t

i = A, B t r >

(e it, e ijt k it i i π it = (α o e it + αe jit + θ it + vθ jt k it e it i e jit j i θ it t i i h(e it, e ijt = 1 2 (e2 it + e2 ijt k it i c(k it = 1 2 k2 it α o α v v > α α v i ϑ i Θ (ϑ, ϑ G i ϑ i t Z it (Z it t θ it t i θ it = ϕ i (t, ϑ i, Z it ϕ i : R + Θ R R i 1 2 ψ(k it + k jt 2 i j

(θ it i (θ it t θ it (θ is s<t Z it ϑ i θ i (θ it (θ it t θ i ϑ i G i g i Θ ψ i (ϑ i 1 G i(ϑ i g i (ϑ i ϑ i ϕ i ϑ i z i ϕ i ϑ (t, ϑ i, z i ϕi (t, ϑ i, z i ; ϕ i ϑ z(t, ϑ i, z i ϕi (t, ϑ i, z i z ( ϕ i ϑ (t, ϑ i, Z it t ϑ i θ it Z it (t, ϑ i, z i ϑ i ϕ i ϑ (t, ϑ i, z i θ it ϑ i Z it ϕ i z(t, ϑ i, z i > C > q (, r E [ ϕ i ϑ (t, ϑ i, Z it ] Ce qt t ϑ i Θ Z At Z Bt

Γ = (w it, e it, e ijt, k it t i=a,b i w it (e it, e ijt k it ϑ (θ s s<t w it (π s s<t (e it, e ijt, k it t i=a,b ( ˆϑ, (ˆθ s s<t (π s s<t [t, t + dt (θ At, θ Bt i θ it ˆθ it (k At, k Bt (e it, e ijt (π At, π Bt i w it Γ E ˆ e rt i=a,b (π it c(k it w it dt i ϑ i E ϑi [ˆ ] e rt (w it h(e it, e ijt dt i j i

Γ Γ E ˆ e rt i=a,b (π it c(k it h(e it, e ijt dt H 1 α 2 α 2 o > t i j k F B it (θ t = 1 H [θ it + vθ jt ] ; e F it B (θ t = α o H [θ it + vθ jt ] ; e F ijt B (θ t = α H [θ jt + vθ it ] i

1 > αo 2 + α 2 θ it Z it ϑ i θ i θ it + vθ jt

Γ i V Γ i (ϑ i E ϑi [ˆ ] e rt (w it h(e it, e ijt dt Γ i ϑ i Vi Γ(ϑ [ˆ ( i = E ϑi e rt ϕ i ϑ ϑ (t, ϑ eit (ϑ, (θ s s t i, Z it + ve ] ijt(ϑ, (θ s s t dt i α o α i ˆ ϑi Vi Γ (ϑ i = Vi Γ (ϑ + ϑ [ˆ E ϑi e rt ϕ i ϑ (t, ϑ i, Z it ( e it ( ϑ, (θ s s t + ve ijt( ϑ, ] (θ s s t dt d α o α ϑ i ϑ i ϑ ϑ i i ϑ i θ it θ it t θ it < θ it θ it e it α o ve ijt α

t ϑ i ϑ i ϑ i i ϑ i θ i = ϕ i (, ϑ i, Z i ˆθ i = ϕ i (, ϑ i, Z i ˆθ it = ϕ i (t, ϑ i, z(t, ϑ i, θ it θ it t > z θ it = ϕ i (t, ϑ i, z(t, ϑ i, θ it t ϑ i ϑ i z(t, ϑ i, θ it ϑ i Z it ϑ i z ϕ i z i

(k it, e it, e ijt t i=a,b Vi Γ (ϑ = Γ ˆ Θ 2 E ϑ ˆ e rt π it c(k it h(e it, e ijt ( eit α o + ve ijt α ϕ i ϑ (t, ϑ i, Z it 1 G i(ϑ i g i (ϑ i dt dg(ϑ i V Γ i (ϑ i k it (ϑ, θ t = 1 H ( (θit ϕ i ϑ (t, ϑ i, Z it ψ i (ϑ i ( + v θ jt ϕ j ϑ (t, ϑ j, Z jt ψ j (ϑ j e it (ϑ, θ t = α o H e ijt (ϑ, θ t = α H ((θ it 1 α2 α 2 o ( ϕ i ϑ (t, ϑ i, Z it ψ i (ϑ i + v θ jt ϕ j ϑ (t, ϑ j, Z jt ψ j (ϑ j ( ( v θ it 1 α2 o α 2 ϕ i ϑ (t, ϑ i, Z it ψ i (ϑ i Vi Γ (ϑ = i = A, B ( + θ jt ϕ j ϑ (t, ϑ j, Z jt ψ j (ϑ j t θ it t V Γ i (θ

