Spectral methods for bivariate Markov processes
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Markov processes Bivariate Markov processes An example Spectral methods for bivariate Markov processes Manuel Dom´ınguez de la Iglesia Departamento de An´alisis Matem´atico, Universidad de Sevilla Hong Kong, May 30, 2013
Markov processes Bivariate Markov processes An example Outline 1Markov processes Preliminaries Spectral methods 2Bivariate Markov processes Preliminaries Spectral methods 3An example A quasi-birth-and-death process A variant of the Wright-Fisher model
Markov processes Bivariate Markov processes An example Outline 1Markov processes Preliminaries Spectral methods 2Bivariate Markov processes Preliminaries Spectral methods 3An example A quasi-birth-and-death process A variant of the Wright-Fisher model
Markov processes Bivariate Markov processes An example 1-D Markov processes AMarkov process with state space S ⊂ Ris a collection of random variables {Xt∈ S :t∈ T } indexed by time T(discrete or continuous) such that they have the Markov property: the future event only depends on the present, not on the past (no memory). Sdiscrete (Markov chains) The transition probabilities come in terms of a matrix P= p11 p12 ··· p21 p22 ··· . . .. . .... ,Pij (t)≡Pr(Xt=j|X0=i) Scontinuous (Markov processes) The probabilities are described in terms of a density p(t;x,y)≡∂ ∂yPr(Xt≤y|X0=x),x,y∈ S
Markov processes Bivariate Markov processes An example 1-D Markov processes AMarkov process with state space S ⊂ Ris a collection of random variables {Xt∈ S :t∈ T } indexed by time T(discrete or continuous) such that they have the Markov property: the future event only depends on the present, not on the past (no memory). Sdiscrete (Markov chains) The transition probabilities come in terms of a matrix P= p11 p12 ··· p21 p22 ··· . . .. . .... ,Pij (t)≡Pr(Xt=j|X0=i) Scontinuous (Markov processes) The probabilities are described in terms of a density p(t;x,y)≡∂ ∂yPr(Xt≤y|X0=x),x,y∈ S
Markov processes Bivariate Markov processes An example 1-D Markov processes AMarkov process with state space S ⊂ Ris a collection of random variables {Xt∈ S :t∈ T } indexed by time T(discrete or continuous) such that they have the Markov property: the future event only depends on the present, not on the past (no memory). Sdiscrete (Markov chains) The transition probabilities come in terms of a matrix P= p11 p12 ··· p21 p22 ··· . . .. . .... ,Pij (t)≡Pr(Xt=j|X0=i) Scontinuous (Markov processes) The probabilities are described in terms of a density p(t;x,y)≡∂ ∂yPr(Xt≤y|X0=x),x,y∈ S
Markov processes Bivariate Markov processes An example Three important cases 1Random walks:S={0,1,2,...},T={0,1,2,...}. P= b0a0 c1b1a1 c2b2a2 ......... ,bi≥0,ai,ci>0,ai+bi+ci= 1 2Birth-and-death processes:S={0,1,2,...},T= [0,∞). P′(t) = AP(t) y P′(t) = P(t)Awhere A= −λ0λ0 µ1−(λ1+µ1)λ1 µ2−(λ2+µ2)λ2 ......... , λi, µi>0 3Diffusion processes:S= (a,b)⊆R,T= [0,∞). ∂ ∂tp(t;x,y) = Ap(t;x,y) y ∂ ∂tp(t;x,y) = A∗p(t;x,y) donde A=1 2σ2(x)d2 dx2+τ(x)d dx
Markov processes Bivariate Markov processes An example Three important cases 1Random walks:S={0,1,2,...},T={0,1,2,...}. P= b0a0 c1b1a1 c2b2a2 ......... ,bi≥0,ai,ci>0,ai+bi+ci= 1 2Birth-and-death processes:S={0,1,2,...},T= [0,∞). P′(t) = AP(t) y P′(t) = P(t)Awhere A= −λ0λ0 µ1−(λ1+µ1)λ1 µ2−(λ2+µ2)λ2 ......... , λi, µi>0 3Diffusion processes:S= (a,b)⊆R,T= [0,∞). ∂ ∂tp(t;x,y) = Ap(t;x,y) y ∂ ∂tp(t;x,y) = A∗p(t;x,y) donde A=1 2σ2(x)d2 dx2+τ(x)d dx
Markov processes Bivariate Markov processes An example Three important cases 1Random walks:S={0,1,2,...},T={0,1,2,...}. P= b0a0 c1b1a1 c2b2a2 ......... ,bi≥0,ai,ci>0,ai+bi+ci= 1 2Birth-and-death processes:S={0,1,2,...},T= [0,∞). P′(t) = AP(t) y P′(t) = P(t)Awhere A= −λ0λ0 µ1−(λ1+µ1)λ1 µ2−(λ2+µ2)λ2 ......... , λi, µi>0 3Diffusion processes:S= (a,b)⊆R,T= [0,∞). ∂ ∂tp(t;x,y) = Ap(t;x,y) y ∂ ∂tp(t;x,y) = A∗p(t;x,y) donde A=1 2σ2(x)d2 dx2+τ(x)d dx
