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Randolph Griffin
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6 N := {1, 2,...} N 0 := {0} N R ++ := (0, ) R + := [0, ) a, b R a b := max{a, b} f g (f g)(x) := f(x) g(x) (Z, Z ) bz Z Z R f := sup z Z f(z) κ: Z R ++ κ f : Z R f(z) f κ := sup z Z κ(z).

7 f κ < f κ b κ Z Z R Z κ (b κ Z, κ ) P (Z, Z ) P : Z Z [0, 1] z P (z, B) Z B Z B P (z, B) z Z P (z, B) z Z B Z n N P n (z, B) := P (z, B)P n 1 (z, dz ) z B Z n Z h : Z R (P n h)(z) :=: E z h(z n ) := h(z )P n (z, dz ) n N 0, P 0 h := h P h := P 1 h P P h h bz Z R m Z f : Z Z R + Z f(z z)dz = 1 z Z P f P (z, B) = B f(z z) dz z Z B Z (Z n ) n 0 (Ω, F, P) (Z, Z ) P {F n } n 0 F (Z n ) n 0 P z Z 0 = z E z τ N 0 {F n } n 0 P{τ < } = 1 {τ n} F n n 0 τ = n n M {F n } n 0 Z R m (Ω, F ) Ω = n=0z F σ Z

8 t 0 Z t r(z t ) c(z t ) t+1 Z t+1 r : Z R c: Z R β (0, 1) v z Z { τ 1 } v (z) := sup E z β t c(z t ) + β τ r(z τ ). τ M t=0 τ M σ Z {0, 1} 0 1 σ τ := inf{t 0 σ(z t ) = 1} z Z ψ (z) := c(z) + β v (z )P (z, dz ). Z g : Z R + n N 0 a 1,, a 4, m, d R + βm < 1 z Z r(z ) P n (z, dz ) a 1 g(z) + a 2, c(z ) P n (z, dz ) a 3 g(z) + a 4, g(z )P (z, dz ) mg(z) + d. n = 0

9 E z r(z n ) E z c(z n ) g E z g(z t ) E z g(z t ) r c n := 0 g := r c m := 1 d := 0 v { } v (z) = max r(z), c(z) + β v (z )P (z, dz ) = max {r(z), ψ (z)}. ψ ψ (z) = c(z) + β max {r(z ), ψ (z )} P (z, dz ). Q Qψ(z) = c(z) + β max{r(z ), ψ(z )}P (z, dz ). Q ψ Q m d l: Z R ( n 1 ) n 1 l(z) := m E z r(z t ) + E z c(z t ) + g(z) + d, t=1 t=0 Q (b l Z, l ) Q b l Z ψ σ (z) = 1{r(z) ψ (z)} n = 0 l(z) = g(z) + d n = 1 l(z) = m c(z) + g(z) + d n n > n n 1 N 0 n 2 N 0 g

10 w t c 0 w t = w(z t ) (Z t ) t 0 r(z) = u(w(z))/(1 β) z u β c 0 := u( c 0 ) (Z t ) t 0 Z := R Z t+1 = ρz t + b + ε t+1, (ε t ) t 1 N(0, σ 2 ), ρ [ 1, 1]. { } u(w(z )) Qψ(z) = c 0 + β max 1 β, ψ(z ) f(z z) dz, f(z z) = N(ρz + b, σ 2 ) w(z) := e z u(w) = w1 δ 1 δ ( δ 0 δ 1) u(w) = ln w ( δ = 1). {ε t } δ 0 δ 1 ρ [0, 1) n N 0 β e ρnξ < 1 ξ := ξ 1 + ξ 2 ξ 1 := (1 δ)b ξ 2 := (1 δ) 2 σ 2 /2 e (1 δ)z P n (z, dz ) = b n e ρn (1 δ)z, b n := e ξ t 1 1 i=0 ρi +ξ t 1 2 i=0 ρ2i. {K j } Z = j=1 K j Γ(K j ) K j+1 Γ : Z 2 Z {Z t } Γ X N(µ, σ 2 ) E e sx = e sµ+s2 σ 2 /2 s R ( Z t Z 0 = z N b t 1 i=0 ρi, σ 2 t 1 i=0 ) ρ2i

