Additive subordination and its applications in finance

Abstract

This paper studies additive subordination, which we show is a useful technique for constructing time-inhomogeneous Markov processes with analytical tractability. This technique is a natural generalization of Bochner’s subordination that has proved to be extremely useful in financial modeling. Probabilistically, Bochner’s subordination corresponds to a stochastic time change with respect to an independent Lévy subordinator, while in additive subordination, the Lévy subordinator is replaced by an additive one. We generalize the classical Phillips theorem for Bochner’s subordination to the additive subordination case, based on which we provide Markov and semimartingale characterizations for a rich class of jump-diffusions and pure jump processes obtained from diffusions through additive subordination, and obtain spectral decomposition for them. To illustrate the usefulness of additive subordination, we develop an analytically tractable cross-commodity model for spread option valuation that is able to calibrate the implied volatility surface of each commodity. Moreover, our model can generate implied correlation patterns that are consistent with market observations and economic intuitions.

Appendices

Appendix A: Proofs

Proof of Proposition 2.1

$T$ is a semimartingale since it is a nondecreasing process which implies that it has finite variation over finite time intervals. Let $(B_{t},A_{t},\Pi_{t})$ be the generating triplet (see [69, Definition 8.2]) of the infinitely divisible distribution $q_{t}=q_{0,t}$ (i.e., the distribution of $T_{t}$) for $t\geq0$, i.e.,

$$ E\big[e^{i\lambda T_{t}}\big]=e^{i\lambda B_{t}-\frac{1}{2}\lambda ^{2}A_{t}+\int_{(0,\infty)}(e^{i\lambda\tau}-1-i\lambda\tau1_{\{|\tau |\le1\}})\Pi_{t}(d\tau)}. $$

(A.1)

From [69, Theorem 9.8], (i) $B_{0}=0$ and $B_{t}$ is continuous in $t$, (ii) $\Pi_{0}=0$, and for all $B\in \mathcal {B}(\mathbb{R}_{+})$, $\Pi _{s}(B)\le\Pi_{t} (B)$ and $\Pi_{s}(B')\to\Pi_{t} (B')$ as $s\to t$, where $B'\subseteq (\varepsilon,\infty)$, $\varepsilon>0$. Since $T_{t}$ is nonnegative, [69, Theorem 24.11] implies that $A_{t}=0$, $\Pi_{t}((-\infty ,0))=0$ and $\int_{(0,\infty)}(\tau\wedge1)\Pi_{t}(d\tau)<\infty$, and $E[e^{i\lambda T_{t}}]$ can also be written as

$$ E\big[e^{i\lambda T_{t}}\big]=e^{i\lambda\Gamma_{t}+\int_{(0,\infty )}(e^{i\lambda\tau}-1)\Pi_{t}(d\tau)}, $$

(A.2)

where $\Gamma_{t}=B_{t}-\int_{(0,\infty)}(\tau\wedge1)\Pi_{t}(d\tau)$ is nondecreasing in $t$ and $\Gamma_{t}\geq0$ for all $t\geq0$.

Using [69, Remark 9.9], there exists a unique measure $\mathcal {V}$ on $[0,\infty)\times \mathbb {R}_{+}$ with $\mathcal{V}([0,t]\times B)=\Pi_{t}(B), t\geq0\,\, \text{and} \,\, B\in \mathcal {B}(\mathbb {R}_{+})$, which satisfies $\mathcal{V}(\{t\}\times \mathbb {R}_{+})=0$ and $\int_{[0,t]\times(0,\infty )}(\tau\wedge1)\mathcal{V}(dsd\tau)<\infty$. From (A.1) and [39, Theorem II.4.15], the deterministic triplet $(B,0,\mathcal{V})$ gives the semimartingale characteristics for $T$. Furthermore, since $T$ is stochastically continuous, it has no fixed time of discontinuity, and hence it is quasi-left-continuous [39, Corollary II.4.18]. Applying Proposition II.2.9 of [39] to $T$, $B$ and $\mathcal{V}$ can be written as

$$ B_{t}=\int_{0}^{t}b(s)F(ds),\quad\mathcal{V}(dsd\tau)=\nu(s,d\tau)F(ds) $$

for some nonnegative continuous nondecreasing deterministic function $F(s)$. Define $\gamma(s)=b(s)-\int_{(0,\infty)}(\tau\wedge1)\nu (s,d\tau)$; then $\Gamma_{t}=\int_{0}^{t}\gamma(s)F(ds)$. Since $\Gamma$ is nondecreasing, $\gamma(s)\geq0$ for $F(ds)$-almost all $s$. Moreover, $\int_{0}^{t}\int_{(0,\infty)}(\tau\wedge1)\nu (s,d\tau)F(ds)<\infty$ for all $t\geq0$ implies that $\int_{(0,\infty)}(\tau\wedge1)\nu (s,d\tau)<\infty$ for $F(ds)$-almost all $s$. The expression for the Laplace transform follows from (A.2). □

Proof of Theorem 3.1

(i) Each $\mathcal {P}^{\psi}_{s,t}$ is a contraction because

$$ \|\mathcal {P}^{\psi}_{s,t}f\|\le\int_{[0,\infty)}\|\mathcal {P}_{u} f\| q_{s,t}(du)\le\int_{[0,\infty)}\|f\| q_{s,t}(du)=\|f\|. $$

For $0\le s\le t\le r$,

$$\begin{aligned} \mathcal {P}^{\psi}_{s,t}\mathcal {P}^{\psi}_{t,r}f&=\mathcal {P}^{\psi}_{s,t}\int _{[0,\infty)}\mathcal {P}_{u_{1}} f q_{t,r}(du_{1})=\int_{[0,\infty)}\mathcal {P}^{\psi}_{s,t}\mathcal {P}_{u_{1}}f q_{t,r}(du_{1})\\ &=\int_{[0,\infty)}\int_{[0,\infty)}\mathcal {P}_{u_{2}}\mathcal {P}_{u_{1}}f q_{s,t}(du_{2})q_{t,r}(du_{1})\\ &=\int_{[0,\infty)}\int_{[0,\infty)}\mathcal {P}_{u_{1}+u_{2}}f q_{s,t}(du_{2})q_{t,r}(du_{1})=\int_{[0,\infty)} \mathcal {P}_{u}f q_{s,r}(du)\\ &=\mathcal {P}^{\psi}_{s,r}f. \end{aligned}$$

