On the Transition Density and First Hitting Time Distributions of the Doubly Skewed CIR Process

Abstract

In this paper, we study doubly skewed CIR processes, which are extensions of skew Brownian motion. We use modified spectral expansion to obtain some properties, including the transition densities and first hitting time distributions, of doubly skewed CIR processes.

Appendices

Appendix A: Proof of Proposition 2.1

First we note the fact that,

$$ p_{X}(t;x_{0},x)=G^{\prime}(x)p_{Y}(t;G(x_{0}),G(x)), $$

then the problem turns to computing the transition density of Y_t whose general form is defined in Eq. 9. That is to say, we need to solve the SL equation $-(\mathcal {G} u)(y)=\lambda u(y),\forall y\in (l_{1},\infty )$, i.e., the following ordinary differential equations (ODEs)

$$ \begin{array}{@{}rcl@{}} \left\{ \begin{array}{ll} -\frac{1}{2}{\sigma_{1}^{2}}(x-l_{1})u_{xx}-k(\theta_{1}-x)u_{x}=\lambda u(x), & x\in(l_{1},\nu_{1}),\\ -\frac{1}{2}{\sigma_{2}^{2}}(x-l_{2})u_{xx}-k(\theta_{2}-x)u_{x}=\lambda u(x), & x\in[\nu_{1},\nu_{2}),\\ -\frac{1}{2}{\sigma_{3}^{2}}(x-l_{3})u_{xx}-k(\theta_{3}-x)u_{x}=\lambda u(x), & x\in[\nu_{2},\infty). \end{array} \right. \end{array} $$

When solving the above ODEs, we adopt the approach of Gorovoi and Linetsky (2004). Let $\xi (x,\lambda ), x\in (l_{1},\infty )$ be the unique (up to multiple independent of x) solution such that

$$ {\int}_{l_{1}}^{\nu_{1}}|\xi_(x,\lambda)|^{2}\mathfrak{m}_{Y}(x)\mathrm{d} x<\infty, $$

and

$$ \lim_{x\downarrow l_{1}}\frac{\xi_{x}^{\prime}(s,\lambda)}{\mathfrak{s}_{Y}(x)}=0, $$

for each $\lambda \in \mathbb {C}$, and ξ(x, λ) and $\xi _{x}^{\prime }(x,\lambda )$ are continuous in x and λ in $(l_{1},\infty )\times \mathbb {C}$ and entire in λ for each fixed $x\in (l_{1},\infty )$. And let $\eta (x,\lambda ), x\in (l_{1},\infty )$ be the unique (up to multiple independent of x) solution such that

$$ {\int}_{\nu_{2}}^{\infty}|\eta_(x,\lambda)|^{2}\mathfrak{m}_{Y}(x)\mathrm{d} x<\infty, $$

and

$$ \lim_{x\uparrow +\infty}\frac{\eta_{x}^{\prime}(x,\lambda)}{\mathfrak{s}_{Y}(x)}=0, $$

for each $\lambda \in \mathbb {C}$, and ξ(x, λ) and $\xi _{x}^{\prime }(x,\lambda )$ are continuous in x and λ in $(l_{1},\infty )\times \mathbb {C}$ and entire in λ for each fixed $x\in (l_{1},\infty )$.

When considering each interval (l₁,ν₁), [ν₁,ν₂) and $[\nu _{2},\infty )$ separately, we know ϕ_i(x, λ) = F(α, β, z_i) and ψ_i(x, λ) = U(α, β, z_i) are two linearly independent solutions of each ODE on each interval for i = 1, 2, 3 from Xu et al. (2016b) with α, z_i defined in Eq. 10 and β showed in Eq. 5. Then we can assume ξ(x, λ) and η(x, λ) on interval $(l_{1},\infty )$ to have the following forms,