11 g(z) = e ρn (1 δ)z m = d = e ρnξ, a 1 = b n (1 β)(1 δ), a 2 = a 3 = 0, a 4 = c 0. r(z) = e (1 δ)z /((1 β)(1 δ)) r(z ) P n (z, dz ) = b n e ρn (1 δ)z 1 (1 β)(1 δ) = a 1g(z) + a 2, g(z )P (z, dz ) = e ρn+1 (1 δ)z e ρn ξ 1 +ρ 2n ξ 2 (g(z) + 1) e ρnξ = mg(z) + d. ρ = 1 δ = 1 ρ [ 1, 0] c r r c r c (w t ) t 0 ln w t = ξ + ε t, (ε t ) t 0 N(0, γ ε ). ξ ξ N(µ, γ) f(w µ, γ) = LN(µ, γ+γ ε ) w ξ w N(µ, γ ) γ = 1/ (1/γ + 1/γ ε ) µ = γ (µ/γ + ln w /γ ε ).

12 h h(z )P (z, dz ) = h(w, µ, γ )f(w µ, γ) dw, µ γ { } u(w ) Qψ(µ, γ) = c 0 + β max 1 β, ψ(µ, γ ) f(w µ, γ) dw. δ 0 δ 1 n := 1 g(µ, γ) := e (1 δ)µ+(1 δ)2 γ/2 m := 1 d := 0 w 1 δ f(w µ, γ) dw = e (1 δ)2 γ ε/2 e (1 δ)µ+(1 δ)2 γ/2. r(w) := w 1 δ /((1 δ)(1 β)) c c 0 E µ,γ g(µ, γ ) := g(µ, γ )f(w µ, γ) dw = g(µ, γ) = mg(µ, γ) + d. l Q (b l Y, l ) ψ Y := R R ++ δ = 1 (µ, γ) w (w, µ, γ) r(w )f(w µ, γ) dw = e (1 δ)2 γ ε/2 e (1 δ)µ+(1 δ)2γ/2 /((1 β)(1 δ)) a 1 := e (1 δ)2 γ ε/2 /((1 β)(1 δ)) a 2 := 0 a 3 := 0 a 4 := c 0

13 Z := R p t = p(z t ) = e Zt γ > 0 β = e γ K r(z) = (p(z) K) + c 0 Qψ(z) = e γ max{(p(z ) K) +, ψ(z )}f(z z) dz. ρ ( 1, 1) ξ := b + σ 2 /2 n N 0 e γ+ ρn ξ < 1 g(z) := e ρnz + e ρnz m := d := e ρn ξ e γ+ξ < 1 n = 0 ρ [ 1, 1] Z t Z := R + r(z t ) = Z t c 0 > 0 θ > 0 F (z z) := P(Z t+1 z Z t = z) = 1 e θ(z z) (z z). Qψ(z) = c 0 + β max{z, ψ(z )} df (z z). n := 0 g(z) := z m := 1 d := 1/θ a t a t = a(z t ) = e Zt Z t Z := R c f > 0 q(a, l) = al α α (0, 1) l

14 p w r(z) = c(z) = Ga(z) 1 1 α cf G = (αp/w) 1 1 α (1 α)w/α ) } Qψ(z) = (Ga(z) 1 1 α cf + β max {Ga(z ) 1 1 α cf, ψ(z ) f(z z) dz. ρ [0, 1) n N 0 βe ρn ξ < 1 ξ := b 1 α + σ2 2(1 α) 2 g(z) := e ρn z/(1 α) m := d := e ρnξ ρ = 1 P z h(z )P (z, dz ) Z h c r l z r(z ) P (z, dz ) z l(z )P (z, dz ) n = 0 P c r g z E z g(z 1 ) g z E z g(z 1 ) z E z r(z t ), E z c(z t ) t = 0,..., n ψ P r P f(z z) z c l z r(z ) f(z z) dz, l(z )f(z z) dz ψ n = 0 r(z) a 1 g(z) + a 2 r, g z E z g(z 1 ) z E z r(z 1 )

15 r c r c ψ P f c z f(z z) ψ ψ P f(z z) = N(ρz + b, σ 2 ) z c c 0 g z E z g(z 1 ) z E z r(z t ) t N l z E z l(z 1 ) n = 1 P c c 0 r g (µ, γ) E µ,γ r(w ), E µ,γ g(µ, γ ) E µ,γ r(w ) := r(w ) f(w µ, γ) dw ψ ψ r(z) e z + K z (e z + K)f(z z) dz z E z r(z 1 ) & n = 0 h : Z R z h(z ) df (z z) P r c g z E z g(z 1 ) z P (z, dz ) = z θe θ(z z) dz = z + 1/θ. [z, ) ψ ψ