Markov processes Bivariate Markov processes An example Random walks ··· a0a1a2a3a4a5 c1c2c3c4c5c6 b0 b1b2b3b4b5 b2 0 1 2 3 4 5 b 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 T S b
Markov processes Bivariate Markov processes An example Random walks ··· a0a1a2a3a4a5 c1c2c3c4c5c6 b0 b1b2b3b4b5 a2 0 1 2 3 4 5 b 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 T S b
Markov processes Bivariate Markov processes An example Random walks ··· a0a1a2a3a4a5 c1c2c3c4c5c6 b0 b1b2b3b4b5 a3 0 1 2 3 4 5 b 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 T S b
Markov processes Bivariate Markov processes An example Random walks ··· a0a1a2a3a4a5 c1c2c3c4c5c6 b0 b1b2b3b4b5 c4 0 1 2 3 4 5 b 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 T S b
Markov processes Bivariate Markov processes An example Random walks ··· a0a1a2a3a4a5 c1c2c3c4c5c6 b0 b1b2b3b4b5 a3 0 1 2 3 4 5 b 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 T S b
Markov processes Bivariate Markov processes An example Random walks ··· a0a1a2a3a4a5 c1c2c3c4c5c6 b0 b1b2b3b4b5 a4 0 1 2 3 4 5 b 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 T S b
Markov processes Bivariate Markov processes An example Random walks ··· a0a1a2a3a4a5 c1c2c3c4c5c6 b0 b1b2b3b4b5 b5 0 1 2 3 4 5 b 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 T S b
Markov processes Bivariate Markov processes An example Birth-and-death processes ··· λ0λ1λ2λ3λ4λ5 µ1µ2µ3µ4µ5µ6 0 1 2 3 4 5 b T S t0 b
Markov processes Bivariate Markov processes An example Birth-and-death processes ··· λ0λ1λ2λ3λ4λ5 µ1µ2µ3µ4µ5µ6 λ0 0 1 2 3 4 5 b T S t0t1 b
Markov processes Bivariate Markov processes An example Birth-and-death processes ··· λ0λ1λ2λ3λ4λ5 µ1µ2µ3µ4µ5µ6 λ1 0 1 2 3 4 5 b T S t0t1t2 b
Markov processes Bivariate Markov processes An example Diffusion processes Ornstein-Uhlenbeck diffusion process:S=Rand σ2(x) = 1, τ(x) = −x It describes the velocity of a massive Brownian particle under the influence of friction. It is the only nontrivial process which is stationary, Gaussian and Markovian. 0 5 10 −2 −1 0 1 2 3 X0=0 0 5 10 −2 −1 0 1 2 3 X0=3 0 5 10 −4 −2 0 2 4 X0=−3 0 5 10 −5 0 5 10 15 X0=10
Markov processes Bivariate Markov processes An example Spectral methods Given a infinitesimal operator A, if we can find a measure ω(x) associated with A, and a set of orthogonal eigenfunctions f(i,x) such that Af(i,x) = λ(i,x)f(i,x), then it is possible to find spectral representations of Transition probabilities Pij (t) (discrete case) or densities p(t;x,y) (continuous case). Invariant measure or distribution π= (πj) (discrete case) with πj= lim t→∞ Pij (t) or ψ(y) (continuous case) with ψ(y) = lim t→∞ p(t;x,y).
Markov processes Bivariate Markov processes An example Spectral methods Given a infinitesimal operator A, if we can find a measure ω(x) associated with A, and a set of orthogonal eigenfunctions f(i,x) such that Af(i,x) = λ(i,x)f(i,x), then it is possible to find spectral representations of Transition probabilities Pij (t) (discrete case) or densities p(t;x,y) (continuous case). Invariant measure or distribution π= (πj) (discrete case) with πj= lim t→∞ Pij (t) or ψ(y) (continuous case) with ψ(y) = lim t→∞ p(t;x,y).
Markov processes Bivariate Markov processes An example Spectral methods Given a infinitesimal operator A, if we can find a measure ω(x) associated with A, and a set of orthogonal eigenfunctions f(i,x) such that Af(i,x) = λ(i,x)f(i,x), then it is possible to find spectral representations of Transition probabilities Pij (t) (discrete case) or densities p(t;x,y) (continuous case). Invariant measure or distribution π= (πj) (discrete case) with πj= lim t→∞ Pij (t) or ψ(y) (continuous case) with ψ(y) = lim t→∞ p(t;x,y).