ϕ i ϑ θ it + vθ jt > t ϑ t θ t t ϕi ϑ (t,ϑ i,z i ϕ i (t,ϑ i,z i z i (t, ϑ i, z i ϕ i ϑ (t,ϑ i,z i ϕ i z(t,ϑ i,z i ϑ i (t, ϑ i, z i C > i θ it Cϕ i ϑ (t, ϑ i, z(t, ϑ i, θ it ψ i (ϑ i C 1 α2 1 α2 α 2 o o α 2 ϑ i t θ it ϑ t θ it ϑ i

ϕ i (t, ϑ i, z i i t k it (ϑ, θ t e it (ϑ, θ t e ijt (ϑ, θ t ϑ θ t i j t i t ϑ i ˆ e rt ϕ i ϑ (t, ϑ i, Z it (k it (ϑ, θ t, e it (ϑ, θ t, e ijt (ϑ, θ t t ( e it ( ˆϑ, ˆθ t + ve ijt( ˆϑ, ˆθ t dt α o α ˆϑ i ˆθ it ( ˆϑ j, (ˆθ jt t j i=a,b i ˆϑ i ˆθ it e it ( ˆϑ, ˆθ e ijt ( ˆϑ, ˆθ t ˆϑ i ˆθ it (k it (ϑ, θ t, e it (ϑ, θ t, e ijt (ϑ, θ t t (w t (ϑ, (θ s s t ; (π s s t t i=a,b w it (ϑ, θ t, π t = B it (ϑ, θ t + S it (ϑ, θ t π it + S ijt (ϑ, θ t π jt i = A, B i B it (ϑ, θ t S it (ϑ, θ t S ijt (ϑ, θ t Γ

ϑ i θ it t ˆϑ (ˆθ s s<t t α o k it (ϑ, θ t S it (ϑ, θ t = e it (ϑ, θ t αk jt (ϑ, θ t S ijt (ϑ, θ t = e ijt (ϑ, θ t S it S ijt b it t b it i t t t t P i (ϑ ϕ i ϑ (t, ϑ i, Z it P i (ϑ P i (ϑ ˆθ t B it (ϑ, θ t b it (ϑ, θ t + β i (ϑ β i (ϑ = rp i (ϑ ˆϑ i ˆθ it i t θ it ϑ i i

c (θ it t dθ it θ it = µ i dt + σ i dz it, i = A, B µ i > σ i > Z it θ it = θ i ((µ i 12 σ it + σ i Z it

θ i ϕ i (t, ϑ i, Z it = θ i ( (µ i 1 2 σ it + σ i Z it ϑi ϑ i = θ i ϕ i (t, θ i, Z it θ i ϕ i θ (t, θ i, Z it = ((µ i 12 σ it + σ i Z it = θ it θ i θ i k it (θ, θ t = 1 H ( [θ it 1 ψ ( i(θ i + vθ jt 1 ψ ] j(θ j θ i θ j θ it > { } (θ A, θ B 1 ψ A(θ A θ A, 1 ψ B(θ B θ B, dk it (θ, θ t = (K i θ it + vk j θ jt dt + σk i θ it dz it + vk j θ jt dz jt K i 1 H ( 1 ψ i(θ i θ i θ i ψ i (θ i K i θ i (θ A, θ B

v K j > v k it (θ, θ t e it (θ, θ t e ijt (θ, θ t S it (θ, θ t = ( ( θ it 1 1 α2 ψ i (θ i α 2 θ i + vθ jt 1 ψ j(θ j θ j ( ( θ it 1 ψ i(θ i θ i + vθ jt 1 ψ j(θ j θ j v = S it (θ, θ t = 1 α 2 ψ 1 i (θ i α 2 θ i 1 ψ i (θ i θ i θ it v θ j > ψ j (θ j w it (ϑ, θ t = b it (ϑ, θ t + S ijt (ϑ, θ t (π it + π jt + (S it (ϑ, θ t S ijt (ϑ, θ t π it

