INTRO

Contex

• 1930

  • Norbert Wiener

  • Kiyosi Itô

  • Orthogonal Polynomials and Stochastic Processes

• 1950

  • Mark Kac

  • Einar Hille

  • et alia

  • Important connections between transition probability functions, birth-death processes, and discrete-time birth-death chains, all in terms of a spectral representation.

Main Goal

• Define the fundamental concepts to develop a methodology and thus carry out spectral analysis of switching diffusion processes.

• Coexistence of \(X(t)\in\mathbb{R}^r\) and \(Y(t)\in\mathbb{N}\) such that \(t\ge 0\).

• Applications in wireless communications, financial engineering, and large-scale system optimization, all under the influence of random environments.

Theoretical Framework Part I: DP’s + OP’s + MM’s [2]

Diffusion processes Part 1

• Definition: \({X_t:t\ge 0}|X_t\in\mathbb{S}\subset\mathbb{R}\) with probability space \((\Omega,\mathcal{F},\mathbb{P})\) homogeneous in time is a diffusion process if there exist \(b(x),\sigma^2(x)\) such that:

  • \(\begin{aligned}\mathbb{E}&\left[X_{t+s}-X_{s}|X_{s}=x\right]=tb(x)+o(t),\\ \mathbb{E}&\left[(X_{t+s}-X_{s})^2|X_{s}=x\right]=t\sigma^2(x)+o(t),\\ \mathbb{E}&\left[|X_{t+s}-X_{s}|^3|X_{s}=x\right]=o(t)\end{aligned}\)

• Transition operator of \({X_t:t\ge 0}\). Given \(p(t;x,dy)\) the probability distribution of \({X_t}\) and let \(f\in\mathbb{B}(\mathbb{S})\) then:

  • \((T_{t}f)(x)=\mathbb{E}_{x}\left[f(X_t)\right]=\int_{\mathbb{S}}f(y)p(t;x,dy),~\text{where }t>0,x\in\mathbb{S}.\)

Diffusion processes Part 2

• Infinitesimal operator of \({T_t:t>0}\). Let \(f\in\mathbb{B}(\mathbb{S})\) then:

• \((\mathcal{L}f)(x)=\lim_{t\downarrow 0}\frac{(T_tf)(x)-f(x)}{t}=b(x)f^{'}(x)+\frac{1}{2}\sigma^2(x)f^{''}(x)\)

• Important properties:

  • \(\frac{\partial}{\partial t}(T_tf)(x)=\mathcal{L}(T_tf),~t>0,~x\in\mathbb{S}\)
  • \(\mathcal{L}(T_tf)=T_t(\mathcal{L}f)\)
  • Kolmogorov Backward: \(\frac{\partial p(t;x,y)}{\partial t}=b(x)\frac{\partial p(t;x,y)}{\partial x}+\frac{1}{2}\sigma^2(x)\frac{\partial^2 p(t;x,y)}{\partial x^2}\)
  • Kolmogorov Forward: \(\frac{\partial p(t;x,y)}{\partial t}=-\frac{\partial}{\partial y}[b(y)p(t;x,y)]+\frac{1}{2}\frac{\partial^2}{\partial y^2}[\sigma^2(y)p(t;x,y)]\)

• Recurrence in terms of the entrance times to state \(y\) such that \(\tau_y=\inf\{t\ge0:X_t=y\}\) y \(\rho_{xy}=\mathbb{P}[\tau_y<\infty|X_0=x]\)

• Ergodicity: existence of an invariant distribution: \(\lim_{t\rightarrow\infty}p(t;x,y)=\pi(y),~\pi(y)dy=\int_{\mathbb{S}}\pi(x)p(t;x,dy)dx\)

Orthogonal polynomials and matrix measures

• Preliminary definitions of orthogonal polynomial theory:

  • Moments of a measure \(\psi\)
  • Hilbert space \(L^2_{\psi}(\mathbb{R})\)
  • Orthogonal sequences \((p_n)_{n\in\mathbb{N}}\): \(\left(p_n,p_m\right)_{\psi}=\int_{\mathbb{R}}p_n(x)p_m(x)d\psi(x)=d^2_n\delta_{nm}~,\)

• Jacobi polynomials

  • \(\left(Q_{n}^{(\alpha,\beta)}(x)|x\in [0,1]\right)_{n\in\mathbb{N}}\), sequence of polynomials that satisfy the following second order differential equation:
    • \(y(1-y)\left[Q_{n}^{(\alpha,\beta)}\right]^{''}(y)+\left[(1-y)(\alpha+1)-y(\beta+1)\right]\left[Q_{n}^{(\alpha,\beta)}\right]^{'}(y)+n(n+\alpha+\beta+1)Q_{n}^{(\alpha,\beta)}(y)=0.\)

• Preliminary definitions of matrix measure theory:

  • Matrix measure: \(\forall X\in\mathbb{B},~~\mu(X)\in\mathcal{M}_{N}~:~\forall x\in\mathbb{R}^{N}~~x^{*}\mu(X)x\ge 0.\)
  • Inner product: \(P,Q\in\mathbb{P}\) and \(\mu\) matrix measure \(\langle P,Q\rangle_{\mu}=\int_{\mathbb{R}}Q^*(x)d\mu(x)P(x)\in\mathbb{C}^{N\times N}.\)
  • Orthonormality: a set \(\{\psi_j\}_{j=1}^{n}\) in \(\mathcal{H}\) such that \(\langle \psi_j,\psi_k \rangle_{\mu}=\delta_{jk}\text{I}_{N\times N}\)

Theoretical Framework Part II: SDP [4]

Switching diffusion processes and important properties Part 1

• SDP
  • \((\Omega,\mathcal{F},\mathbb{P})\) with a given filtration \(\{\mathcal{F}_t\}\). \((X(t),Y(t))\) homogeneous Markov process in \(\mathbb{S}\times S=(a,b)\times\{1,\cdots,N\}\subset\mathbb{R}\times\mathbb{N}\) (usually \(\mathbb{S}\subset\mathbb{R}^r:r\in\mathbb{N}\))
  • \(Y(\cdot)\) jump process with a generator (transition intensity matrix) \(Q(x)\) which acts on vector-valued functions \(f(x,\cdot)\)
    • \(Q(x)f(x,\cdot)(i)=\sum_{j=1}^Nq_{ij}(x)f(x,j)\)
  • \(w(\cdot)\in\mathbb{R}^r\) a regular Brownian motion defined on \((\Omega,\mathcal{F},\{\mathcal{F}_t\},\mathbb{P})\)
  • \((X(t),Y(t))\) is an SDP if there exist \(b(\cdot,\cdot):\mathbb{S}\times S\rightarrow \mathbb{R}\) and \(\sigma(\cdot,\cdot):\mathbb{S}\times S\rightarrow \mathbb{R}^{r}\) such that:
    • \(\begin{aligned}&dX(t)=b(X(t),Y(t))dt+\sigma(X(t),Y(t))dw(t),\\&(X(0),Y(0))=(x,\alpha):\alpha\in S,~x\in\mathbb{S},\end{aligned}\)
    • \(\mathbb{P}\left[Y(t+\Delta)=j|Y(t)=i,X(s),Y(s),s\le t\right]=q_{ij}(X(t))\Delta+o(\Delta)~:~i\neq j\in S\)
    • \(\lim_{h\rightarrow 0}{\frac{1}{h}\left[P(h;x,\mathbb{S})-I_{N\times N}\right]} = Q(x)\)

Switching diffusion processes and important properties Part 2

• Transition operator and infinitesimal operator: let \(f(x)\in\mathbb{B}(\mathbb{S}^N)\) therefore