Markov processes Bivariate Markov processes An example Random walks S=T={0,1,2,...}. Spectral theorem: there exists a measure ωassociated with Pwhich orthogonal polynomials (qn)nsatisfy Pq = b0a0 c1b1a1 ......... q0(x) q1(x) . . . =x q0(x) q1(x) . . . ,x∈[−1,1] Transition probabilities Pr(Xn=j|X0=i) = Pn ij =1 kqik2Z1 −1 xnqi(x)qj(x)dω(x) Invariant measure Non-null vector π= (π0, π1,...)≥0 such that πP=π⇒πi=a0a1· · · ai−1 c1c2···ci =1 kqik2 Examples: Jacobi polynomials (Legendre, Gegenbauer)
Markov processes Bivariate Markov processes An example Random walks S=T={0,1,2,...}. Spectral theorem: there exists a measure ωassociated with Pwhich orthogonal polynomials (qn)nsatisfy Pq = b0a0 c1b1a1 ......... q0(x) q1(x) . . . =x q0(x) q1(x) . . . ,x∈[−1,1] Transition probabilities Pr(Xn=j|X0=i) = Pn ij =1 kqik2Z1 −1 xnqi(x)qj(x)dω(x) Invariant measure Non-null vector π= (π0, π1,...)≥0 such that πP=π⇒πi=a0a1· · · ai−1 c1c2···ci =1 kqik2 Examples: Jacobi polynomials (Legendre, Gegenbauer)
Markov processes Bivariate Markov processes An example Random walks S=T={0,1,2,...}. Spectral theorem: there exists a measure ωassociated with Pwhich orthogonal polynomials (qn)nsatisfy Pq = b0a0 c1b1a1 ......... q0(x) q1(x) . . . =x q0(x) q1(x) . . . ,x∈[−1,1] Transition probabilities Pr(Xn=j|X0=i) = Pn ij =1 kqik2Z1 −1 xnqi(x)qj(x)dω(x) Invariant measure Non-null vector π= (π0, π1,...)≥0 such that πP=π⇒πi=a0a1· · · ai−1 c1c2···ci =1 kqik2 Examples: Jacobi polynomials (Legendre, Gegenbauer)
Markov processes Bivariate Markov processes An example Random walks S=T={0,1,2,...}. Spectral theorem: there exists a measure ωassociated with Pwhich orthogonal polynomials (qn)nsatisfy Pq = b0a0 c1b1a1 ......... q0(x) q1(x) . . . =x q0(x) q1(x) . . . ,x∈[−1,1] Transition probabilities Pr(Xn=j|X0=i) = Pn ij =1 kqik2Z1 −1 xnqi(x)qj(x)dω(x) Invariant measure Non-null vector π= (π0, π1,...)≥0 such that πP=π⇒πi=a0a1· · · ai−1 c1c2···ci =1 kqik2 Examples: Jacobi polynomials (Legendre, Gegenbauer)
Markov processes Bivariate Markov processes An example Birth-and-death processes S={0,1,2,...},T= [0,∞). Spectral theorem: there exists a measure ωassociated with Awhich orthogonal polynomials (qn)nsatisfy Aq= −λ0λ0 µ1−(λ1+µ1)λ1 ......... q0(x) q1(x) . . . =−x q0(x) q1(x) . . . Transition probabilities Pr(Xt=j|X0=i) = Pij (t) = 1 kqik2Z∞ 0 e−xtqi(x)qj(x)dω(x) Invariant measure Non-null vector π= (π0, π1,...)≥0 such that πA= 0 ⇒πi=λ0λ1· · · λi−1 µ1µ2···µi =1 kqik2 Examples: Laguerre, Hahn, Krawtchouk, Charlier polynomials
Markov processes Bivariate Markov processes An example Birth-and-death processes S={0,1,2,...},T= [0,∞). Spectral theorem: there exists a measure ωassociated with Awhich orthogonal polynomials (qn)nsatisfy Aq= −λ0λ0 µ1−(λ1+µ1)λ1 ......... q0(x) q1(x) . . . =−x q0(x) q1(x) . . . Transition probabilities Pr(Xt=j|X0=i) = Pij (t) = 1 kqik2Z∞ 0 e−xtqi(x)qj(x)dω(x) Invariant measure Non-null vector π= (π0, π1,...)≥0 such that πA= 0 ⇒πi=λ0λ1· · · λi−1 µ1µ2···µi =1 kqik2 Examples: Laguerre, Hahn, Krawtchouk, Charlier polynomials
Markov processes Bivariate Markov processes An example Outline 1Markov processes Preliminaries Spectral methods 2Bivariate Markov processes Preliminaries Spectral methods 3An example A quasi-birth-and-death process A variant of the Wright-Fisher model
Markov processes Bivariate Markov processes An example 2-D Markov processes Now we have a bivariate or 2-component Markov process of the form {(Xt,Yt) : t∈ T } indexed by a parameter set T(time) and with state space C=S × {1,2,...,N}, where S ⊂ R. The first component is the level while the second component is the phase. Now the transition probabilities can be written in terms of a matrix-valued function P(t;x,A), defined for every t∈ T ,x∈ S, and any Borel set Aof S, whose entry (i,j) gives Pij (t;x,A) = Pr{Xt∈A,Yt=j|X0=x,Y0=i}. Every entry must be nonnegative and P(t;x,A)eN≤eN,eN= (1,1,...,1)T The infinitesimal operator Ais now matrix-valued. Ideas behind: random evolutions (Griego-Hersh-Papanicolaou-Pinsky-Kurtz...60’s and 70’s).