S ijt (#,3 t S it (#,3 t -S ijt (#,3 t k it (#,3 t S it (#,3 t #1 4 9 8 7 6 5.9.85.8.75 4 3 2 1 1 2 3 4 5 6 7 8 time.7.65.6.55 1 2 3 4 5 6 7 8 time 1.9.8.5.4.3.7.6.5.4.3.2.2.1 -.1 -.2.1 1 2 3 4 5 6 7 8 time -.3 1 2 3 4 5 6 7 8 time θ i =.55 θ i =.6 i = A, B (, 1 Z it α o =.8 α =.55 v =.2 µ A = 18 µ B = 14 σ = 2 S ijt (S it S ijt µ A > µ B A B v

k it (#,3 t S it (#,3 t k it (#,3 t S it (#,3 t #1 4 2.5 1 2.98.96 1.5.94.92 1.9.5.88.86 1 2 3 4 5 6 7 8 time #1 4 7.84 1 2 3 4 5 6 7 8 time 1 6.95 5 4 3 2.9.85.8 1.75 1 2 3 4 5 6 7 8 time.7 1 2 3 4 5 6 7 8 time v α v =.2 v =.5 v = 1 α =.5 α =.55 α =.58 α o =.8 µ A = µ B = 14 σ = 2 α

ϑ i = µ i ϕ i µ(t, µ i, Z it = θ it t θ it k it (µ, θ t = 1 H [θ it (1 ψ i (µ i t + vθ jt (1 ψ j (µ j t] θ it, θ jt > { } t = ga (µ A 1 G A (µ A, g B (µ B g 1 G B (µ B µ i i (µ i 1 G i (µ i µ i dk it (µ, θ t = ((K i µ i ψ i (µ i θ it + v(k j µ j ψ j (µ j θ jt dt + σ i K i θ it dz it + σ j vk j θ jt dz jt K i = 1 H (1 ψ i(µ i t µ i v = i t = g i(µ i 1 G(µ i = 1 ψ i (µ i t K i = k it (µ i, θ it ψ i (µ i θ it ( θ it 1 1 α2 ψ α S it (µ, θ t = 2 i (µ i t + vθ jt (1 ψ j (µ j t o θ it (1 ψ i (µ i t + vθ jt (1 ψ j (µ j t

v = S it (µ i, θ it = 1 α 2 1 α 2 ψ i (µ i t o 1 ψ i (µ i t θ it S it (µ, θ t (θ it t dθ it = µ i dt + σ i dz it, i = A, B µ i > σ i > θ i θ it = θ i + µ t t + σ i Z it ϕ i (t, ϑ i, Z it = θ i + µ t t + σ i Z it ϑ i = θ i ϑ i = µ i ϕ i θ (t, θ i, Z it = 1 ϕ i µ(t, µ i, Z it = t t θ it ϕ i θ (t, θ i, Z it = 1 k it (θ, θ t = 1 H [θ it ψ i (θ i + v (θ jt ψ j (θ j ]

dk it (θ, θ t = 1 H (µ i + vµ j dt + 1 H (σ idz it + vσ j dz jt k it > t {(θ A, θ B θ A + vθ B ψ A (θ A + vψ B (θ B } (θ A, θ B k it (θ, θ t kit F B (θ t ϕ i µ(t, µ i, Z it = t k it (θ, θ t = 1 H [θ it ψ i (θ i t + v (θ jt ψ j (θ j t] dk it (µ, θ t = 1 H (dθ it + vdθ jt 1 H [ψ i(µ i + vψ j (µ j ] dt k it > µ A µ B {(θ A, θ B θ A + vθ B (ψ A (µ A + vψ B (µ B t} t (θ A, θ B θ A θ B