  • \(\underline{T_{t}f(x)=\mathbb{E}\left[f(X_t)|X_0=x\right]:=\int_{\mathbb{S}}P(t;x,y)f(y)dy}\)

  • There exist matrices \(A(x),B(x):x\in\mathbb{S}\)such that for \(\epsilon>0\):

    • \(\lim_{h\rightarrow 0}\frac{1}{h}\left[\int_{y:|y-x|\le\epsilon}(y-x)^2P(h;x,y)dy\right]=\frac{1}{2}A(x)=\frac{1}{2}\begin{bmatrix}\sigma^2(x,1)&0&0&\dots&0\\0&\sigma^2(x,2)&0&\dots&0\\\vdots&\ddots&\vdots&\ddots&\vdots\\0&\dots&0&\dots&\sigma^2(x,N) \end{bmatrix},\)

    • \(\lim_{h\rightarrow 0}\frac{1}{h}\left[\int_{y:|y-x|\le\epsilon}(y-x)P(h;x,y)dy\right]=B(x)=\begin{bmatrix}b(x,1)&0&0&\dots&0\\0&b(x,2)&0&\dots&0\\\vdots&\ddots&\vdots&\ddots&\vdots\\0&\dots&0&\dots&b(x,N) \end{bmatrix}.\)

  • \(\underline{(\mathcal{L}f)(x)=\lim_{h\rightarrow 0}\frac{1}{h}\left[T_hf(x)-f(x)\right]=\frac{1}{2}A(x)f''(x)+B(x)f'(x)+Q(x)f(x)}\)

Switching diffusion processes and important properties Part 3

• Important properties
  • There exists a unique solution to the equation
    • \(\begin{aligned}&dX(t)=b(X(t),Y(t))dt+\sigma(X(t),Y(t))dw(t),\\&(X(0),Y(0))=(x,\alpha):\alpha\in S,~x\in\mathbb{S},\end{aligned}\\\mathbb{P}\left[Y(t+\Delta)=j|Y(t)=i,X(s),Y(s),s\le t\right]=q_{ij}(X(t))\Delta+o(\Delta)~:~i\neq j\in S\)
  • The process is càdlàg.” (right-continuous with left limits)
• Example of an SDP
  • \(\begin{cases}dX(t)=b(X(t),Y(t))dt+\sigma(X(t),Y(t))dw(t): &t\in[0,+\infty),\\(X(0),Y(0))=(0,2):&(X(t),Y(t))\in\mathbb{R}\times\{1,2,3\}.\end{cases}\)
  • \(\sigma^2(x,i)=i^2,b(x,i)=-ix:(x,i)\in\mathbb{R}\times\{1,2,3\}.\)
  • \(Q(x)=\begin{bmatrix}-\frac{2+x^2}{1+x^2}&\frac{5+3x^2}{6(1+x^2)}&\frac{7+3x^2}{6(1+x^2)}\\ 1+\frac{x^2}{400}&-2-\frac{x^2}{100}&1+\frac{3x^2}{400}\\ \frac{1}{2}+\frac{1}{2}e^{-x^2}&\frac{1}{2}+\frac{1}{2}e^{-x^2}&-1-e^{-x^2}\end{bmatrix}.\)

Switching diffusion processes and important properties Part 4

Regularity of a switching diffusion process Part 1

• \(\beta_n=\beta_{n}^{x,\alpha}:=\inf\{t\ge 0: |X^{x,\alpha}(t)|\ge n\}\)
• \((X^{x,\alpha}(t),Y^{x,\alpha}(t))\) is a regular process if \(\beta_{\infty}=\lim_{n\rightarrow\infty}\beta_n=\infty~\text{a.s.}\)
• Important Properties:
  • \((X(t),Y(t))\) is a regular process if \(b(x,i),\sigma(x,i)\) are locally Lipschitz and satisfy linear growth conditions.
• Example of regularity
  • \(dX(t)=X(t)\left[b(Y(t))-a(Y(t))X^2(t)dt+\sigma(Y(t))dw(t)\right],\)
  • \(\begin{aligned}b(1)&=3,a(1)=2,\sigma(1)=1\\b(2)&=10,a(2)=1,\sigma(2)=-1,\end{aligned}\)
  • \(Q(x)=\begin{bmatrix}-3-\sin(x)\cos(x)&3+\sin(x)\cos(x)\\2&-2\end{bmatrix},\)