16 r P c z max{r(z ), ψ(z )}P (z, dz ) ψ b l Z ψ ρ 0 f(z z) = N(ρz + b, σ 2 ) z r c ψ ρ 0 ψ ψ r(w) µ µ f(w µ, γ) = N(µ, γ + γ ε ) µ E µ,γ (r(w ) ψ(µ, γ )) µ ψ µ c c 0 ψ µ Z = Z 1 Z m R m z = (z 1,..., z m ) h : Z R D j i h(z) j h z i f D j i f(z z) := j f(z z)/ (z i ) j D i c(z) z int(z) i = 1,..., m z i := (z 1,..., z i 1, z i+1,..., z m ) z 0 Z δ > 0 B δ (z0) i := {z i Z i : z i z0 i < δ} B δ (z0) i P f i = 1,..., m Di 2 f(z z) (z, z ) int(z) Z

17 (z, z ) D i f(z z) z i D 2 i f(z z) = 0 z i (z, z i ) z 0 int(z) δ > 0 A Z z / A z i (z, z i 0 ) / B δ (z i 0) z i D 2 i f(z z) = 0 z 0 int(z) z z i (z, z i 0 ) k k(z )D i f(z z) dz < z int(z) k {r, l} i = 1,..., m ψ ψ z int(z) i = 1,..., m D i ψ (z) = D i c(z) + max{r(z ), ψ (z )}D i f(z z) dz. i = 1,..., m z D i c(z) int(z) k z k(z )D i f(z z) dz int(z) k {r, l} z D i ψ (z) int(z) i = 1,..., m h(z, a) := e a(ρz+b)+a2 σ 2 /2 / 2πσ 2 2 f(z z) z 2 f(z z) z dz = ρ 2 σ π e az f(z z) z h(z, a) exp = 0 z (z ) := z b±σ ρ } { [z (ρz+b+aσ 2 )] 2 ρz + ρ(ρz+b) 2σ 2 σ 2

18 z z z ψ ψ v ψ v ψ v z = 3 ψ Θ R k P θ, r θ c θ vθ ψ θ θ Θ γ = 0.04 K = 20 b = 0.2 σ = 1 ρ = ±0.65

19 n θ a iθ m θ d θ g θ θ n := sup n θ m := sup m θ d := sup d θ ā := 4 i=1 sup a iθ θ Θ θ Θ θ Θ θ Θ θ Θ βm < 1 n, d, ā < m d l : Z Θ R ( n 1 ) n 1 l(z, θ) := m E θ z r θ (Z t ) + E θ z c θ (Z t ) + g θ (z) + d, t=1 t=0 Q b l (Z Θ) (z, θ) ψθ (z) E θ z P θ (z, ) P θ (z, ) (z, θ) h(z )P θ (z, dz ) h : Z R (z, θ) c θ (z), r θ (z), l(z, θ), rθ (z ) P θ (z, dz ), l(z, θ)p θ (z, dz ) (z, θ) ψθ (z) Θ := [ 1, 1] A B C A, B R ++, R C R Θ θ = (ρ, σ, b, c 0 ) (θ, z) ψθ (z)

20 Z R m Z = X Y X R Y R m 1 (Z t ) t 0 {(X t, Y t )} t 0 (X t ) t 0 (Y t ) t 0 X Y (X t ) t 0 (Y t ) t 0 Y t (X t+1, Y t+1 ) X t P (z, dz ) (x, y ) y F y (x, y ) P (z, dz ) = P ((x, y), d(x, y )) = df y (x, y ) c : Y R r X y Y x X r(x, y) = c(y) + β v (x, y ) df y (x, y ) X t Y t X Y x : Y X x x r( x(y), y) = ψ (y) y Y { 1{x x(y)}, r x σ (x, y) = 1{x x(y)}, r x r x x θ θ Θ (y, θ) x θ (y)

21 Y y = (y 1,..., y m 1 ) h : Y R l : X Y R D i h(y) := h(y)/ y i D i l(x, y) := l(x, y)/ y i D x l(x, y) := l(x, y)/ x r int(z) x int(y) D i x(y) = D ir( x(y), y) D i ψ (y) D x r( x(y), y) y int(y) i = 1,..., m. (x, y) r(x, y) ψ (y) x y x(y) y i y i x r X x { } u(w) v (w, µ, γ) = max 1 β, c 0 + β v (w, µ, γ )f(w µ, γ) dw,