θ it = e x it dx it = η i ( x i x t dt + σdz it x it η i > x i σ > Z it θ it dθ it = θ it ( η i ( x i θ it + 1 2 σ2 dt + σθ it dz it θ it = ( e ηit θ i + x ( 1 e η it + σe ηit B η i it B η i it = Z e 2η i t 1 2η i θ it ϕ i θ (t, θ i, B η i it = θ it e η it θ i [ ] ψa (µ A + ψ v B(µ B t θ A v t v B η i it dm t = e ηit dz it M M t = 1 2η i (e 2ηit 1 Z e 2η i t 1 M t 2η i

e η it θ it θ i e ηit θ it θ it θ i η i θ i k it (θ, θ t = 1 H [θ it ( 1 ψ i(θ i e η it θ i + vθ jt ( { θ it θ jt k it (θ, θ t = 1 ψ i(θ i 1 ψ ] j(θ j e η jt θ j } e η jt θ i e ηit, 1 ψ j(θ j θ j v = θ i t = 1 η i ψ i(θ i θ i t θ i η i η S it (ϑ, θ t S ijt (ϑ, θ t S it (θ, θ t = ( θ it 1 1 α2 α 2 ( θ it ψ i (θ i θ i 1 ψ i(θ i θ i ( e η it + vθ jt ( + vθ jt e η it 1 ψ j(θ j θ j 1 ψ j(θ j θ j e η jt e η jt S it (θ, θ t θ it+vθ jt θ it +vθ jt = 1 S ijt (θ, θ t S it S ijt

S ijt (#,3 t S it (#,3 t - S ijt (#,3 t k it (#,3 t S it (#,3 t 4 1 35 3.95 25 2 15.9.85 1 5.8 1 2 3 4 5 6 7 8 time.75 1 2 3 4 5 6 7 8 time 1.1.95.8.9.85.8.75.6.4.2.7.65 1 2 3 4 5 6 7 8 time -.2 1 2 3 4 5 6 7 8 time η =.1 η =.5 η =.25 (, 1 θ i =.6 i = A, B Z it x = 5 α o =.8 α =.5 v =.5 σ = 2

v i α t E [dk it (ϑ, θ t ] = k it (ϑ, θ t dt

i ϵ > t ϵ t > t ϵ ϕ i ϑ (t, ϑ i, z(t, ϑ i, θ i ψ i (ϑ i < ϵ (ϑ i, θ i Θ R (θ i, θ it t i i S it (ϑ, ϑ t S ijt (ϑ, θ t 1 t (θ it t S it (ϑ, θ t S ijt (ϑ, θ t E [ ϕ i ϑ(t, ϑ i, Z it ψ i (ϑ i ] = t

e η it j i x ϕ ī x(t, x i, B η i it = θ it(1 e η it θ it

( ˆϑ j, (ˆθ js s t j t i ϑ i ˆϑ i i α o e it ( ˆϑ, (ˆθ s s t + ˆθ it = α ê it + θ it αe ijt ( ˆϑ, (ˆθ s s t + vˆθ it = αê ijt + vθ it (e it ( ˆϑ, (ˆθ s s t, e ijt ( ˆϑ, (ˆθ s s t i α o e it ( ˆϑ, (ˆθ s s t + ˆθ it αe ijt ( ˆϑ, (ˆθ s s t + vˆθ it i j i i ê it =e it ( ˆϑ, (ˆθ s s t + 1 (ˆθit θ it α o ê ijt =e ijt ( ˆϑ, (ˆθ s s t + v (ˆθit θ it α i = A, B t ˆθ it = θ it t ( ˆϑ, (ˆθ s s t π it (( ˆϑ, (ˆθ s s t = (α o e it ( ˆϑ, (ˆθ s s t + e jit ( ˆϑ, (ˆθ s s t + ˆθ it + vˆθ jt k it ( ˆϑ, (ˆθ s s<t i U i (ϑ i ; ˆϑ i i ϑ i ˆϑ i ˆπ t = U i (ϑ i ; ˆϑ i = E ϑ i [ˆ ( ( ] e rt w it ˆϑ, (ˆθs s t, (ˆπ s s t h(ê it, ê ijt dt ( π At ( ˆϑ, (ˆθ s s t, π Bt ( ˆϑ, (ˆθ s s t θ it = ϕ i (t, ϑ i, Z it U i (ϑ i ; ˆϑ i