Regularity of a switching diffusion process Part 2

Continuity of a Switching Diffusion Process

• Continuity in probability and mean square \(Z(t)=(X(t),Y(t))\), it is enough for \(Z(t)\) to be regular.
  • \(\begin{aligned}&\mathbb{P}[|Z(t+\Delta)-Z(t)|\ge 0]\rightarrow 0,\text{ whenever }\Delta\rightarrow 0, \text{ y}\\&\mathbb{E}[|Z(t+\Delta)-Z(t)|^2]\rightarrow 0,\text{ whenever }\Delta\rightarrow 0.\end{aligned}\)
• Property of Feller and strong Feller property \(f(\cdot,\cdot)\) measurable, continuous and bounded:
  • Feller: \(T_t f(x,i)\) is continuous w.r.t \((x,i)\) and \(\lim_{t\downarrow 0}T_t f(x,i)=\mathbb{E}\left[f(X(0),Y(0))|(X(0),Y(0))=(x,i)\right]=f(x,i).\)
    • It is sufficient for \((X(t),Y(t))\) to be regular.
  • Strong Feller: \(T_t f(x,i)\) is continuous w.r.t \(f,t\)
    • The coefficients \(b, \sigma, q\) must be Hölder continuous, bounded, and \(Q(x)\) irreducible.

Recurrence Part 1

• \((X(t),Y(t))\) is recurrent if for any \(U=D\times J\subset \mathbb{R}^r\times S\), given \(\sigma_{U}^{x,\alpha}=\inf{t\ge 0:\left(X^{x,\alpha}(t),Y^{x,\alpha}(t)\right)\in U}\), then:

  • \(\mathbb{P}\left[\sigma_{U}^{x,\alpha}<\infty\right]=1~:~(x,\alpha)\in D^c\times S\)

• \((X(t),Y(t))\) is positive recurrent if \(\mathbb{E}\left[\sigma_{U}^{x,\alpha}\right]<\infty~:~(x,\alpha)\in D^c\times S\)

• Important Properties:

  • \((X(t),Y(t))\) will be recurrent if there exists a Lyapunov function \(V(\cdot,\cdot):D^c\times S\rightarrow\mathbb{R}^+\) such that:
    • \(\begin{aligned}&V_n:=\inf_{|x|\ge n, i\in S}V(x,i),~\lim_{n\rightarrow\infty}{V_n}=\infty, \\ &\mathcal{L}V(x,i)\le 0,\text{ for all }(x,i)\in D^c\times S.\end{aligned}\)
  • \((X(t),Y(t))\) will be positive recurrent if there exists a Lyapunov function \(V(\cdot,\cdot):D^c\times S\rightarrow\mathbb{R}^+\) and \(\gamma>0\) such that:
    • \(\mathcal{L}V(x,i)\le -\gamma,~\text{for all } (x,i)\in D^c\times S.\)

Recurrence Part 2

• Example of recurrence

  • \(\begin{cases}dX(t)=b(X(t),Y(t))dt+\sigma(X(t),Y(t))dw(t),\\\mathbb{P}\left[Y(t+\Delta)=j|Y(t)=i,X(s),Y(s),s\le t, t\ge 0\right]=q_{ij}(X(t))\Delta+o(\Delta):(X(t),Y(t))\in\mathbb{R}\times\{1,2,3\},\end{cases}\)