Markov processes Bivariate Markov processes An example 2-D Markov processes Now we have a bivariate or 2-component Markov process of the form {(Xt,Yt) : t∈ T } indexed by a parameter set T(time) and with state space C=S × {1,2,...,N}, where S ⊂ R. The first component is the level while the second component is the phase. Now the transition probabilities can be written in terms of a matrix-valued function P(t;x,A), defined for every t∈ T ,x∈ S, and any Borel set Aof S, whose entry (i,j) gives Pij (t;x,A) = Pr{Xt∈A,Yt=j|X0=x,Y0=i}. Every entry must be nonnegative and P(t;x,A)eN≤eN,eN= (1,1,...,1)T The infinitesimal operator Ais now matrix-valued. Ideas behind: random evolutions (Griego-Hersh-Papanicolaou-Pinsky-Kurtz...60’s and 70’s).
Markov processes Bivariate Markov processes An example 2-D Markov processes Now we have a bivariate or 2-component Markov process of the form {(Xt,Yt) : t∈ T } indexed by a parameter set T(time) and with state space C=S × {1,2,...,N}, where S ⊂ R. The first component is the level while the second component is the phase. Now the transition probabilities can be written in terms of a matrix-valued function P(t;x,A), defined for every t∈ T ,x∈ S, and any Borel set Aof S, whose entry (i,j) gives Pij (t;x,A) = Pr{Xt∈A,Yt=j|X0=x,Y0=i}. Every entry must be nonnegative and P(t;x,A)eN≤eN,eN= (1,1,...,1)T The infinitesimal operator Ais now matrix-valued. Ideas behind: random evolutions (Griego-Hersh-Papanicolaou-Pinsky-Kurtz...60’s and 70’s).
Markov processes Bivariate Markov processes An example 2-D Markov processes Now we have a bivariate or 2-component Markov process of the form {(Xt,Yt) : t∈ T } indexed by a parameter set T(time) and with state space C=S × {1,2,...,N}, where S ⊂ R. The first component is the level while the second component is the phase. Now the transition probabilities can be written in terms of a matrix-valued function P(t;x,A), defined for every t∈ T ,x∈ S, and any Borel set Aof S, whose entry (i,j) gives Pij (t;x,A) = Pr{Xt∈A,Yt=j|X0=x,Y0=i}. Every entry must be nonnegative and P(t;x,A)eN≤eN,eN= (1,1,...,1)T The infinitesimal operator Ais now matrix-valued. Ideas behind: random evolutions (Griego-Hersh-Papanicolaou-Pinsky-Kurtz...60’s and 70’s).
Markov processes Bivariate Markov processes An example 2-D Markov processes Now we have a bivariate or 2-component Markov process of the form {(Xt,Yt) : t∈ T } indexed by a parameter set T(time) and with state space C=S × {1,2,...,N}, where S ⊂ R. The first component is the level while the second component is the phase. Now the transition probabilities can be written in terms of a matrix-valued function P(t;x,A), defined for every t∈ T ,x∈ S, and any Borel set Aof S, whose entry (i,j) gives Pij (t;x,A) = Pr{Xt∈A,Yt=j|X0=x,Y0=i}. Every entry must be nonnegative and P(t;x,A)eN≤eN,eN= (1,1,...,1)T The infinitesimal operator Ais now matrix-valued. Ideas behind: random evolutions (Griego-Hersh-Papanicolaou-Pinsky-Kurtz...60’s and 70’s).
Markov processes Bivariate Markov processes An example Discrete time quasi-birth-and-death processes Now we have C={0,1,2, . . .} × {1,2,...,N},T={0,1,2, . . .}and (Pii′)jj′= Pr(Xn+1 =i,Yn+1 =j|Xn=i′,Yn=j′) = 0 for |i−i′|>1 i.e. a N×Nblock tridiagonal transition probability matrix P= B0A0 C1B1A1 C2B2A2 ......... (An)ij ,(Bn)ij ,(Cn)ij ≥0,det(An),det(Cn)6= 0 X j (An)ij + (Bn)ij + (Cn)ij = 1,i= 1,...,N Similar for continuous time quasi-birth-and-death processes but now we have C={0,1,2, . . .} × {1,2,...,N},T= [0,+∞) and the transition probability matrix Asatisfies (An)ij ,(Bn)ij ,i6=j,(Cn)ij ≥0,(Bn)ii ≤0 X j (An)ij + (Bn)ij + (Cn)ij = 0,i= 1,...,N