In the second to last equality, we used the convolution property (2.1). This shows $\mathcal {P}^{\psi}_{s,r}=\mathcal {P}^{\psi}_{s,t}\mathcal {P}^{\psi}_{t,r}$. By (2.2), $q_{t,t}=\delta_{0}$, hence $\mathcal {P}^{\psi}_{t,t}f=f$. Together we have shown that $(\mathcal {P}^{\psi}_{s,t})_{0\le s\le t}$ is a backward propagator. From Fubini’s theorem for Bochner integrals (see [30, Theorem E.8]),

$$\begin{aligned} &\int_{[0,\infty)}\int_{[0,\infty)}\mathcal {P}_{u_{1}+u_{2}}f q_{s,t}(du_{2})q_{t,r}(du_{1})\\ &=\int_{[0,\infty)}\int_{[0,\infty)}\mathcal {P}_{u_{1}+u_{2}}f q_{t,r}(du_{1})q_{s,t}(du_{2}). \end{aligned}$$

Hence we also have

$$ \mathcal {P}^{\psi}_{s,r}=\mathcal {P}^{\psi}_{s,t}\mathcal {P}^{\psi}_{t,r}=\mathcal {P}^{\psi}_{t,r}\mathcal {P}^{\psi}_{s,t}. $$

(A.3)

This implies that $(\mathcal {P}^{\psi}_{s,t})_{0\le s\le t}$ is also a propagator. Finally, we want to show the strong continuity of $(\mathcal {P}^{\psi}_{s,t})_{0\le s\le t}$. From Theorem 2.1 in [37], the latter is equivalent to $(\mathcal {P}^{\psi}_{s,t})_{0\le s\le t}$ being separately strongly continuous (i.e., for fixed $t$ and $f\in\mathfrak{B}$, $s\mapsto \mathcal {P}^{\psi}_{s,t}f$ is continuous on $[0,t]$, and for fixed $s$ and $f\in\mathfrak{B}$, $t\mapsto \mathcal {P}^{\psi}_{s,t}f$ is continuous on $[s,\infty)$) and locally uniformly bounded (i.e., for compact set $K$ in $\{(s,t):0\le s\le t<\infty\}$, $\|\mathcal {P}^{\psi}_{s,t}\|$ is bounded uniformly in $(s,t)\in K$). The locally uniform boundedness is obvious, since every $\mathcal {P}^{\psi}_{s,t}$ is a contraction. Next, we consider the separate strong continuity. First, we prove that for every $f\in\mathfrak{B}$,

$$ \mathcal {P}^{\psi}_{s,t}f\longrightarrow f\ \text{as}\ s\uparrow t,\qquad \mathcal {P}^{\psi}_{s,t}f\longrightarrow f\ \text{as}\ t\downarrow s. $$

(A.4)

We have

$$\|\mathcal {P}^{\psi}_{s,t}f-f\|=\left\|\int_{[0,\infty )}(\mathcal {P}_{u}f-f)q_{s,t}(du)\right\|\le\int_{[0,\infty)}\|\mathcal {P}_{u}f-f\|q_{s,t}(du). $$

Since $\|\mathcal {P}_{u}f-f\|$ is a bounded continuous function in $u$, applying (2.3) shows (A.4). For fixed $t$ and $f$, to show that $s\mapsto \mathcal {P}^{\psi}_{s,t}f$ is continuous on $[0,t]$, we need to show for every $s< t$ that $\mathcal {P}^{\psi}_{s-h,t}f\to \mathcal {P}^{\psi}_{s,t}f$ and $\mathcal {P}^{\psi}_{s+h,t}f\to \mathcal {P}^{\psi}_{s,t}f$ as $h\downarrow0$. Note that by (A.3) and the contraction property,

$$\begin{aligned} \|\mathcal {P}^{\psi}_{s-h,t}f-\mathcal {P}^{\psi}_{s,t}f\|&=\|\mathcal {P}^{\psi}_{s,t}(\mathcal {P}^{\psi}_{s-h,s}-I)f\|\le\|(\mathcal {P}^{\psi}_{s-h,s}-I)f\|,\\ \|\mathcal {P}^{\psi}_{s+h,t}f-\mathcal {P}^{\psi}_{s,t}f\|&=\|\mathcal {P}^{\psi}_{s+h,t}(I-\mathcal {P}^{\psi}_{s,s+h})f\|\le\|(I-\mathcal {P}^{\psi}_{s,s+h})f\|. \end{aligned}$$

Hence, (A.4) implies the continuity of $\mathcal {P}^{\psi}_{s,t}f$ in $s$. The continuity of $\mathcal {P}^{\psi}_{s,t}f$ in $t$ can be proved similarly.

(ii) Recall that for every $t\geq0$, $T^{\phi_{t}}$ is the Lévy subordinator whose drift and Lévy measure are given by $\gamma(t)$ and $\nu(t,\cdot)$. $(\mathcal {P}^{\phi_{t}}_{u})_{u\geq0}$ is the Lévy subordinate semigroup of $(\mathcal {P}_{s})_{s\geq0}$ with respect to $T^{\phi_{t}}$. Its generator is denoted by $\mathcal {G}^{\phi_{t}}$, and we have (3.3) from the classical Phillips theorem. We want to show for any $s,t\geq0$ that $(\mathcal {P}^{\phi_{s}}_{u})_{u\ge 0}$ and $(\mathcal {P}^{\phi_{t}}_{u})_{u\ge0}$ commute, that is, for any $u,v\geq0$, $\mathcal {P}^{\phi_{s}}_{u}\mathcal {P}^{\phi_{t}}_{v}f=\mathcal {P}^{\phi_{t}}_{v}\mathcal {P}^{\phi _{s}}_{u}f$ for $f\in\mathfrak{B}$. Denote by $\pi^{\phi_{s}}_{u}(\cdot)$ and $\pi^{\phi_{t}}_{v}(\cdot)$ the laws of $T^{\phi_{s}}_{u}$ and $T^{\phi _{t}}_{v}$. Then we have

$$\begin{aligned} \mathcal {P}^{\phi_{s}}_{u}\mathcal {P}^{\phi_{t}}_{v}f&=\int_{[0,\infty)} \mathcal {P}_{r} \bigg(\int_{[0,\infty)} \mathcal {P}_{\tau} f\pi^{\phi_{t}}_{v}(d\tau )\bigg)\pi^{\phi_{s}}_{u}(dr)\\ &=\int_{[0,\infty)} \int_{[0,\infty)}( \mathcal {P}_{r} \mathcal {P}_{\tau} f)\pi^{\phi_{t}}_{v}(d\tau)\pi^{\phi_{s}}_{u}(dr)\\ &=\int_{[0,\infty)} \int_{[0,\infty)} (\mathcal {P}_{r+\tau} f)\pi ^{\phi_{t}}_{v}(d\tau)\pi^{\phi_{s}}_{u}(dr)\\ &=\int_{[0,\infty)} \int_{[0,\infty)} (\mathcal {P}_{\tau+r} f)\pi ^{\phi_{s}}_{u}(dr)\pi^{\phi_{t}}_{v}(d\tau)\\ &=\int_{[0,\infty)} \mathcal {P}_{\tau} \bigg(\int_{[0,\infty)} \mathcal {P}_{r} f\pi^{\phi_{s}}_{u}(dr)\bigg)\pi^{\phi_{t}}_{v}(d\tau)=\mathcal {P}^{\phi _{t}}_{v}\mathcal {P}^{\phi_{s}}_{u}f. \end{aligned}$$