$$ \begin{array}{@{}rcl@{}} \xi(x,\lambda)&=\left\{ \begin{array}{ll} \phi_{1}(x,\lambda), & l_{1}<x<\nu_{1},\\ A_{1}\phi_{2}(x,\lambda)+B_{1}\psi_{2}(x,\lambda), & \nu_{1}\leqslant x<\nu_{2},\\ A_{2}\phi_{3}(x,\lambda)+B_{2}\psi_{3}(x,\lambda), & x\geqslant\nu_{2}, \end{array} \right. \\ \eta(x,\lambda)&=\left\{ \begin{array}{ll} A_{3}\phi_{1}(x,\lambda)+B_{3}\psi_{1}(x,\lambda), & l_{1}<x<\nu_{1},\\ A_{4}\phi_{2}(x,\lambda)+B_{4}\psi_{2}(x,\lambda), & \nu_{1}\leqslant x<\nu_{2},\\ \psi_{3}(x,\lambda), & x\geqslant\nu_{2}. \end{array} \right. \end{array} $$

The continuity of ξ(x, λ), $\xi _{x}^{\prime }(x,\lambda )$, η(x, λ) and $\eta _{x}^{\prime }(x,\lambda )$ at x = ν₁ and x = ν₂ uniquely implies the following facts:

$$ \begin{array}{@{}rcl@{}} \left\{\begin{array}{l} \phi_{1}(\nu_{1},\lambda)=A_{1}\phi_{2}(\nu_{1},\lambda)+B_{1}\psi_{2}(\nu_{1},\lambda),\\ \phi_{1x}^{\prime}(\nu_{1},\lambda)=A_{1}\phi_{2x}^{\prime}(\nu_{1},\lambda)+B_{1}\psi_{2x}^{\prime}(\nu_{1},\lambda),\\ A_{1}\phi_{2}(\nu_{2},\lambda)+B_{1}\psi_{2}(\nu_{2},\lambda)=A_{2}\phi_{3}(\nu_{2},\lambda)+B_{2}\psi_{3}(\nu_{2},\lambda),\\ A_{1}\phi_{2x}^{\prime}(\nu_{2},\lambda)+B_{1}\psi_{2x}^{\prime}(\nu_{2},\lambda)=A_{2}\phi_{3x}^{\prime}(\nu_{2},\lambda)+B_{2}\psi_{3x}^{\prime}(\nu_{2},\lambda),\\ A_{3}\phi_{1}(\nu_{1},\lambda)+B_{3}\psi_{1}(\nu_{1},\lambda)=A_{4}\phi_{2}(\nu_{1},\lambda)+B_{4}\psi_{2}(\nu_{1},\lambda),\\ A_{3}\phi_{1x}^{\prime}(\nu_{1},\lambda)+B_{3}\psi_{1x}^{\prime}(\nu_{1},\lambda)=A_{4}\phi_{2x}^{\prime}(\nu_{1},\lambda)+B_{4}\psi_{2x}^{\prime}(\nu_{1},\lambda),\\ A_{4}\phi_{2}(\nu_{2},\lambda)+B_{4}\psi_{2}(\nu_{2},\lambda)=\psi_{3}(\nu_{2},\lambda),\\ A_{4}\phi_{2x}^{\prime}(\nu_{2},\lambda)+B_{4}\psi_{2x}^{\prime}(\nu_{2},\lambda)=\psi_{3x}^{\prime}(\nu_{2},\lambda). \end{array} \right. \end{array} $$

Solving the above equations, we can derive the unknown coefficients A_i,B_i,i = 1, 2, 3, 4. Then the Wronskian of ξ(x, λ) and η(x, λ) is¹

$$ \begin{array}{@{}rcl@{}} \omega(\lambda)&=&\xi(x,\lambda)\frac{\eta_{x}^{\prime}(x,\lambda)}{\mathfrak{s}_{Y}(x)}-\eta(x,\lambda)\frac{\xi_{x}^{\prime}(x,\lambda)}{\mathfrak{s}_{Y}(x)}\\ &=&\frac{1}{\mathfrak{s}_{Y}(\nu_{2})}(\xi(\nu_{2},\lambda)\eta_{x}^{\prime}(\nu_{2},\lambda)-\xi_{x}^{\prime}(\nu_{2},\lambda)\eta(\nu_{2},\lambda)), \end{array} $$

which yields the value ω(λ) in Proposition 2.1. At an eigenvalue λ = λ_n (zero of the function ω(λ)), the Wronskian vanishes and ξ(x, λ_n) and η(x, λ_n) become linearly dependent, i.e.,²

$$ \eta(x,\lambda_{n})=A_{n}\xi(x,\lambda_{n}), \ \ \ A_{n}=\frac{\psi_{3}(\nu_{2},\lambda_{n})}{A_{2}\phi_{3}(\nu_{2},\lambda_{n})+B_{2}\psi_{3}(\nu_{2},\lambda_{n})}. $$