Markov processes Bivariate Markov processes An example Discrete time quasi-birth-and-death processes Now we have C={0,1,2, . . .} × {1,2,...,N},T={0,1,2, . . .}and (Pii′)jj′= Pr(Xn+1 =i,Yn+1 =j|Xn=i′,Yn=j′) = 0 for |i−i′|>1 i.e. a N×Nblock tridiagonal transition probability matrix P= B0A0 C1B1A1 C2B2A2 ......... (An)ij ,(Bn)ij ,(Cn)ij ≥0,det(An),det(Cn)6= 0 X j (An)ij + (Bn)ij + (Cn)ij = 1,i= 1,...,N Similar for continuous time quasi-birth-and-death processes but now we have C={0,1,2, . . .} × {1,2,...,N},T= [0,+∞) and the transition probability matrix Asatisfies (An)ij ,(Bn)ij ,i6=j,(Cn)ij ≥0,(Bn)ii ≤0 X j (An)ij + (Bn)ij + (Cn)ij = 0,i= 1,...,N
Markov processes Bivariate Markov processes An example N= 4 phases ··· ··· ··· ··· 1 5 9 13 17 21 2 6 10 14 18 22 3 7 11 15 19 23 4 8 12 16 20 24 b
Markov processes Bivariate Markov processes An example N= 4 phases ··· ··· ··· ··· 1 5 9 13 17 21 2 6 10 14 18 22 3 7 11 15 19 23 4 8 12 16 20 24 b
Markov processes Bivariate Markov processes An example N= 4 phases ··· ··· ··· ··· 1 5 9 13 17 21 2 6 10 14 18 22 3 7 11 15 19 23 4 8 12 16 20 24 b
Markov processes Bivariate Markov processes An example N= 4 phases ··· ··· ··· ··· 1 5 9 13 17 21 2 6 10 14 18 22 3 7 11 15 19 23 4 8 12 16 20 24 b
Markov processes Bivariate Markov processes An example Switching diffusion processes We have C= (a,b)× {1,2,...,N},T= [0,∞). The transition probability density is now a matrix which entry (i,j) gives Pij (t;x,A) = Pr(Xt∈A,Yt=j|X0=x,Y0=i) for any t>0, x∈(a,b) and Aany Borel set. The infinitesimal operator Ais now a matrix-valued differential operator (Berman, 1994) A=1 2A(x)d2 dx2+B(x)d1 dx1+Q(x)d0 dx0 We have that A(x) and B(x) are diagonal matrices and Q(x) is the infinitesimal operator of a continuous time Markov chain, i.e. Qii (x)≤0,Qij (x)≥0,i6=j,Q(x)eN=0
Markov processes Bivariate Markov processes An example Switching diffusion processes We have C= (a,b)× {1,2,...,N},T= [0,∞). The transition probability density is now a matrix which entry (i,j) gives Pij (t;x,A) = Pr(Xt∈A,Yt=j|X0=x,Y0=i) for any t>0, x∈(a,b) and Aany Borel set. The infinitesimal operator Ais now a matrix-valued differential operator (Berman, 1994) A=1 2A(x)d2 dx2+B(x)d1 dx1+Q(x)d0 dx0 We have that A(x) and B(x) are diagonal matrices and Q(x) is the infinitesimal operator of a continuous time Markov chain, i.e. Qii (x)≤0,Qij (x)≥0,i6=j,Q(x)eN=0
Markov processes Bivariate Markov processes An example Switching diffusion processes We have C= (a,b)× {1,2,...,N},T= [0,∞). The transition probability density is now a matrix which entry (i,j) gives Pij (t;x,A) = Pr(Xt∈A,Yt=j|X0=x,Y0=i) for any t>0, x∈(a,b) and Aany Borel set. The infinitesimal operator Ais now a matrix-valued differential operator (Berman, 1994) A=1 2A(x)d2 dx2+B(x)d1 dx1+Q(x)d0 dx0 We have that A(x) and B(x) are diagonal matrices and Q(x) is the infinitesimal operator of a continuous time Markov chain, i.e. Qii (x)≤0,Qij (x)≥0,i6=j,Q(x)eN=0
Markov processes Bivariate Markov processes An example An illustrative example N= 3 phases and S=Rwith Aii (x) = i2,Bii (x) = −ix,i= 1,2,3. 0 0.5 1 1.5 2 2.5 3 3.5 −8 −6 −4 −2 0 2 4 T S Bivariate Ornstein−Uhlenbeck process 2
Markov processes Bivariate Markov processes An example An illustrative example N= 3 phases and S=Rwith Aii (x) = i2,Bii (x) = −ix,i= 1,2,3. 0 0.5 1 1.5 2 2.5 3 3.5 −8 −6 −4 −2 0 2 4 T S Bivariate Ornstein−Uhlenbeck process 2 1
Markov processes Bivariate Markov processes An example An illustrative example N= 3 phases and S=Rwith Aii (x) = i2,Bii (x) = −ix,i= 1,2,3. 0 0.5 1 1.5 2 2.5 3 3.5 −8 −6 −4 −2 0 2 4 T S Bivariate Ornstein−Uhlenbeck process 2 1 3
Markov processes Bivariate Markov processes An example An illustrative example N= 3 phases and S=Rwith Aii (x) = i2,Bii (x) = −ix,i= 1,2,3. 0 0.5 1 1.5 2 2.5 3 3.5 −8 −6 −4 −2 0 2 4 T S Bivariate Ornstein−Uhlenbeck process 2 1 3 2