The interchange of the order of integration is justified by Fubini’s theorem. We next verify the following statement: for $f\in{\mathfrak {D}}(\mathcal {G})$, $\mathcal {G}^{\phi_{t-}}f=\lim_{s\to t-}\mathcal {G}^{\phi _{s}}f$ and $\mathcal {G}^{\phi_{t+}}f=\lim_{s\to t+}\mathcal {G}^{\phi_{s}}f$ exists, and $\mathcal {G}^{\phi_{t+}}f=\mathcal {G}^{\phi_{t}}f$ for every $t\geq0$. Condition (a) implies that for any $f\in D(\mathcal{G})$, $\gamma (t)\mathcal{G}f$ is right-continuous with left limits. For the second part of the RHS of (3.3), we note that

$$\int_{(0,\infty)} (\mathcal {P}_{\tau}f-f)\nu(t,d\tau )=\int_{(0,\infty)}\frac{\mathcal {P}_{\tau}f-f}{\tau\wedge1}\nu _{F}(t,d\tau) . $$

It is easy to see that $\frac{\mathcal {P}_{\tau}f-f}{\tau\wedge1}$ is a continuous function in $\tau$, and bounded in the Banach space norm since $\|\mathcal {P}_{\tau}f-f\|\le\min\{\tau\|\mathcal {G}f\|,2\|f\|\}$ [70, Eq. (13.3)]. Therefore, by [63, Theorem 2], the weak convergence of $\nu_{F}(t,\cdot)$ assumed in condition (b) implies that the second part of the RHS of (3.3) is also right-continuous with left limit (Nielsen [63] deals with probability measures, but his result also applies to weakly convergent finite measures.). From condition (c), we also have $\mathcal {G}^{\phi_{t-}}f=\mathcal {G}^{\phi _{t+}}f=\mathcal {G}^{\phi_{t}}f$ on ${\mathfrak{D}}(\mathcal {G})$ for all but a finite number of $t$ in any bounded interval.

Recall $R^{\Pi}_{s,t}$ defined in (3.4). We have verified all the conditions of Theorem 3.1 in [36], which implies that $U_{s,t}f:=\lim_{|\Pi|\to0}R^{\Pi}f$ for $f\in\mathfrak{B}$ exists and $(U_{s,t})_{0\le s\le t}$ is a strongly continuous contraction propagator on $\mathfrak{B}$. Furthermore, for $f\in {\mathfrak{D}}(\mathcal {G})$, the family of generators of $(U_{s,t})_{0\le s\le t}$ is given by (3.5).

We now prove that $U_{s,t}=\mathcal {P}^{\psi}_{s,t}$ on $\mathfrak{B}$ for $0\le s< t$. For a partition $\Pi$ given by $s=t_{0}< t_{1}<\cdots<t_{n}=t$, define $q^{\Pi}_{s,t}:=\pi^{\phi_{t_{0}}}_{t_{1}-t_{0}}*\pi^{\phi _{t_{1}}}_{t_{2}-t_{1}}*\cdots{}*\pi^{\phi_{t_{n-1}}}_{t_{n}-t_{n-1}}$, where ∗ denotes convolution. The properties of convolutions give $R^{\Pi}_{s,t}f=\int_{[0,\infty)}\mathcal {P}_{u}f q^{\Pi}_{s,t}(du)$. The Laplace transform of $q^{\Pi}_{s,t}$ is

$$ \int_{(0,\infty)}e^{-\lambda u}q^{\Pi}_{s,t}(du)=e^{-\sum _{i=0}^{n-1}\psi(\lambda,t_{i})(t_{i+1}-t_{i})}, $$

(A.5)

where $\psi(\lambda,\cdot)$ is defined in (2.5). Under the assumed conditions (a)–(c), $\psi(\lambda,t)$ is piecewise continuous in $t$. Hence as $|\Pi|\to0$, (A.5) converges to the Laplace transform of $q_{s,t}$. Therefore, $q^{\Pi }_{s,t}$ converges to $q_{s,t}$ weakly, which implies that for any continuous linear functional $\ell$ on $\mathfrak{B}$ and any $f\in \mathfrak{B}$,

$$ \ell(R^{\Pi}f)=\int_{(0,\infty)}\ell(\mathcal {P}_{u}f)q^{\Pi }_{s,t}(du)\longrightarrow \int_{(0,\infty)}\ell(\mathcal {P}_{u}f)q_{s,t}(du)=\ell (\mathcal {P}^{\psi}_{s,t}f), $$

since $\ell(\mathcal {P}_{u}f)$ is a continuous bounded function in $u$. Recall that $U_{s,t}f$ is the strong limit of $R^{\Pi}f$, hence $U_{s,t}f=\mathcal {P}^{\psi}_{s,t}f$. This allows us to conclude from (3.5) that

$$ \lim_{h\to0+}h^{-1}(\mathcal {P}^{\psi}_{t,t+h}f-f)=\mathcal {G}^{\phi_{t}}f \qquad \text{for }f\in{\mathfrak{D}}(\mathcal {G}). $$

Hence ${\mathfrak{D}}(\mathcal {G})\subseteq{\mathfrak{D}}(\mathcal {G}^{\psi}_{t})$, and (3.3) gives (3.1). (3.2) follows from Theorem 3.1 in [36]. □

Proof of Proposition 3.2

We only need to show that $\int_{\mathbb{R}^{d}}|\psi(-\eta_{X}(\theta ),t)|^{2}|\hat{f}(\theta)|^{2}d\theta$ is finite when $\int_{\mathbb {R}^{d}}|\eta_{X}(\theta)|^{2}|\hat{f}(\theta)|^{2}d\theta$ is finite. Notice that for $\lambda\in\mathbb{C}$ with its real part $\Re(\lambda)\ge0$,

$$ |\psi(\lambda,t)|\le\gamma(t)|\lambda| +2(1+|\lambda|)\int _{(0,\infty)}(1\wedge\tau)\nu(t,d\tau)\le c(t)(1+|\lambda|) $$

for some suitable $c(t)>0$ ($c(t)$ is a constant that only depends on $t$), which follows from the inequality $1\wedge(|\lambda|\tau)\le (1+|\lambda|)(1\wedge\tau)$. Hence $|\psi(\lambda,t)|^{2}\le 2c^{2}(t)(1+|\lambda|^{2})$, and the claim is implied by $\int_{\mathbb {R}^{d}}|\hat{f}(\theta)|^{2}d\theta<\infty$ and $\int_{\mathbb {R}^{d}}|\eta_{X}(\theta)|^{2}|\hat{f}(\theta)|^{2}d\theta<\infty$. □