Finally, according to Theorem 5 in Linetsky (2004b), the corresponding normalized eigenfunctions can take the form

$$ \varphi_{n}(x)= \sqrt{\frac{A_{n}}{{\Delta}_{n}}}\xi(x,\lambda_{n}), \ \ \text{or},\ \ \frac{1}{\sqrt{A_{n}{\Delta}_{n}}}\eta(x,\lambda_{n}), $$

where ${\Delta }_{n}=\frac {d\omega (\lambda )}{d\lambda }|_{\lambda =\lambda _{n}}$ and we arrive at the result in Proposition 2.1.

Appendix B: Proof of Proposition 3.1

The proof is similar to the proof of Proposition 2.1. The different point is that we need to solve the SL equation $-(\mathcal {G}_{|(l_{1},y]} u)(x)=\lambda u(x)$ for all x ∈ (l₁,y] and the solution η(x, λ) satisfy the boundary conditions

$$ \eta(y,\lambda)=0,\ \ \eta_{x}^{\prime}(y,\lambda)=\frac{\partial\eta(x,\lambda)}{\partial x}|_{x=y}=-1. $$

(B.1)

For the case when y > ν₂, we assume that ξ(x, λ) and η(x, λ) have the following form

$$ \begin{array}{@{}rcl@{}} \xi(x,\lambda)&=\left\{ \begin{array}{ll} \phi_{1}(x,\lambda), & l_{1}<x<\nu_{1},\\ \tilde{A}_{1}\phi_{2}(x,\lambda)+\tilde{B}_{1}\psi_{2}(x,\lambda), & \nu_{1}\leqslant x<\nu_{2},\\ \tilde{A}_{2}\phi_{3}(x,\lambda)+\tilde{B}_{2}\psi_{3}(x,\lambda), & \nu_{2}\leqslant x\leqslant y, \end{array} \right. \\ \eta(x,\lambda)&=\left\{ \begin{array}{ll} \tilde{A}_{3}\phi_{1}(x,\lambda)+\tilde{B}_{3}\psi_{1}(x,\lambda), & l_{1}<x<\nu_{1},\\ \tilde{A}_{4}\phi_{2}(x,\lambda)+\tilde{B}_{4}\psi_{2}(x,\lambda), & \nu_{1}\leqslant x<\nu_{2},\\ \tilde{A}_{5}\phi_{3}(x,\lambda)+\tilde{B}_{5}\psi_{3}(x,\lambda), & \nu_{2}\leqslant x\leqslant y. \end{array} \right. \end{array} $$

For the case when $\nu _{1}<y\leqslant \nu _{2}$, we assume that

$$ \begin{array}{@{}rcl@{}} \xi(x,\lambda)&=\left\{ \begin{array}{ll} \phi_{1}(x,\lambda), & l_{1}<x<\nu_{1},\\ \hat{A}_{1}\phi_{2}(x,\lambda)+\hat{B}_{1}\psi_{2}(x,\lambda), & \nu_{1}\leqslant x\leqslant y, \end{array} \right. \\ \eta(x,\lambda)&=\left\{ \begin{array}{ll} \hat{A}_{2}\phi_{1}(x,\lambda)+\hat{B}_{2}\psi_{1}(x,\lambda), & l_{1}<x<\nu_{1},\\ \hat{A}_{3}\phi_{2}(x,\lambda)+\hat{B}_{3}\psi_{2}(x,\lambda), & \nu_{1}\leqslant x\leqslant y. \end{array} \right. \end{array} $$

For the case when $y\leqslant \nu _{1}$, ξ(x, λ) = ϕ₁(x, λ) and we assume that $\eta (x,\lambda )=\check {A}_{1}\phi _{1}(x,\lambda )+\check {B}_{1}\psi _{1}(x,\lambda )$ which should satisfy the boundary conditions presented in Eq. B.1.