Markov processes Bivariate Markov processes An example An illustrative example N= 3 phases and S=Rwith Aii (x) = i2,Bii (x) = −ix,i= 1,2,3. 0 0.5 1 1.5 2 2.5 3 3.5 −8 −6 −4 −2 0 2 4 T S Bivariate Ornstein−Uhlenbeck process 2 1 3 2 3
Markov processes Bivariate Markov processes An example Switching diffusion models C= (a,b)× {1,2,...,N},T= [0,∞) If there exists a weight matrix Wsymmetric w.r.t. Awhich matrix-valued orthogonal functions (Φn)nsatisfies AΦn(x) = 1 2A(x)Φ′′ n(x) + B(x)Φ′ n(x) + Q(x)Φn(x) = Φn(x)Γn Transition probability density matrix (MdI, 2012) P(t;x,y) = ∞ X n=0 Φn(x)eΓntΦ∗ n(y)W(y) Invariant distribution (MdI, 2012) ψ(y) = (ψ1(y), ψ2(y),...,ψN(y)) such that A∗ψ(y) = 0 ⇒ψ(y) = Zb a eT NW(x)eNdx−1 eT NW(y)
Markov processes Bivariate Markov processes An example Switching diffusion models C= (a,b)× {1,2,...,N},T= [0,∞) If there exists a weight matrix Wsymmetric w.r.t. Awhich matrix-valued orthogonal functions (Φn)nsatisfies AΦn(x) = 1 2A(x)Φ′′ n(x) + B(x)Φ′ n(x) + Q(x)Φn(x) = Φn(x)Γn Transition probability density matrix (MdI, 2012) P(t;x,y) = ∞ X n=0 Φn(x)eΓntΦ∗ n(y)W(y) Invariant distribution (MdI, 2012) ψ(y) = (ψ1(y), ψ2(y),...,ψN(y)) such that A∗ψ(y) = 0 ⇒ψ(y) = Zb a eT NW(x)eNdx−1 eT NW(y)
Markov processes Bivariate Markov processes An example Switching diffusion models C= (a,b)× {1,2,...,N},T= [0,∞) If there exists a weight matrix Wsymmetric w.r.t. Awhich matrix-valued orthogonal functions (Φn)nsatisfies AΦn(x) = 1 2A(x)Φ′′ n(x) + B(x)Φ′ n(x) + Q(x)Φn(x) = Φn(x)Γn Transition probability density matrix (MdI, 2012) P(t;x,y) = ∞ X n=0 Φn(x)eΓntΦ∗ n(y)W(y) Invariant distribution (MdI, 2012) ψ(y) = (ψ1(y), ψ2(y),...,ψN(y)) such that A∗ψ(y) = 0 ⇒ψ(y) = Zb a eT NW(x)eNdx−1 eT NW(y)
Markov processes Bivariate Markov processes An example Outline 1Markov processes Preliminaries Spectral methods 2Bivariate Markov processes Preliminaries Spectral methods 3An example A quasi-birth-and-death process A variant of the Wright-Fisher model
Markov processes Bivariate Markov processes An example An example coming from group representation Let N∈ {1,2,...},α, β > −1, 0 <k< β + 1 and Eij will denote the matrix with 1 at entry (i,j) and 0 otherwise. For x∈(0,1), we have a symmetric pair {W,A} (Gr¨unbaum-Pacharoni-Tirao, 2002) where W(x) = xα(1 −x)β N X i=1 β−k+i−1 i−1N+k−i−1 N−ixN−iEii A=1 2A(x)d2 dx2+B(x)d dx +Q(x)d0 dx0 A(x) = 2x(1 −x)I,B(x) = N X i=1 [α+ 1 + N−i−x(α+β+ 2 + N−i)]Eii Q(x) = N X i=2 µi(x)Ei,i−1− N X i=1 (λi(x) + µi(x))Eii + N−1 X i=1 λi(x)Ei,i+1, λi(x) = 1 1−x(N−i)(i+β−k), µi(x) = x 1−x(i−1)(N−i+k).
Markov processes Bivariate Markov processes An example An example coming from group representation Let N∈ {1,2,...},α, β > −1, 0 <k< β + 1 and Eij will denote the matrix with 1 at entry (i,j) and 0 otherwise. For x∈(0,1), we have a symmetric pair {W,A} (Gr¨unbaum-Pacharoni-Tirao, 2002) where W(x) = xα(1 −x)β N X i=1 β−k+i−1 i−1N+k−i−1 N−ixN−iEii A=1 2A(x)d2 dx2+B(x)d dx +Q(x)d0 dx0 A(x) = 2x(1 −x)I,B(x) = N X i=1 [α+ 1 + N−i−x(α+β+ 2 + N−i)]Eii Q(x) = N X i=2 µi(x)Ei,i−1− N X i=1 (λi(x) + µi(x))Eii + N−1 X i=1 λi(x)Ei,i+1, λi(x) = 1 1−x(N−i)(i+β−k), µi(x) = x 1−x(i−1)(N−i+k).
Markov processes Bivariate Markov processes An example An example coming from group representation Let N∈ {1,2,...},α, β > −1, 0 <k< β + 1 and Eij will denote the matrix with 1 at entry (i,j) and 0 otherwise. For x∈(0,1), we have a symmetric pair {W,A} (Gr¨unbaum-Pacharoni-Tirao, 2002) where W(x) = xα(1 −x)β N X i=1 β−k+i−1 i−1N+k−i−1 N−ixN−iEii A=1 2A(x)d2 dx2+B(x)d dx +Q(x)d0 dx0 A(x) = 2x(1 −x)I,B(x) = N X i=1 [α+ 1 + N−i−x(α+β+ 2 + N−i)]Eii Q(x) = N X i=2 µi(x)Ei,i−1− N X i=1 (λi(x) + µi(x))Eii + N−1 X i=1 λi(x)Ei,i+1, λi(x) = 1 1−x(N−i)(i+β−k), µi(x) = x 1−x(i−1)(N−i+k).