Proof of Proposition 4.1

By conditioning on the sample path of $X^{0}$ between 0 and $h$ and using the independence of the exponential random variable and $X^{0}$, we obtain that for any measurable and bounded function $f$ on $I$,

$$ E_{x}\left[f(X_{h})1_{\{\zeta>h\}}\right]=E_{x}\left[e^{-\int_{0}^{h} k(X^{0}_{u})du}f(X^{0}_{h})1_{\{\zeta_{0}>h\}}\right]. $$

We want to show that

$$\begin{aligned} &h^{-1}E_{x}\big[f(X^{0}_{h})1_{\{\zeta_{0}>h\}}\big]-h^{-1}E_{x}\left [f(X_{h})1_{\{\zeta>h\}}\right]\\ &=E_{x}\Big[h^{-1}\Big(1-e^{-\int_{0}^{h} k(X^{0}_{u})du}\Big)f(X^{0}_{h})1_{\{ \zeta_{0}>h\}}\Big]\xrightarrow{\ \mathit{bp}\ }k(x)f(x), \end{aligned}$$

where the convergence is bounded pointwise on compact intervals of $I$. Let $J$ be such an interval. Pick $\delta$ small enough such that for all $x\in J$, $[x-\delta,x+\delta]\subseteq\hat{J}\subset(\ell ,r)$, where $\hat{J}$ is a compact interval. Let $\tau_{x}^{\delta}:=\inf\{t\ge0: |X^{0}_{t}-x|\ge\delta\}$. We have

$$\begin{aligned} &E_{x}\Big[h^{-1}\Big(1-e^{-\int_{0}^{h} k(X^{0}_{u})du}\Big)f(X^{0}_{h})1_{\{ \zeta_{0}>h\}}\Big]\\ &=E_{x}\Big[h^{-1}\Big(1-e^{-\int_{0}^{h} k(X^{0}_{u})du}\Big)f(X^{0}_{h})1_{\{ \tau_{x}^{\delta}>h\}}\Big]\\ &\phantom{=:}{}+E_{x}\Big[h^{-1}\Big(1-e^{-\int_{0}^{h} k(X^{0}_{u})du}\Big)f(X^{0}_{h})1_{\{\tau_{x}^{\delta}\le h, \zeta_{0}>h\}}\Big]. \end{aligned}$$

The second term is bounded by $\|f\|_{\infty}E_{x}[h^{-1}1_{\{\tau _{x}^{\delta}\le h, \zeta_{0}>h\}}]$ ($\|f\|_{\infty}$ is the $L^{\infty}$-norm of $f$), which converges to 0 boundedly pointwise on $J$ as shown in the proof of Theorem 16.84 of [11] (see its claim ($\text{i}^{\text{o}}$)). For the first term, notice that on $\{\tau_{x}^{\delta}> h\}$, $k(X^{0}_{u})$ is bounded (say by $M$) for all $0\le u\le h$ as $k(x)$ is continuous. Thus $|h^{-1}(1-e^{-\int_{0}^{h} k(X^{0}_{u})du})|\le h^{-1}(1-e^{-Mh})$, which is bounded for $h$ sufficiently small. It follows that the first term is also bounded for $h$ sufficiently small. Applying the dominated convergence theorem shows that it converges to $k(x)f(x)$ boundedly pointwise on $J$.

Now setting $f(y)=1_{\{|y-x|>\varepsilon\}}$, $(y-x)1_{\{|y-x|\le \varepsilon\}}$, $(y-x)^{2}1_{\{|y-x|\le\varepsilon\}}$, 1, respectively and applying (4.2)–(4.5) give us (4.6)–(4.9). □

Proof of Theorem 4.2

Theorem 3.1 (ii) implies that for $f\in C^{2}_{c}(I)$,

$$\begin{aligned} \mathcal {G}^{\psi}_{t} f(x) &= \gamma(t)\frac{1}{2}\sigma^{2}(x)f''(x)+\gamma (t)\mu(x)f'(x)-\gamma(t)k(x)f(x)\\ &\phantom{=:}+\int_{(0,\infty)}\big(\mathcal {P}_{\tau}f(x)-f(x)\big)\nu(t,d\tau), \end{aligned}$$

where $(\mathcal {P}_{t})_{t\geq0}$ is the transition semigroup of the underlying diffusion. We write the last term as

$$\begin{aligned} &\int_{(0,\infty)}\big(\mathcal {P}_{\tau}f(x)-f(x)\big)\nu(t,d\tau)\\ &=\int_{(0,\infty)}\left(\int_{\mathbb{R}}p(\tau ,x,x+y)f(x+y)dy-f(x)\right)\nu(t,d\tau)\\ &=\int_{(0,\infty)}\bigg(\int_{\mathbb{R}}p(\tau,x,x+y)\Big(\big(f(x+y)-f(x)-1_{\{|y|\le1\}}yf'(x)\big)+f(x)\\ &\phantom{=:}{}+1_{\{|y|\le1\}}yf'(x)\Big)dy-f(x)\bigg)\nu(t,d\tau )\\ &=\int_{\mathbb{R}}\left(f(x+y)-f(x)-1_{\{|y|\le1\}}yf'(x)\right )\left(\int_{(0,\infty)}p(\tau,x,x+y)\nu(t,d\tau)\right)dy\\ &\phantom{=:}{}+f(x)\int_{(0,\infty)}\left(1-\int_{\mathbb {R}}p(\tau,x,x+y)dy\right)\nu(t,d\tau)\\ &\phantom{=:}{}+f'(x)\int_{(0,\infty)}\left(\int_{\{|y|\le1\} }yp(\tau,x,x+y)dy\right)\nu(t,d\tau). \end{aligned}$$

