ J \ √ 6 u(xA , xB ) = tAi xA + tBj xB % i = {L, H} j = {L, H} + ∆k = tkH − tkL > 0% k = A, B αij > 0 ! " C(xA , xB ) = cA xA + cB xB 8
+ B # # S t = 1 t = 1/2% S # σ ∈ [0, 1] %
2 t = 1 c = 1/2 t = 1 B % p , # F (v) = v + σ(v 2 − v) 2 # σ/24 $*& - 6 ;_`=:@
# ;_*>@; $.& + "% t = 0%
# t = 1 b p% " B 1 B B S 1 B
(1 − γ) : γ 1 $<& + "% (1 − γ) : γ 0" # #1 CcL;>?;LO*:@ *>? M@=P;G !
A B 6 (a) (b) 9 i
i t = 1(i = a, b) 2 12 q 2 + % A % B $, # A
uA (xa , xb ) = vA xb % , # B ' uB (xa , xb ) = vB xa ,& +/ % + "% t = 1/2 #% % 9 % vi = (3kiIi) , i = A, B % Ii ' # % ki ' $ vi 3k1 vi3 &q kA = α, kB = β t = 0 $ & $2 % " , & 1 3
i
$*& R - # IA , IB qa , qb t = 0 $ # & $.& ( " + , vA , vB -/ $p 6 A B qa , qb, TAB %
TAB A B % uA = (vA qb − TAB − 1 2 q ), uB = (vB qa + TAB − 12 qb2 )& 2 a $<& - + , vA , vB -/ $p 6 A B qa, qb % " " TAB = 0&q
uA + uB ;_`=:@ " $?& ! # " + " " t = 1% # t = 1/2 $ -/ # t = 1/2& $;& ! # + "% t = 1 $ -/ #
t = 1/2& + α/β → 0 ( 6 (a) (d) α/β = 1/2 $[& 2 $.& $<&% $?& $;&% " 1 W,, ;_`=:@1 a_*>@; /,, 1 + 1
H C / # E 2 c% / # v C E " W # %
/ # Q " % # $y = 1& -
" c > c b
% / # " " $x = 1& B # % / # v > v B / # # $x = 0&% #% ! # % ,# + "% v − c > 0% v − c > v − c + 2% p∗% c < c < p∗ < v < v y=0 y=1
x=0 v = v, c = c v = v, c = c
x=1 v = v, c = c v = v, c = c
$*& - # $.& + "% x y ,# - pxy % $<& + "% x " % y 0" 1
$?& + "% y " % x 0" 1 $ - % " & $;& b p% x% y 1 -
# , # $ # p∗& 2 / # %
% # ;_`=:@ $ % &