22 x := w R ++ =: X y := (µ, γ) R R ++ =: Y w : Y R w µ µ

23 Q β = 0.95 γ ε = 1.0 c 0 = 0.6 δ = 3, 4, 5, 6 (µ, γ) [ 10, 10] [10 4, 10] 200 µ 100 γ µ γ µ γ µ µ γ µ 178 f t R {f t } h f h(f) df < x t x t = ξ t +ε x t {ε x t } N(0, γ x ) ξ t ε x t 10 4 σ = 3, 4, 5, 6 w [10 4, 10] 50 µ γ f t < 0

24 ξ t = ρξ t 1 + ε ξ t, {ε ξ t} N(0, γ ξ ). y t+1 t y t = ξ t + ε y t {ε y t } N(0, γ y ) ξ N(µ, γ) y ξ y N(µ, γ ) γ = 1/ ( 1/γ + ρ 2 /(γ ξ + γ y ) ) µ = γ (µ/γ + ρy /(γ ξ + γ y )). u(x) = (1 e ax ) /a a > 0 ( ) r(f, µ, γ) := E µ,γ [u(x)] f = 1 e aµ+a2 (γ+γ x)/2 /a f c 0 x := f R =: X y := (µ, γ) R R ++ =: Y Qψ(µ, γ) = β max {E µ,γ [u(x )] f, ψ(µ, γ )} p(f, y µ, γ) d(f, y ), p(f, y µ, γ) = h(f )l(y µ, γ) l(y µ, γ) = N(ρµ, ρ 2 γ +γ ξ +γ y ) n := 1 g(µ, γ) := e µ+a2 γ/2 m := 1 d := 0 l f : Y R Q (b l Y, l ) ψ f(µ, γ) = E µ,γ [u(x)] ψ (µ, γ) ψ f ρ 0 ψ µ ρ {ξ t } β = 0.95 a = 0.2 γ x = 0.1 γ y = 0.05 h = LN(0, 0.01) ρ = 1 γ ξ = 0 (µ, γ) [ 2, 10] [10 4, 10] 100 {f t } R

25 { P f f(µ, γ) } µ γ µ γ 921 w t = η t + θ t ξ t, ln θ t = ρ ln θ t 1 + ln u t, ρ [ 1, 1]. {ξ t } h {η t } v η 1 v(η) dη < {u t } LN(0, γ u ) {ξ t } {η t } {θ t } 10 4 f [10 4, 10] µ γ

26 θ t ξ t η t Qψ(θ) = c 0 + β { } u(w ) max 1 β, ψ(θ ) f(θ θ)h(ξ )v(η ) d(θ, ξ, η ), w = η + θ ξ f(θ θ) = LN(ρ ln θ, γ u ) w R ++ =: X θ R ++ =: Y w δ δ = 1 ρ ( 1, 1) n N 0 βe ρ2n γ u < 1 g(θ) := θ ρn + θ ρn m := d := e ρ2n γ u ρ ( 1, 1) Q (b l Y, l ) ψ w(θ) = e (1 β)ψ (θ) ψ w w (θ) = (1 β) w(θ)ψ (θ). ρ 0 ψ w θ w w θ w θ θ w ψ (θ) [(1 β) w(θ)] 1 βe γu/2 < 1 ρ = 1 ρ = 1 δ 0 δ 1 β = 0.95 c 0 = 0.6 γ u = 10 4 v = LN(0, 10 6 ) h = LN(0, ) ρ [0, 1] 100

27 θ [10 4, 10] 200 θ ρ = 0 {θ t } LN(0, γ u ) θ θ θ ρ > 0 θ θ ρ ρ θ < 1 ρ θ > w π

28 ρ = 0.75 β = 0.95 c 0 = 0.6 γ u = 10 4 v = LN(0, 10 6 ) h = LN(0, ) (θ, w) [10 4, 10] 2 (θ, w) (200, 200) (200, 300) (200, 400) (300, 200) (300, 300) (300, 400) (400, 200) (400, 300) (400, 400) 10 4 t Z t c(z t ) t + 1 s(z t ) t + 1 α Z g : Z R + n N 0 a 1,, a 4 m, d R + βm < 1 z Z s(z ) P n (z, dz ) a 1 g(z) + a 2