  • \(Q(x)=\begin{bmatrix}-\frac{2+|x|}{1+x^2}-\frac{1+3x^2}{2+x^2}&\frac{2+|x|}{1+x^2}&\frac{1+3x^2}{2+x^2}\\ 1-\frac{\sin(x)}{1+x^2}&\frac{\sin(x)}{1+x^2}-\frac{\cos(x)}{2+|x|}-3&2+\frac{\cos(x)}{2+|x|}\\ \frac{1+\cos^2(x)}{2+x^2}&2&-\frac{1+\cos^2(x)}{2+x^2}-2\end{bmatrix},\)

  • \(\begin{aligned}b(x,1)&=3x-1,b(x,2)=\frac{3}{2}x+1,b(x,3)=-x+\frac{x}{1+|x|}\\ \sigma(x,1)&=\sqrt{3+x^2},\sigma(x,1)=\sqrt{2-\sin(x)+2x^2},\sigma(x,3)=3+\sqrt{4+x^2}.\end{aligned}\)

  • \(\begin{cases}dX(t)&=(3X(t)-1)dt+\sqrt{3+X(t)^2}dw(t)\\dX(t)&=\left(\frac{3}{2}X(t)+1\right)dt+\sqrt{2-\sin(X(t))+2X(t)^2}dw(t)\\dX(t)&=\left(-X(t)+\frac{X(t)}{1+|X(t)|}\right)dt+\left(3+\sqrt{4+X(t)^2}\right)dw(t)\end{cases}\)

Recurrence Part 3

Ergodicity

• Under the condition of positive recurrence, the process \((X(t),Y(t))\) is ergodic

  • There exists a distribution \(\hat{\nu}(dx,i)\) such that for any continuous \(f(\cdot,\cdot)\):

    • \(\sum_{i=1}^N\int_{\mathbb{R}^r}\mathbb{E}_{(x,i)}\left[f(X(t),Y(t))\hat{\nu}(dx,i)\right]=\sum_{i=1}^N\int_{\mathbb{R}^r}f(x,i)\hat{\nu}(dx,i).\)

  • There exists an stationary distribution \(\nu(A,i)=\frac{\hat{\nu}(A,i)}{\sum_{j=1}^N\hat{\nu}(\mathbb{R}^r,j)}, ~i\in S\)

  • Let \(\mu(\cdot,\cdot)\) be the density function of \(\nu(\cdot,\cdot)\) thus whenever \(T\rightarrow\infty\):

    • \(\frac{1}{T}\int_0^T\mathbb{E}_{(x,i)}\left[f(X(t),Y(t))\right]dt\rightarrow \sum_{i=1}^{N}\int_{\mathbb{R}^r}f(x,i)\mu(x,i)dx\)

Example of spectral analysis of SDP [1] [3]

Wright-Fisher diffusion model Part I.

• Population of fixed size \(N\) composed of two types: \(A\) and \(B\).

• The mutation rate from \(A\) to \(B\) is \(\frac{1+\beta}{2}\), where \(\beta>-1\).

• The mutation rate from \(B\) to \(A\) is \(\frac{1+\alpha}{2}\), where \(\alpha>-1\).

• When \(N\rightarrow\infty\), \(X(t)\in[0,1]\), the proportion of the population of type \(A\) is modeled as a DP with coefficients:

  • \(b(x)=\alpha+1-x(\alpha+\beta+2),~\sigma^2(x)=2x(1-x),~\beta,\alpha>-1\)

• \(b(x),\sigma^2(x)\) correspond to the Jacobi diffusion coefficients:

  • The probability density of \(X(t)\) will be described through Jacobi polynomials on \([0,1]\).

• Simulating the Wright-Fisher model: the effects of mutation will be determined by a SDP with state space \([0,1]\times{1,\cdots,N}\).

• The parameter \(k\) is considered to measure the way in which the process moves between phases: \(0<k<\beta+1\).

Wright-Fisher diffusion model Part II.