Similarly, using the continuity of ξ(x, λ), $\xi _{x}^{\prime }(x,\lambda )$, η(x, λ) and $\eta _{x}^{\prime }(x,\lambda )$ at x = ν₁ and x = ν₂, we can derive the unknown coefficients in ξ(x, λ) and η(x, λ) for each case. After obtaining ξ(x, λ) and η(x, λ) in each case, we can know each Wronskian ω(λ), eigenvalues, eigenfunctions and then derive our results.

Appendix C: Proof of Proposition 3.2

The proof is also similar to the proof of Proposition 2.1. The different point is that we need to solve the SL equation $-(\mathcal {G}_{|[y,\infty )} u)(x)=\lambda u(x)$ for all $x\in [y,\infty )$ and the solution ξ(x, λ) satisfy the boundary conditions

$$ \xi(y,\lambda)=0,\ \ \xi_{x}^{\prime}(y,\lambda)=\frac{\partial\xi(x,\lambda)}{\partial x}|_{x=y}=1. $$

(C.1)

For the case when y < ν₁, we suppose ξ(x, λ) and η(x, λ) to have the following form

$$ \begin{array}{@{}rcl@{}} \xi(x,\lambda)&=\left\{ \begin{array}{ll} \tilde{C}_{1}\phi_{1}(x,\lambda)+\tilde{D}_{1}\psi_{1}(x,\lambda), & y\leqslant x<\nu_{1},\\ \tilde{C}_{2}\phi_{2}(x,\lambda)+\tilde{D}_{2}\psi_{2}(x,\lambda), & \nu_{1}\leqslant x<\nu_{2},\\ \tilde{C}_{3}\phi_{3}(x,\lambda)+\tilde{D}_{3}\psi_{3}(x,\lambda), & x\geqslant\nu_{2}, \end{array} \right. \\ \eta(x,\lambda)&=\left\{ \begin{array}{ll} \tilde{C}_{4}\phi_{1}(x,\lambda)+\tilde{D}_{4}\psi_{1}(x,\lambda), &y\leqslant x<\nu_{1},\\ \tilde{C}_{5}\phi_{2}(x,\lambda)+\tilde{D}_{5}\psi_{2}(x,\lambda), & \nu_{1}\leqslant x<\nu_{2},\\ \psi_{3}(x,\lambda), & x\geqslant\nu_{2}. \end{array} \right. \end{array} $$

For the case when $\nu _{1}\leqslant y<\nu _{2}$, we suppose that

$$ \begin{array}{@{}rcl@{}} \xi(x,\lambda)&=\left\{ \begin{array}{ll} \hat{C}_{1}\phi_{2}(x,\lambda)+\hat{D}_{1}\psi_{2}(x,\lambda), & y\leqslant x<\nu_{2},\\ \hat{C}_{2}\phi_{3}(x,\lambda)+\hat{D}_{2}\psi_{3}(x,\lambda), & x\geqslant\nu_{2}, \end{array} \right. \\ \eta(x,\lambda)&=\left\{ \begin{array}{ll} \hat{C}_{3}\phi_{2}(x,\lambda)+\hat{D}_{3}\psi_{2}(x,\lambda), & y\leqslant x<\nu_{2},\\ \psi_{3}(x,\lambda), & x\geqslant\nu_{2}. \end{array} \right. \end{array} $$

For the case when $y\leqslant \nu _{1}$, η(x, λ) = ψ₃(x, λ) and we suppose that ξ(x, λ) = C₁ϕ₃(x, λ) + D₁ψ₃(x, λ) which should satisfy the boundary conditions showed in Eq. C.1.

Similarly, the continuity of ξ(x, λ), $\xi _{x}^{\prime }(x,\lambda )$, η(x, λ) and $\eta _{x}^{\prime }(x,\lambda )$ at x = ν₁ and x = ν₂ uniquely determines the unknown coefficients in each ξ(x, λ) and η(x, λ), and then we can get the Wronskian ω(λ), eigenvalues, eigenfunctions and our results.