ϑ i U(ϑ i ; ϑ ˆϑ i = [ˆ E ϑi i ϑ i [ˆ =E ϑi ( ( e rt w it ˆϑ, (ˆθs s t, (ˆπ s s t h(ê it, ê ijt ] ] dt ( e rt h(ê it, ê ijt dt ϑ i [ˆ ( =E ϑi e rt ϕ i e it ( ϑ(t, ϑ i, Z it ˆϑ, (ˆθ s s t + ve ijt( ˆϑ, ] (ˆθ s s t dt α o α e i e ij ϑ i U(ϑ i ; ˆϑ i U i (ϑ i ; ˆϑ i ϑ i Vi Γ(ϑ i = ˆϑi U i (ϑ i ; ˆϑ i E ϑi [ˆ ] e rt w it dt = V Γ i (ϑ i + E ϑi [ˆ ] e rt h(e it, e ijt dt ˆ Θ 2 E ϑ [ˆ e rt i (π it c(k it h(e it, e ijt dt ] dg(ϑ i ˆ Θ V Γ i (ϑ i dg i (ϑ i ˆ Θ ˆ ϑ Vi Γ (ϑ i dg i (ϑ i =Vi Γ ( ϑ ϑ Vi Γ(ϑ ˆ ϑ i G i (ϑ i dϑ i = ϑ i ϑ V Γ i (ϑ i 1 G i (ϑ i ϑ i g i (ϑ i g i (ϑ i dϑ i + V Γ i (ϑ α o e it + αe ijt + θ it + vθ jt = k it α o k it = e it + 1 α o ϕ i ϑ(t, ϑ i, Z it ψ i (ϑ i αk jt = e ijt + v α ϕi ϑ(t, ϑ i, Z it ψ i (ϑ i i = A, B (k it (ϑ, θ t, e it (ϑ, θ t, e ijt (ϑ, θ t t ϑ t θ t

θ it t > w it (ϑ, θ t, π t = b it (ϑ, θ t + S it (ϑ, θ t π it + S ijt (ϑ, θ t π jt e it (ϑ,θ t α S it (ϑ, θ t = o k it (ϑ,θ t k it (ϑ, θ t > e ijt (ϑ,θ t αk ; S ijt (ϑ, θ t = jt (ϑ,θ t k jt (ϑ, θ t > b it (ϑ, θ t i t t π it (θ t ; ˆθ t = b it (ϑ, θ t =h (e it (ϑ, θ t, e ijt (ϑ, θ t S it (ϑ, θ t π it (θ t ; θ t S ijt (ϑ, θ t π jt (θ t ; θ t ˆ ( ˆθit e it (ϑ, + θ it, θ jt + ve ijt(ϑ, θ it, θ jt d θ it α o α θ (α o e it (ϑ, ˆθ t + αe ijt (ϑ, ˆθ t + θ it + vθ jt k it (ϑ, ˆθ t t ˆθ t θ t ϑ u it (θ it ; ˆθ it θ it i t ˆθ it u it (θ it ; ˆθ it = b it (ˆθ t + S it (ˆθ t π it (θ it, ˆθ jt ; ˆθ t + S ijt (ˆθ t π jt (θ it, ˆθ jt ; ˆθ t ˆθ jt j t j ˆθ jt i t > θ it, ˆθ it u it (θ it ; θ it u it (θ it ; ˆθ it j ˆ θit u it (θ it ; θ it u it (θ it ; θ it + ˆ ( θit e it (ˆθ t = ˆθ it = θ + ve ijt(ˆθ t α o α ( e it (ˆθ t + ve ijt(ˆθ t α o α [ 1 ( e it (ˆθ t e it ( θ it, α ˆθ jt + v ( e ijt (ˆθ t e ijt ( θ it, o α ˆθ jt ] d θ it ˆ ( ˆθit e it ( θ it, d θ it + ˆθ jt + ve ijt( θ it, ˆθ jt d θ it θ α o α (θ it ˆθ it + u it (ˆθ it ; ˆθ it =S it (ˆθ t (θ it ˆθ it k it (ˆθ it + vs ijt (ˆθ it (θ it ˆθ it + u it (ˆθ it ; ˆθ it =u it (θ it ; ˆθ it ˆθ it t