Markov processes Bivariate Markov processes An example The orthogonal eigenfunctions Φi(x) of Aare called matrix-valued spherical functions associated with the complex projective space. There are many structural formulas available studied in the last years (Gr¨unbaum-Pacharoni-Tirao-Rom´an-MdI). Bispectrality:Φi(x) satisfy a three-term recurrence relation xΦi(x) = AiΦi+1(x) + BiΦi(x) + CiΦi−1(x),i= 0,1,... whose Jacobi matrix describes a discrete-time quasi-birth-and-death process (Gr¨unbaum-MdI, 2008). It was recently connected with urn and Young diagram models (Gr¨unbaum-Pacharoni-Tirao, 2011). The infinitesimal operator Adescribes a nontrivial switching diffusion process from which we can give a description of the matrix-valued probability density P(t;x,y) and invariant distribution ψ(y) in terms of the eigenfunctions Φi(x) , among other properties (MdI, 2012).
Markov processes Bivariate Markov processes An example The orthogonal eigenfunctions Φi(x) of Aare called matrix-valued spherical functions associated with the complex projective space. There are many structural formulas available studied in the last years (Gr¨unbaum-Pacharoni-Tirao-Rom´an-MdI). Bispectrality:Φi(x) satisfy a three-term recurrence relation xΦi(x) = AiΦi+1(x) + BiΦi(x) + CiΦi−1(x),i= 0,1,... whose Jacobi matrix describes a discrete-time quasi-birth-and-death process (Gr¨unbaum-MdI, 2008). It was recently connected with urn and Young diagram models (Gr¨unbaum-Pacharoni-Tirao, 2011). The infinitesimal operator Adescribes a nontrivial switching diffusion process from which we can give a description of the matrix-valued probability density P(t;x,y) and invariant distribution ψ(y) in terms of the eigenfunctions Φi(x) , among other properties (MdI, 2012).
Markov processes Bivariate Markov processes An example The orthogonal eigenfunctions Φi(x) of Aare called matrix-valued spherical functions associated with the complex projective space. There are many structural formulas available studied in the last years (Gr¨unbaum-Pacharoni-Tirao-Rom´an-MdI). Bispectrality:Φi(x) satisfy a three-term recurrence relation xΦi(x) = AiΦi+1(x) + BiΦi(x) + CiΦi−1(x),i= 0,1,... whose Jacobi matrix describes a discrete-time quasi-birth-and-death process (Gr¨unbaum-MdI, 2008). It was recently connected with urn and Young diagram models (Gr¨unbaum-Pacharoni-Tirao, 2011). The infinitesimal operator Adescribes a nontrivial switching diffusion process from which we can give a description of the matrix-valued probability density P(t;x,y) and invariant distribution ψ(y) in terms of the eigenfunctions Φi(x) , among other properties (MdI, 2012).
Markov processes Bivariate Markov processes An example Particular case α=β= 0,k= 1/2 An= (2n+ 1)(n+ 2)2 2(2n+ 3)2(n+ 1) 0 2(n+ 2) (2n+ 5)(2n+ 3)2 n+ 3 2(2n+ 5) Bn= 1 2−4n2+ 8n−1 2(2n+ 1)2(2n+ 3)2 n+ 2 (2n+ 3)2(n+ 1) 2(n+ 1) (2n+ 1)(2n+ 3)2 1 2−1 (2n+ 3)2 Cn= n2(2n+ 3) 2(2n+ 1)2(n+ 1) n (n+ 1)(2n+ 1)2 0n 2(2n+ 1)
Markov processes Bivariate Markov processes An example Particular case α=β= 0,k= 1/2 Pentadiagonal Jacobi matrix: P= 5 9 2 9 2 9 2 9 7 18 4 45 3 10 5 36 1 18 107 225 3 50 27 100 1 6 4 75 23 50 6 175 2 7 14 75 2 75 597 1225 4 147 40 147 1 5 6 245 47 98 8 441 5 18 81 392 3 196 1955 3969 5 324 175 648 ...............