Combining this with the other terms yields (4.10). We now justify that the interchange of the order of integration in the above derivation is valid. Notice that for $f\in C^{2}_{c}(I)$, $|f(x+y)-f(x)-1_{\{|y|\le1\}}yf'(x)|\le C_{x}(1\wedge y^{2})$ for some positive constant $C_{x}$ which only depends on $x$. Thus, if we can show that

$$ \int_{(0,\infty)}\int_{\mathbb{R}}(1\wedge y^{2})p(\tau,x,x+y)dy\nu (t,d\tau)< \infty, $$

(A.6)

we can apply the dominated convergence theorem to justify the interchange. This also implies that $\Pi^{\psi}(t,x,dy)$ is a Lévy-type measure. To prove (A.6), we notice that

$$\begin{aligned} &\int_{(0,\infty)}\int_{\mathbb{R}}(1\wedge y^{2})p(\tau,x,x+y)dy\nu (t,d\tau)\\ &=\int_{(0,\infty)}\int_{\{|y|\le1\}}y^{2}p(\tau,x,x+y)dy\nu (t,d\tau)\\ &\phantom{=:}{}+\int_{(0,\infty)}\int_{\{|y|>1\}}p(\tau,x,x+y)dy\nu (t,d\tau). \end{aligned}$$

Note that $\int_{\{|y|\le1\}}y^{2}p(\tau,x,x+y)dy$ and $\int_{\{ |y|>1\}}p(\tau,x,x+y)dy$ are bounded by 1 for all $\tau>0$. (4.8) implies that $\int_{\{|y|\le1\}}y^{2}p(\tau,x,x+y)dy\sim \sigma^{2}(x)\tau$ as $\tau\to0$. From (4.6), $\int_{\{|y|>1\}}p(\tau,x,x+y)dy=o(\tau)$ as $\tau\to0$. These facts together with $\int_{(0,\infty)}(\tau\wedge1)\nu(t,d\tau )<\infty$ show (A.6). Similar arguments also show that the terms $\int_{(0,\infty)}(\int_{\{|y|\le1\} }yp(\tau,x,x+y)dy)\nu(t,d\tau)$ and $\int_{(0,\infty)}P(\tau,x,\{ \Delta\})\nu(t,d\tau)$ are well defined by noticing that (4.7) implies $\int_{\{|y|\le1\}}yp(\tau,x,x+y)dy\sim\mu (x)\tau$ and (4.9) implies $P(\tau,x,\{\Delta\})\sim k(x)\tau$ as $\tau\to0$. □

Proof of Theorem 4.4

If we can show for $f\in C^{2}_{b}(I)$ (bounded and twice continuously differentiable functions on $I$) that

$$\begin{aligned} M^{f} :=& f(\hat{X}^{\psi})-f(x)-f'(\hat{X}^{\psi}_{-})\cdot B^{\psi}- \frac {1}{2}f''(\hat{X}^{\psi}_{-})\cdot C^{\psi}\\ &{} - \big(f(\hat{X}^{\psi}_{-}+y)-f(\hat{X}^{\psi}_{-})-f'(\hat{X}^{\psi}_{-})y1_{\{|y|\le1\}}\big)*\nu^{\psi} \end{aligned}$$

is a local martingale (“⋅” and “∗” denote stochastic integration with respect to a semimartingale and a random measure, respectively; see [39]), then Theorem II.2.42 in [39] implies that $\hat{X}^{\psi}$ is a semimartingale with $(B^{\psi},C^{\psi},\nu^{\psi})$ as characteristics. To show this, notice the following. First, (3.2) and Theorem 4.2 imply that for $f\in C^{2}_{c}(I)$, $\mathcal {P}^{\psi}_{s,t}f-f=\int_{s}^{t}\mathcal {P}^{\psi}_{s,u}\mathcal {G}^{\psi}_{u}fdu$. Thus under $P_{s,x}$, $f(X_{t})-f(X_{s})-\int_{s}^{t}\mathcal {G}^{\psi}_{u}f(X_{u})du$ is a martingale with respect to $(\mathcal {F}^{0}_{s,t})_{t\ge s}$. From [21, Remark 2.3, No.2], it is also a martingale with respect to $(\mathcal {F}^{0}_{s,t+})_{t\ge s}$. Second, from the bounded pointwise convergence on compacts for (4.6)–(4.9) and using arguments from the proof of Theorem 4.2, we can show that for $x$ in any compact interval in $I$,

$$\begin{aligned} \bigg|\int_{(0,\infty)}\bigg(\int_{\{|y|\le1\}}yp(\tau ,x,x+y)dy\bigg)\nu(t,d\tau)\bigg|&\le C_{1}\int_{(0,\infty)}(\tau \wedge1)\nu(t,d\tau),\\ \bigg|\int_{(0,\infty)}P(\tau,x,\{\Delta\})\nu(t,d\tau)\bigg|&\le C_{2}\int_{(0,\infty)}(\tau\wedge1)\nu(t,d\tau),\\ \bigg|\int_{\{y\neq0\}}(y^{2}\wedge1)\Pi^{\psi}(t,x,dy)\bigg|&\le C_{3}\int_{(0,\infty)}(\tau\wedge1)\nu(t,d\tau), \end{aligned}$$

for some positive constants $C_{1}$, $C_{2}$ and $C_{3}$ which do not depend on $t$ and $x$. Furthermore, from conditions (a)–(c) of Theorem 3.1 (ii), for $t$ in any compact interval, $\int_{(0,\infty)}(\tau\wedge1)\nu(t,d\tau)$ and $\gamma(t)$ are continuous in $t$ except for a finite number of points, hence bounded. Also note that $\mu(x)$, $\sigma(x)$ and $k(x)$ are continuous. These imply that

$$ \mu^{\psi}(t,x), \sigma^{\psi}(t,x), k^{\psi}(t,x)\quad \text{and}\quad \int_{\{y\neq0\}}(y^{2}\wedge1)\Pi^{\psi}(t,x,dy) $$

are bounded for $t$ and $x$ in any compact set.