ϑ i β i (ϑ B it (ϑ, θ t b it (ϑ, θ t + β i (ϑ w it (ϑ, θ t, π t w it (ϑ, θ t, π t + β i (ϑ ˆ ϑi β i (ϑ = ϑ [ˆ E re rt ϕ i ϑ(t, ϑ i, Z it [ˆ (ˆ θit E re rt θ ( e it ( ϑ i, ϑ j, θ t + ve ijt( ϑ i, ϑ j, θ t α o α d θ it ( e it (ϑ, θ it, θ jt + ve ijt(ϑ, θ it, θ jt α o α u i (ϑ i ; ˆϑ i ϑ i ˆϑ i θ it i ϑ i Θ R u i : Θ Θ R u i(ϑ i, ˆϑ i ϑ i u i(ϑ i, ˆϑ i ϑ i ˆ ϑi u i ( P i (ϑ = ϑ i, ϑ d ϑ ϑ i u i (ϑ i, ϑ i i ϑ j ϑ dt dt ] ] d ϑ i w it θ t u i (ϑ i ; ˆϑ i u i (ϑ i ; ˆϑ i = E ˆ e rt w it ( ˆϑ i, ϑ j, θ t, π it h(e it ( ˆϑ i, ϑ j, θ t, e ijt ( ˆϑ i, ϑ j, θ t dt }{{} =u it(ϕ i (t,ϑ i,z it;θ it t > u it (ϕ i (t, ϑ i, Z it ; θ it u it θ it u it θ ( it = S it ( θ it ϑ ˆϑ i, ϑ j, θ t k it ( ˆϑ i, ϑ j, θ t + vs ijt ( ˆϑ i, ϑ j, θ t k jt ( ˆϑ i, ϑ j, θ t ϕ i ϑ(t, ϑ i, Z it i ( =ϕ i e it ( ϑ(t, ϑ i, Z it ˆϑ i, ϑ j, θ t + ve ijt( ˆϑ i, ϑ j, θ t α o α ϑ i θ it ϕ i t

ϑ i u i (ϑ i ; ˆϑ [ˆ ( i =E e rt ϕ i e it ( ϑ ϑ(t, ϑ i, Z it ˆϑ i, ϑ j, θ t + ve ijt( ˆϑ ] i, ϑ j, θ t dt i α o α ˆϑ i u i(ϑ i ; ˆϑ i ϑ i u it (ϕ i (t, ϑ i, Z it ; θ it = ( θ it eit (ϑ, θ it,θ jt θ α o + ve ijt(ϑ, θ it,θ jt α d θ it P i (ϑ = βi(ϑ r ϑ j β i (ϑ θ t i w it (ϑ, θ t ϑ, θ t ϑ i = ϑ θ it = θ t [ˆ ˆ θit w it = E ϑ re rt θ ( e it (ϑ, ϑ j, θ t + ve ijt(ϑ, ϑ j, θ ] t d θ it dt α o α c [ˆ ˆ θit c E ϑ re rt θ ( e it ( ϑ, ϑ j, θ t + ve ijt( ϑ, ϑ j, θ ] t d θ it dt α o α w it c v = ϕ i ϑ(t, ϑ i, z(t, ϑ i, θ it = θ it Φ i (t, ϑ i Φ i (t, ϑ i ϑ i [ Φ i t(t, ϑ i > < k it (ϑ i, θ it = θit H 1 Φ i (t, ϑ i ψ i (ϑ i ] dk it (ϑ i, θ it = µ(θ it H [ 1 Φ i (t, ϑ i ψ i (ϑ i ] dt + σ(θ it H [ 1 Φ i (t, ϑ i ψ i (ϑ i ] dz it µ(θ it σ(θ it θ it Φ i (t, ϑ i ϑ i ϑ i S it (ϑ, θ t

1 S it (ϑ, θ t = ϕi ϑ (t, ϑ i, Z it ψ i (ϑ i α o k it (ϑ, θ t k it (ϑ, θ t ϕ i ϑ (t, ϑ i, Z it S it (ϑ, θ t 1 1 S ijt (ϑ, θ t = ϕi ϑ (t, ϑ i, Z it ψ i (ϑ i αk jt (ϑ, θ t S ijt (ϑ, θ t 1