Markov processes Bivariate Markov processes An example Associated network ··· ··· .22 .27 .27 .14 .23 .20 .56 .39 .48 .49 .46 .48 .22 .05 .02.22 .06 .03 .09 .03 .02 .06 .03 .02 .30 .29 .28 .17 .20 .21 1 3 5 2 4 6
Markov processes Bivariate Markov processes An example The invariant measure Invariant measure The row vector π= (π0;π1;···) πn=1 kQnk2 e W1,1 ,1 kQnk2 e W2,2 ,··· ,1 kQnk2 e WN,N,n≥0 is an invariant measure of P Particular case N= 2, α=β= 0, k= 1/2: πn=2(n+ 1)3 (2n+ 3)(2n+ 1),(n+ 1)(n+ 2) 2n+ 3 ,n≥0 π=2 3,2 3;16 15,6 5;54 35,12 7;128 63 ,20 9;250 99 ,30 11;432 143,42 13;686 195,56 15;···
Markov processes Bivariate Markov processes An example The invariant measure Invariant measure The row vector π= (π0;π1;···) πn=1 kQnk2 e W1,1 ,1 kQnk2 e W2,2 ,··· ,1 kQnk2 e WN,N,n≥0 is an invariant measure of P Particular case N= 2, α=β= 0, k= 1/2: πn=2(n+ 1)3 (2n+ 3)(2n+ 1),(n+ 1)(n+ 2) 2n+ 3 ,n≥0 π=2 3,2 3;16 15,6 5;54 35,12 7;128 63 ,20 9;250 99 ,30 11;432 143,42 13;686 195,56 15;···
Markov processes Bivariate Markov processes An example A variant of the Wright-Fisher model The Wright-Fisher diffusion model involving only mutation effects considers a big population of constant size Mcomposed of two types Aand B. A 1+β 2 −−→ B,B 1+α 2 −−→ A, α, β > −1 As M→ ∞, this model can be described by a diffusion process whose state space is S= [0,1] with drift and diffusion coefficient τ(x) = α+ 1 −x(α+β+ 2), σ2(x) = 2x(1 −x), α, β > −1 The Nphases of our bivariate Markov process are variations of the Wright-Fisher model in the drift coefficients: Bii (x) = α+ 1+ N−i−x(α+β+2+ N−i),Aii (x) = 2x(1−x) Now there is an extra parameter k∈(0, β + 1) in Q(x), which measures how the process moves through all the phases.
Markov processes Bivariate Markov processes An example A variant of the Wright-Fisher model The Wright-Fisher diffusion model involving only mutation effects considers a big population of constant size Mcomposed of two types Aand B. A 1+β 2 −−→ B,B 1+α 2 −−→ A, α, β > −1 As M→ ∞, this model can be described by a diffusion process whose state space is S= [0,1] with drift and diffusion coefficient τ(x) = α+ 1 −x(α+β+ 2), σ2(x) = 2x(1 −x), α, β > −1 The Nphases of our bivariate Markov process are variations of the Wright-Fisher model in the drift coefficients: Bii (x) = α+ 1+ N−i−x(α+β+2+ N−i),Aii (x) = 2x(1−x) Now there is an extra parameter k∈(0, β + 1) in Q(x), which measures how the process moves through all the phases.
Markov processes Bivariate Markov processes An example A variant of the Wright-Fisher model The Wright-Fisher diffusion model involving only mutation effects considers a big population of constant size Mcomposed of two types Aand B. A 1+β 2 −−→ B,B 1+α 2 −−→ A, α, β > −1 As M→ ∞, this model can be described by a diffusion process whose state space is S= [0,1] with drift and diffusion coefficient τ(x) = α+ 1 −x(α+β+ 2), σ2(x) = 2x(1 −x), α, β > −1 The Nphases of our bivariate Markov process are variations of the Wright-Fisher model in the drift coefficients: Bii (x) = α+ 1+ N−i−x(α+β+2+ N−i),Aii (x) = 2x(1−x) Now there is an extra parameter k∈(0, β + 1) in Q(x), which measures how the process moves through all the phases.
Markov processes Bivariate Markov processes An example Waiting times and tendency Waiting times We have to take a look to the diagonal entries of Q(x): Qii (x) = −1 1−x[(N−i)(i+β−k) + x(i−1)(N−i+k)] If x→1−⇒all phases are instantaneous. If x→0+or k→0+⇒phase Nis absorbing. If k→β+ 1 ⇒phase 1 is absorbing. Tendency If k→β+ 1 ⇒Backward tendency Meaning: The parameter khelps the population of A’s to survive against the population of B’s. If k→0+⇒Forward tendency Meaning: Both populations Aand B’fight’ in the same conditions.
Markov processes Bivariate Markov processes An example Waiting times and tendency Waiting times We have to take a look to the diagonal entries of Q(x): Qii (x) = −1 1−x[(N−i)(i+β−k) + x(i−1)(N−i+k)] If x→1−⇒all phases are instantaneous. If x→0+or k→0+⇒phase Nis absorbing. If k→β+ 1 ⇒phase 1 is absorbing. Tendency If k→β+ 1 ⇒Backward tendency Meaning: The parameter khelps the population of A’s to survive against the population of B’s. If k→0+⇒Forward tendency Meaning: Both populations Aand B’fight’ in the same conditions.
Markov processes Bivariate Markov processes An example Invariant distribution The invariant distribution ψ(y) (α, β ≥0) comes from the study of lim t→∞ P(t;x,y) = ∞ X n=0 Φn(x)eΓntΦ∗ n(y)W(y) This should be independent of the initial state and phase. Therefore we should expect a row vector invariant distribution ψ(y) = (ψ1(y), ψ2(y),...,ψN(y)) with 0 ≤ψj(y)≤1 and N X j=1 Z1 0 ψj(y)dy = 1 Explicit formula (MdI, 2012) ⇒ψ(y) = Z1 0 eT NW(x)eNdx−1 eT NW(y) where eT= (1,1,...,1). In particular ψj(y) = yα+N−j(1 −y)βN−1 j−1α+β+N α(β+N)(k)N−j(β−k+1)j−1 (α+β−k+2)N−1
Markov processes Bivariate Markov processes An example Study of the invariant distribution