To prove that $M_{f}$ is a local martingale, based on the above two conclusions, one can use the arguments in the proof of Proposition 3.2 in [21]. The details are omitted here. □

Proof of Theorem 5.1

Theorem 3.1 (i) already implies that $(\mathcal {P}^{\psi}_{s,t})_{0\le s\le t}$ is a strongly continuous propagator/backward propagator of contractions on ℋ. We next prove that each $\mathcal {P}^{\psi}_{s,t}$ is symmetric. For $f,g\in \mathcal {H}$,

$$\begin{aligned} \big\langle \mathcal {P}_{s,t}^{\psi}f,g\big\rangle &=\left\langle\int _{[0,\infty)}\mathcal {P}_{u}fq_{s,t}(du),g\right\rangle=\int_{[0,\infty )}\left\langle \mathcal {P}_{u}f,g\right\rangle q_{s,t}(du) \\ &=\int_{[0,\infty)}\left\langle f,\mathcal {P}_{u}g\right\rangle q_{s,t}(du)=\left\langle f, \int_{[0,\infty)}\mathcal {P}_{u}gq_{s,t}(du)\right\rangle=\big\langle f,\mathcal {P}_{s,t}^{\psi}g\big\rangle . \end{aligned}$$

This shows the symmetry. From Fubini’s theorem, we observe that for all $f\in{ \mathcal {H}}$ and $0\le s\le t$,

$$\begin{aligned} \int_{[0,\infty)}\mathcal {P}_{u} f q_{s,t}(du)&=\int_{[0,\infty)} \int _{(-\infty,0]}e^{\lambda u} E(d\lambda)fq_{s,t}(du)\\ &=\int_{(-\infty,0]}\int_{[0,\infty)}e^{\lambda u}q_{s,t}(du) E(d\lambda)f, \end{aligned}$$

which yields the spectral decomposition (5.2). □

Proof of Proposition 5.2

The claim can be proved using arguments similar to those used in the proof of Proposition 2.4 in [46]. We omit the details here. □

Proof of Proposition 6.3

The futures price is the conditional expectation of the spot price under the pricing measure. In our model, $F_{1}(t,T) = a_{1}(T) E[X_{T}^{\psi_{1}}|X_{t}^{\psi_{1}}]$ and $F_{2}(t,T) = a_{2}(T) ( E[X_{T}^{\psi_{1}}|X_{t}^{\psi_{1}}] + E[X_{T}^{\psi_{2}}|X_{t}^{\psi _{2}}] )$ for any $0\leq t \leq T$. Therefore, we just need to calculate $\mathcal {P}^{\psi}_{t,T}f(x)$ with $f(x)=x$ for a generic ASubCIR process. It is easy to verify $x\in L^{2}(\mathbb {R}_{++},\mathfrak{m})$. From [5, p.115], for a function $g(x)$ such that its derivatives up to order $n$ are bounded as $x\rightarrow0$ and of at most polynomial growth as $x\rightarrow\infty$ ($L_{n}^{(\alpha)}(x)$ is the generalized Laguerre polynomial),

$$ \int_{0}^{\infty}g(x)L_{n}^{(\alpha)}(x)x^{\alpha} e^{-x}dx=\frac {(-1)^{n}}{n!}\int_{0}^{\infty} e^{-x}g^{(n)}(x)x^{n+\alpha}dx. $$

Hence for $n>1$, $f_{n}=0$. It is straightforward to find $f_{0}$ and $f_{1}$ using the explicit expression of $\varphi_{0}(x)$ and $\varphi_{1}(x)$ (see (6.2)), as well as some elementary integration. They are given by

$$ f_{0} = \alpha^{-\frac{\beta+1}{2}} \frac{\Gamma(1+\beta)}{\sqrt {\kappa\Gamma(\beta)}},\qquad f_{1} = - \alpha^{-\frac{\beta +1}{2}} \frac{\Gamma(1+\beta)}{\sqrt{\kappa\Gamma(\beta+1)}}. $$

Thus, for any $T>t$,

$$\begin{aligned} E[X_{T}^{\psi}|X_{t}^{\psi}= x] = & e^{-\int_{t}^{T} \psi(0, u)du}\varphi _{0}(x) f_{0} + e^{-\int_{t}^{T} \psi(\kappa,u)du} \varphi_{1}(x) f_{1} \\ = & \frac{\beta}{\alpha} + e^{-\int_{t}^{T} \psi(\kappa,u)du} \left ( x - \frac{\beta}{\alpha} \right) = \theta+ e^{-\int_{t}^{T} \psi (\kappa,u)du} \left( x - \theta\right), \end{aligned}$$

where we have used the definitions of $\alpha$ and $\beta$ in (6.1). Some further simple calculations give us the claim. □

Proof of Proposition 6.4

(1) Recall that $\mathbf{M}(a,c;z)$ is the scaled Kummer confluent hypergeometric function defined in (B.4). We compute the expansion coefficients for the payoff $(K-x)^{+}$ which is in $L^{2}(\mathbb {R}_{++},\mathfrak{m})$. They are

$$\begin{aligned} f_{n}(K)& = \int_{0}^{\infty} (K-x)^{+} \varphi_{n}(x) \mathfrak{m}(dx) \\ & = \sqrt{\kappa} \alpha^{\frac{\beta-1}{2}} \sqrt{\frac{\Gamma (n+\beta)}{n!}} \frac{\alpha^{2-\beta}}{\kappa} \int_{0}^{\infty}\! (K-x)^{+} (\alpha x)^{\beta-1} e^{-\alpha x} \mathbf{M}(-n,\beta ;\alpha x) dx. \end{aligned}$$

Using the Kummer transformation identity (B.7) and the change of variables $x=Ky$, we have

$$\begin{aligned} f_{n}(K) & = \alpha^{-\frac{\beta+1}{2}} \sqrt{\frac{\Gamma(n+\beta )}{n!\kappa}} (\alpha K)^{\beta+1} \int_{0}^{1} \mathbf{M}(n+\beta ,\beta;-\alpha Ky) y^{\beta-1}(1-y) dy \\ & = \alpha^{-\frac{\beta+1}{2}} \sqrt{\frac{\Gamma(n+\beta )}{n!\kappa}} (\alpha K)^{\beta+1} \mathbf{M}(n+\beta,\beta +2;-\alpha K) \\ & = \alpha^{-\frac{\beta+1}{2}} \sqrt{\frac{\Gamma(n+\beta )}{n!\kappa}} e^{-\alpha K} (\alpha K)^{\beta+1} \mathbf {M}(2-n,\beta+2; \alpha K). \end{aligned}$$

(A.7)

Here, we have used (B.8), the integral representation of $\mathbf{M}(a,b,z)$. Moreover, using (B.5), we can write $f_{n}(K)$ as

$$ f_{n}(K) = \frac{1}{\sqrt{n(n-1)\kappa}} \alpha^{-\frac{\beta +1}{2}} e^{-\alpha K} (\alpha K)^{\beta+1} \ell_{n-2}^{(\beta +1)}(\alpha K) \qquad \text{for any } n\geq2. $$

Using the two identities $M(1,a+1;z) = e^{z} a z^{-a} \gamma(a,z)$ [65, Eq. (13.6.5)], and $aM(a+1,b;z) = (a-b+1)M(a,b;z)+(b-1)M(a,b-1;z)$ [65, Eq. (13.3.3)], where $M$ is the Kummer confluent hypergeometric function defined in (B.3), we have

$$\begin{aligned} f_{0}(K) & = \frac{\alpha^{-\frac{\beta+1}{2}}}{\sqrt{\kappa\Gamma (\beta)}} \big(\alpha K\gamma(\beta,\alpha K)-\gamma(\beta +1,\alpha K)\big),\\ f_{1}(K) & = \frac{1}{\sqrt{\kappa\Gamma(\beta+1)}} \alpha^{-\frac {\beta+1}{2}}\gamma(\beta+1,\alpha K). \end{aligned}$$

The claim can be proved by substituting the expression for $f_{n}(K)$ back into the eigenfunction expansion and simplifying.