29 c(z ) P n (z, dz ) a 3 g(z) + a 4 g(z )P (z, dz ) mg(z) + d v (z) r (z) z Z v r { } v (z) = max r (z), c(z) + β v (z )P (z, dz ), r (z) = s(z) + αβ v (z )P (z, dz ) + (1 α)β r (z )P (z, dz ). ψ := c + βp v v = r ψ ψ r ψ = c + βp (r ψ) r = s + αβp (r ψ) + (1 α)βp r. m d κ : Z R + n 1 κ(z) := m E z [ s(z t ) + c(z t ) ] + g(z) + d t=0 (b κ Z b κ Z, ρ κ ) ρ κ b κ Z b κ Z ρ κ ((ψ, r), (ψ, r )) = ψ ψ κ r r κ. (b κ Z b κ Z, ρ κ ) (b κ Z, κ ) L b κ Z b κ Z ( ) ( ) ψ c + βp (r ψ) L =. r s + αβp (r ψ) + (1 α)βp r m d κ

30 L (b κ Z b κ Z, ρ κ ) L b κ Z b κ Z h := (ψ, r ) b 1, b 2 R + z Z v (z) n 1 t=0 βt E z [ r(z t ) + c(z t ) ] + b 1 g(z) + b 2 ψ (z) n 1 t=1 βt E z r(z t ) + n 1 t=0 βt E z c(z t ) + b 1 g(z) + b 2 m 1 E z r(z n ) a 1 g(z) + a 2 E z c(z n ) a 3 g(z) + a 4 E z g(z 1 ) mg(z) + d z Z t 1 E z g(z t ) = E z [E z (g(z t ) F t 1 )] = E z ( E Zt 1 g(z 1 ) ) m E z g(z t 1 ) + d. t 0 E z g(z t ) m t g(z) + 1 mt d. 1 m t n E z r(z t ) = E z [E z ( r(z t ) F t n )] = E z ( E Zt n r(z n ) ) a 1 E z g(z t n ) + a 2.

31 t n ( E z r(z t ) a 1 m t n g(z) + 1 ) mt n 1 m d t n ( E z c(z t ) a 3 E z g(z t n ) + a 4 a 3 m t n g(z) + 1 ) mt n 1 m d + a 2. S(z) := t 1 βt E z [ r(z t ) + c(z t ) ] + a 4. n 1 S(z) β t E z [ r(z t ) + c(z t ) ] + a 1 + a 3 1 βm g(z) + (a 1 + a 3 )d + a 2 + a 4. (1 βm)(1 β) t=1 v r + c + S ψ c + S b 1 := a 1+a 3 1 βm b 2 := (a 1+a 3 )d+a 2 +a 4 (1 βm)(1 β) (X, X, ν) (Y, Y, u) p : Y X R x q : Y X R x q p Y X x q(y, x)u(dy) x p(y, x)u(dy) ( ) E z sup k 1 k 0 t=0 βt c(z t ) + β k r(z k ) < z Z k 1 sup β t c(z t ) + β k r(z k 0 k ) β t [ r(z t ) + c(z t ) ] t=0 t 0 P z z Z

32 d 1 := a 1 + a 3 d 2 := a 2 + a 4 βm < 1 m d m + d 1 m > 1 β(m + d 1 m ) < 1 d (d 2 m + d)/(m + d 1 m 1) Q Qψ Qϕ ψ, ϕ b l Z ψ ϕ Q0 b l Z Qψ Z ψ b l Z Q(ψ +al) Qψ +aβ(m+d 1 m )l a R + ψ b l Z (Q0)(z) c(z) r(z l(z) l(z) + β ) l(z) P (z, dz ) (1 + β)/m < z Z Q0 l < Qψ E z r(z t ) P (z, dz ) = E z r(z t+1 ) E z c(z t ) P (z, dz ) = E z c(z t+1 ). h(z) := n 1 t=1 E z r(z t ) + n 1 t=0 E z c(z t ) n n h(z )P (z, dz ) = E z r(z t ) + E z c(z t ). t=2 m d m+d 1 m > 1 (d 2 m +d+d )/(m+d 1 m ) d ( n ) n l(z )P (z, dz ) = m E z r(z t ) + E z c(z t ) t=2 t=2 t=1 ( n 1 ) n 1 m E z r(z t ) + E z c(z t ) t=1 t=1 + g(z )P (z, dz ) + d + (m + d 1 m )g(z) + d 2 m + d + d ( (m + d 1 m m ) m + d 1 m h(z) + g(z) + d 2m + d + d ) m + d 1 m (m + d 1 m )l(z). ψ b l Z a R + z Z Q(ψ + al)(z) = c(z) + β max { r(z ), ψ(z ) + al(z ) } P (z, dz ) c(z) + β max { r(z ), ψ(z ) } P (z, dz ) + aβ l(z )P (z, dz ) Qψ(z) + aβ(m + d 1 m )l(z).