• \(\begin{aligned}dX(t)&=b(X(t),Y(t))dt+\sigma(X(t))dw(t)\\ b(x,i)&=\alpha+1+N-i-x(\alpha+\beta+2+N-i)\\ \sigma^2(x)&=2x(1-x),~i=1,\cdots,N.\end{aligned}\)

• \(Q(x)\) generator of \(Y(\cdot)\) such that:

  • \(Q(x)=\sum_{i=2}^N\mu_i(x)E_{i,i-1}-\sum_{i=1}^N(\lambda_i(x)+\mu_i(x))E_{ii}+\sum_{i=1}^{N-1}\lambda_i(x)E_{i,i+1}\)

  • \(E_{ij}\) is an \(N\times N\) matrix whose \((i,j)\) entry is \(1\) and the rest are \(0\).

  • \(\begin{aligned}\lambda_i(x)&=\frac{1}{1-x}(N-i)(i+\beta-k),\\ \mu_i(x)&=\frac{x}{1-x}(i-1)(N-i+k).\end{aligned}\)

Wright-Fisher diffusion model Part III

• Consider \(M,J,\hat{J},H\) matrices:

  • \(\begin{aligned}M&=M=\sum_{i=1}^{N-1}E_{i,i+1}~,~J=\sum_{i=1}^{N}(N-i)E_{i,i}\\ \hat{J}&=\sum_{i=1}^{N}(i-1)E_{i,i}=(N-1)I-J~,~H=\sum_{i=1}^{N}\omega_{i} E_{i,i}\end{aligned}\)
  • \(\omega_i=\begin{pmatrix}\beta-k+i-1\\i-1\end{pmatrix}\begin{pmatrix}N+k-i-1\\N-i\end{pmatrix}\)

• Operator \(\mathcal{L}\) is self-adjoint with respect to weighted measure \(W\):

  • \(W(x)=x^{\alpha}(1-x)^{\beta}Hx^{J},x\in(0,1)\)
  • \(\mathcal{L}=\frac{1}{2}A(x)\frac{d^2}{dx^2}+B(x)\frac{d}{dx}+Q(x)\frac{d^0}{dx^0}\)
    • \(\begin{aligned}A(x)&=2x(1-x)I\\B(x)&=(\alpha-1)I+J-x((\alpha+\beta+2)I+J)\end{aligned}\)

Wright-Fisher diffusion model Part IV

• The spectral representation of \(P(t;x,y)\) is given by:

  • \(P(t;x,y)=\Psi(x)\left(\sum_{n=0}^{\infty}R_n(x)e^{t\Gamma_n}||R_n||^{-2}_{\widetilde{W}}R^T_n(y)\widetilde{W}(y)\right)\Psi^{-1}(y)\)

    • \(\begin{cases}\Psi(x)&=e^{\hat{J}M^T}(1-x)^{\hat{J}}T\\ T&=\sum_{i\le j}^{N}(-1)^{i}\frac{(-j)_{i}}{(1-N)_{i}}\frac{(\alpha+\beta-k+j+1)_{i}}{(\beta-k+1)_{i}}E_{ij},\\ \widetilde{W}(x)&=\Psi^T(x)W(x)\Psi(x),\\ \widetilde{\mathcal{L}}F(x)&=\Psi^{-1}(x)\mathcal{L}(\Psi(x)F(x))\\ \widetilde{\mathcal{L}}R_n(x)&=R_n(x)\Gamma_n\end{cases}\)

• \(\sigma^2(x)\) is not affected by the phase change.

• \(b(x,i)\) is.

• The transition \(i\rightarrow i+1\) increases the intensity of \(B\rightarrow A\) by one unit (with respect to \(\alpha\)).

• \(i\rightarrow i-1\) decreases the intensity of \(B\rightarrow A\) by one unit (with respect to \(\alpha\)).

• The intensity of \(A\rightarrow B\) (with respect to \(\beta\)) is not affected by \(\sigma^2(x),b(x,i)\)

Wright-Fisher diffusion model Part V

Wright-Fisher diffusion model Part VI