(2) It is easy to verify $f(x_{1},x_{2}):=(K-\omega_{1} x_{1}-\omega_{2} x_{2})^{+}\in L^{2}(\mathbb {R}_{++}^{2},\mathfrak{M})$. Let $k_{1}(x)=\frac {K}{\omega_{2}}-\frac{\omega_{1}}{\omega_{2}}x$. Then

$$\begin{aligned} f_{n,m} &= \int_{0}^{\infty}\int_{0}^{\infty}(K-\omega_{1} x_{1}-\omega_{2} x_{2})^{+} \varphi_{n}^{1}(x_{1})\varphi_{m}^{2}(x_{2})\mathfrak{m}_{1}(dx_{1})\mathfrak {m}_{2}(dx_{2}) \\ & = \omega_{2} \int_{0}^{\infty}\boldsymbol{1}_{\{k_{1}(x_{1})>0\}} \bigg( \int _{0}^{k_{1}(x_{1})} \big(k_{1}(x_{1})-x_{2}\big) \varphi_{m}^{2}(x_{2}) \mathfrak {m}_{2}(dx_{2}) \bigg) \varphi_{n}^{1}(x_{1}) \mathfrak{m}_{1}(dx_{1}) \\ & = \frac{K^{\beta_{1}+\beta_{2}+1}}{\sqrt{\kappa_{1}\kappa_{2}}\omega _{1}^{\beta_{1}}\omega_{2}^{\beta_{2}}} \alpha_{1}^{\frac{\beta _{1}+1}{2}}\alpha_{2}^{\frac{\beta_{2}+1}{2}} \sqrt{\frac{\Gamma (n+\beta_{1})\Gamma(m+\beta_{2})}{n!m!}} \\ &\phantom{=:}{} \times\int_{0}^{1} y^{\beta_{1}-1}(1-y)^{\beta_{2}+1} \\ &\phantom{=:}{}\times\mathbf{M}(n+\beta_{1},\beta_{1}; -\gamma_{1} y) \mathbf{M} \big(m+\beta_{2},\beta_{2}+2; -\gamma_{2}(1-y) \big)dy, \end{aligned}$$

where we used (A.7). Some simplification gives us the put option formula for the daughter commodity. Using (B.3) and (B.8), we get

$$\begin{aligned} &\pi_{n,m}(\gamma_{1},\gamma_{2}) \\ & = \sum_{p=0}^{\infty} \bigg( \frac{(m+\beta_{2})_{p}(-\gamma _{2})^{p}}{p!\Gamma(\beta_{2}+2+p)} \int_{0}^{1} y^{\beta_{1}-1}(1-y)^{p+\beta _{2}+1} \mathbf{M}(n+\beta_{1},\beta_{1}; -\gamma_{1} y)dy \bigg) \\ & = \sum_{p=0}^{\infty} \frac{(m+\beta_{2})_{p}(-\gamma_{2})^{p}}{p!} \mathbf{M}\big(n+\beta_{1},\beta_{1}+\beta_{2}+2+p;-\gamma_{1} \big). \end{aligned}$$

This proves the claim. □

Proof of Proposition 6.6

We only prove case (b), because all the other cases are similar to Proposition 6.4. We compute the expansion coefficients for the payoff $g(x_{1},x_{2}):=(K + \omega_{1} x_{1}-\omega_{2} x_{2})^{+} \in L^{2}(\mathbb {R}_{++}^{2},\mathfrak{M})$ for $\omega_{1}>0$ and $\omega_{2}>0$. Let $k_{1}(x)=\frac{K}{\omega _{2}}+\frac{\omega_{1}}{\omega_{2}}x$. Using (A.7), (B.6) and (B.7),

$$\begin{aligned} g_{n,m} &= \int_{0}^{\infty}\int_{0}^{\infty}(K + \omega_{1} x_{1}-\omega_{2} x_{2})^{+} \varphi_{n}^{1}(x_{1})\varphi_{m}^{2}(x_{2})\mathfrak{m}_{1}(dx_{1})\mathfrak {m}_{2}(dx_{2}) \\ & = \omega_{2} \int_{0}^{\infty}\boldsymbol{1}_{\{k_{1}(x_{1})>0\}} \bigg( \int _{0}^{k_{1}(x_{1})} \big(k_{1}(x_{1})-x_{2}\big) \varphi_{m}^{2}(x_{2}) \mathfrak {m}_{2}(dx_{2}) \bigg) \varphi_{n}^{1}(x_{1}) \mathfrak{m}_{1}(dx_{1}) \\ & = \frac{\omega_{2}}{\sqrt{\kappa_{1}\kappa_{2}} } \alpha_{1}^{\frac {\beta_{1}+1}{2}}\alpha_{2}^{\frac{\beta_{2}+1}{2}} \sqrt{\frac{\Gamma (n+\beta_{1})\Gamma(m+\beta_{2})}{n!m!}}\int_{0}^{\infty}y^{\beta_{1}-1} \bigg(\frac{K}{\omega_{2}}+\frac{\omega_{1}}{\omega_{2}}y\bigg)^{\beta _{2}+1}\\ &\phantom{=:}{}\times\mathbf{M}(n+\beta_{1},\beta_{1}; -\alpha_{1} y)\mathbf{M} \bigg(m+\beta_{2},\beta_{2}+2; -\alpha_{2}\Big(\frac {K}{\omega_{2}}+\frac{\omega_{1}}{\omega_{2}}y\Big) \bigg)dy, \end{aligned}$$

which can be expressed in terms of $\pi_{n,m}^{1}(\gamma_{1},\gamma_{2})$, $\pi_{n,m}^{2}(\omega_{1},\omega_{2})$ and $\pi_{n,m}^{3}(\gamma_{1},\gamma _{2})$ by a change of variables. From (B.3), we have