33 ψ Q ψ b l Z ψ Q b l Z τ := inf{t 0 : v (Z t ) = r(z t )} b l cz b l Z l b l cz b l Z ψ Q(b l cz) b l cz ψ b l cz h(z) := max{r(z), ψ(z)} G R + h(z) r(z) + Gl(z) =: h(z) z h(z) ± h(z) (z m ) m 0 Z z m z Z ( h(z ) ± h(z )) P (z, dz ) lim inf ( h(z ) ± h(z )) P (z m, dz ). m lim m h(z )P (z m, dz ) = h(z )P (z, dz ) ( ) ± h(z )P (z, dz ) lim inf ± h(z )P (z m, dz ), m (a m ) m 0 (b m ) m 0 R lim a m lim inf (a m + b m ) = lim a m + lim inf b m m m m m lim sup h(z )P (z m, dz ) h(z )P (z, dz ) lim inf h(z )P (z m, dz ), m m z h(z )P (z, dz ) c Qψ b l cz Q(b l cz) b l cz ψ b l iz b l Z ψ Q(b l iz) b l iz µ(z) := max{r(z ), ψ (z )}f(z z) dz µ i (z) := max{r(z ), ψ (z )}D i f(z z) dz

34 i = 1,..., m P f D i f(z z) (z, z ) int(z) Z z 0 int(z) δ > 0 k {r, l} k(z ) sup D i f(z z) dz < z i B δ (z0 i ) D i µ(z) = µ i (z) z int(z) i = 1,..., m (z i = z i 0 ). z 0 int(z) {z n } int(z) z i n z i 0 zi n z i 0 z i n = z i 0 n N δ > 0 N N zn i B δ (z0 i ) n N z i = z0 i ξ i (z, z n, z 0 ) B δ (z0 i ) i (z, z n, z 0 ) := f(z z n ) f(z z 0 ) zn i z0 i = D i f(z z) z i =ξ i (z,z n,z 0 ) sup D i f(z z). z i B δ (z0 i ) ψ Gl G R + n N max{r(z ), ψ (z )} i (z, z n, z 0 ) ( r(z ) + Gl(z )) sup D i f(z z) z i B δ (z0 i ) ( r(z ) + Gl(z )) sup D i f(z z) dz < z i B δ (z0 i ) max{r(z ), ψ (z )} i (z, z n, z 0 ) max{r(z ), ψ (z )}D i f(z z 0 ) n µ(z n ) µ(z 0 ) z i n z i 0 = max{r(z ), ψ (z )} i (z, z n, z 0 ) dz max{r(z ), ψ (z )}D i f(z z 0 ) dz = µ i (z 0 ). D i µ(z 0 ) = µ i (z 0 ) z 0 int(z) δ > 0 A Z z / A z i (z, z i 0 ) / B δ(z i 0 ) z i = z i 0 sup D i f(z z) = D i f(z z) z z i B i δ (z0 i ) =z0 i +δ D if(z z) z i =z0 i δ =: hδ (z, z 0 ).

35 G R + z i = z i 0 sup D i f(z z) sup D i f(z z) 1(z A) + h δ (z, z 0 ) 1(z A c ) z i B δ (z0 i ) z A,z i B δ (z0 i ) ) G 1(z A) + ( D i f(z z) z i =z i0 +δ + D if(z z) z i =z i0 δ 1(z A c ). D i ψ (z) = D i c(z) + µ i (z) z int(z) D i ψ (z) = D i c(z) + µ i (z) int(z) D i c(z) ψ z µ i (z) int(z) ψ Gl G R + max{r(z ), ψ (z )}D i f(z z) ( r(z ) + Gl(z )) D i f(z z), z, z Z. z z [ r(z ) + Gl(z )] D i f(z z) dz z µ i (z) r(f, µ, γ ) ( ) 1/a + e a2 γ x/2 /a e aµ +a 2 γ /2 + f. e aµ +a 2 γ /2 P (z, dz ) = e aµ +a 2 γ /2 l(y µ, γ) dy = e aµ+a2γ/2. µ f := f h(f) df r(f, µ, γ ) ( ) P (z, dz ) (1/a + µ f ) + e a2 γ x/2 /a e aµ+a2γ/2. n := 1 g(µ, γ) := e aµ+a2 γ/2 m := 1 d := 0 P (µ, γ) (µ, γ) (µ, γ) E µ,γ r(z 1 ) g

36 g(µ, γ) = E µ,γ g(µ, γ ) l µ ρ 0 w = η + θξ ln w 1/w + w ln w P (z, dz ) (1/η + η )v(η ) dη + ξ h(ξ ) dξ θ f(θ θ) dθ = µ η + µ + η + µ ξ e γu/2 θ ρ, µ + η := ηv(η) dη µ η := η 1 v(η) dη µ ξ := ξh(ξ) dξ ( ln w P t (z, dz ) a (t) 1 θ ρt + a (t) 2 a (t) 1 θ ρt + θ ρt) + a (t) 2 a (t) 1 a(t) 2 > 0 θ t N n g m d θ ρn+1 + θ ρn+1 θ ρn + θ ρn + 1 θ > 0 ρ [ 1, 1] g(θ )f(θ θ) dθ = ( θ ρn+1 + θ ρn+1) e ρ2n γ u/2 mg(θ) + d. θ f(θ θ) θ (θ, θ ) f(θ θ)/ θ 2 f(θ θ)/ θ 2 = 0 : θ = θ (θ ) = ã i e ln θ /ρ, i = 1, 2 ã 1, ã 2 > 0 ρ > 0 (< 0) θ (θ ) (0) θ θ (θ ) 0 ( ) θ 0 f(θ θ) ρ > 0 X LN(µ, σ 2 ) E X s = e sµ+s2 σ 2 /2 s R

37 d 1 := a 1 + a 3 d 2 := a 2 + a 4 βm < 1 m, d > 0 m + d 1 m > 1 β(m + d 1 m ) < 1 d d 2m +d m+d 1 m 1 κ(z )P (z, dz ) (m + d 1 m )κ(z). z Z L: (b κ Z b κ Z, ρ κ ) (b κ Z b κ Z, ρ κ ) h := (ψ, r) b κ Z b κ Z p(z) := c(z) + β max{r(z ), ψ(z )}P (z, dz ) q(z) := s(z) + αβ max{r(z ), ψ(z )}P (z, dz ) + (1 α)β r(z )P (z, dz ) p q Z G R + z Z p(z) q(z) κ(z) c(z) s(z) κ(z) + βg κ(z )P (z, dz ) κ(z) 1 m + β(m + d 1m )G <. p b κ Z q b κ Z Lh b κ Z b κ Z L (b κ Z b κ Z, ρ κ ) h 1 := (ψ 1, r 1 ) h 2 := (ψ 2, r 2 ) b κ Z b κ Z ρ κ (Lh 1, Lh 2 ) = I J I := βp (r 1 ψ 1 ) βp (r 2 ψ 2 ) κ J := αβ[p (r 1 ψ 1 ) P (r 2 ψ 2 )] + (1 α)β(p r 1 P r 2 ) κ z Z P (r 1 ψ 1 )(z) P (r 2 ψ 2 )(z) r 1 ψ 1 r 2 ψ 2 (z )P (z, dz ) ( ψ 1 ψ 2 r 1 r 2 )(z )P (z, dz ) ρ κ (h 1, h 2 ) κ(z )P (z, dz ), a b a b a a b b I β(m + d 1 m )ρ κ (h 1, h 2 ) J β(m + d 1 m )ρ κ (h 1, h 2 ) L (b κ Z b κ Z, ρ κ ) β(m + d 1 m ) ρ κ (Lh 1, Lh 2 ) = I J β(m + d 1 m )ρ κ (h 1, h 2 ). ψ r h := (ψ, r ) L h b κ Z b κ Z max{ r (z), ψ (z) } β t E z [ s(z t ) + c(z t ) ], t=0

Optimal Timing of Decisions: A General Theory Based on Continuation Values 1

Optimal Timing of Decisions: A General Theory Based on Continuation Values 1 Qingyin Ma a and John Stachurski b a, b Research School of Economics, Australian National University April 17, 2017 ABSTRACT.