$$\begin{aligned} & \pi_{n,m}^{1}(\gamma_{1},\gamma_{2}) \\ & = \sum_{k=0}^{\infty}\frac{(m+\beta_{2})_{k}(-\gamma_{2})^{k}}{k!\Gamma (\beta_{2}+2+k)}\int_{0}^{\infty}x^{\beta_{1}-1}(1+x)^{k+\beta_{2}+1} \mathbf{M}(n+\beta_{1},\beta_{1}; -\gamma_{1} x) dx \\ & = \frac{n!}{\Gamma(n+\beta_{1})} \sum_{k=0}^{\infty}\frac{(m+\beta _{2})_{k}(-\gamma_{2})^{k}}{k!\Gamma(\beta_{2}+2+k)} \\ &\phantom{=:}{}\times\int_{0}^{\infty}x^{\beta_{1}-1}(1+x)^{k+\beta_{2}+1} e^{-\gamma_{1} x}L_{n}^{(\beta_{1}-1)}(\gamma_{1} x) dx \\ & = \sum_{k=0}^{\infty}\frac{(m+\beta_{2})_{k}(-\gamma_{2})^{k}}{k!\Gamma (\beta_{2}+2+k)}\\ &\phantom{=:}{}\times\Biggl(\sum_{\ell=0}^{n} \frac{n!(-\gamma _{1})^{\ell}}{\ell!(n-\ell)!\Gamma(\ell+\beta_{1})}\int_{0}^{\infty}x^{\ell+\beta_{1}-1}(1+x)^{k+\beta_{2}+1} e^{-\gamma_{1} x} dx \Biggr). \end{aligned}$$

Here, we have used the relation $\mathbf{M}(n+\beta_{1},\beta_{1}; -\gamma_{1} x)=e^{-\gamma_{1} x}\frac{n!}{\Gamma(n+\beta _{1})}L_{n}^{(\beta_{1}-1)}(\gamma_{1} x)$ and the series representation of generalized Laguerre polynomials [5, Eq. (4.5.3)]. Using (B.9), we can obtain the formula for $\pi _{n,m}^{1}(\gamma_{1},\gamma_{2})$ after rearranging terms. Similarly, we can compute $\pi_{n,m}^{3}(\gamma_{1},\gamma_{2})$. Finally, we compute $\pi_{n,m}^{2}(\omega_{1},\omega_{2})$ as

$$\begin{aligned} & \pi_{n,m}^{2}(\omega_{1},\omega_{2}) \\ &= \sum_{k=0}^{\infty}\frac{(m+\beta_{2})_{k}(-\alpha_{2}\omega _{1})^{k}}{k!\Gamma(\beta_{2}+2+k)}\int_{0}^{\infty}x^{\beta_{1}+\beta_{2}+k} \mathbf{M}(n+\beta_{1},\beta_{1}; -\alpha_{1}\omega_{2} x) dx \\ & = \frac{n!}{\Gamma(n+\beta_{1})}\\ &\phantom{=:}{}\times\sum_{k=0}^{\infty}\frac{(m+\beta_{2})_{k}(-\alpha _{2}\omega_{1})^{k}}{k!\Gamma(\beta_{2}+2+k)}\int_{0}^{\infty}x^{\beta _{1}+\beta_{2}+k}e^{-\alpha_{1}\omega_{2} x} L_{n}^{(\beta_{1}-1)}(\alpha _{1}\omega_{2} x) dx \\ & = \frac{n!}{\Gamma(n+\beta_{1})}\sum_{k=0}^{\infty}\frac{(m+\beta _{2})_{k}(-\alpha_{2}\omega_{1})^{k}}{k!\Gamma(\beta_{2}+2+k)}\frac{(-\beta _{2}-k-1)_{n}}{n!(\alpha_{1}\omega_{2})^{\beta_{1}+\beta_{2}+k+1}} \Gamma (\beta_{1}+\beta_{2}+k+1) \\ & = \frac{\Gamma(\beta_{1}+\beta_{2}+1)}{\Gamma(n+\beta_{1})(\alpha _{1}\omega_{2})^{\beta_{1}+\beta_{2}+1}}\\ &\phantom{=:}{}\times\sum_{k=0}^{\infty}\frac{(m+\beta_{2})_{k} (\beta _{1}+\beta_{2}+1)_{k}}{k!\Gamma(\beta_{2}+2+k)} \bigg(-\frac{\alpha _{2}\omega_{1}}{\alpha_{1}\omega_{2}}\bigg)^{k} (-\beta_{2}-k-1)_{n}, \end{aligned}$$

where the first equation comes from (B.3), the second from (B.5), and the third one follows from the integral identity [67, Eq. (2.19.3.5)]

$$ \int_{0}^{\infty}x^{\alpha-1}e^{-cx}L_{n}^{(\lambda)}(cx)dx=\frac {(1-\alpha+\lambda)_{n}}{n!c^{\alpha}}\Gamma(\alpha). $$

When $\beta_{2}+2-n\neq0,-1,\dots$, it follows from the identity $(a)_{n}=\frac{(-1)^{n}\Gamma(1-a)}{\Gamma(1-a-n)}$ that

$$\begin{aligned} &\pi_{n,m}^{2}(\omega_{1},\omega_{2}) \\ & = \frac{(-1)^{n}\Gamma(\beta_{1}+\beta_{2}+1)}{\Gamma(n+\beta _{1})\Gamma(\beta_{2}+2-n)(\alpha_{1}\omega_{2})^{\beta_{1}+\beta_{2}+1}}\\ &\phantom{=:}{}\times\sum_{k=0}^{\infty}\frac{(m+\beta_{2})_{k} (\beta _{1}+\beta_{2}+1)_{k}}{k!(\beta_{2}+2-n)_{k}} \bigg(-\frac{\alpha_{2}\omega _{1}}{\alpha_{1}\omega_{2}}\bigg)^{k} \\ & = \frac{(-1)^{n}\Gamma(\beta_{1}+\beta_{2}+1)}{\Gamma(n+\beta _{1})\Gamma(\beta_{2}+2-n)(\alpha_{1}\omega_{2})^{\beta_{1}+\beta_{2}+1}} {}_{2}F_{1}\bigg( \textstyle\begin{array}{c} m+\beta_{2}, \beta_{1}+\beta_{2}+1 \\ \beta_{2}+2-n \end{array}\displaystyle ; -\frac{\alpha_{2}\omega_{1}}{\alpha_{1}\omega_{2}}\bigg), \end{aligned}$$

where the last equation is from the definition of the Gauss hypergeometric function ${}_{2}F_{1}$ (see (B.10)). If $\beta_{2}+2+p=n$ for some nonnegative integer $p$, the formula for $\pi _{n,m}^{2}(\omega_{1},\omega_{2})$ follows from the fact that $(-\beta _{2}-k-1)_{n}=0$ for any positive integer $k>p$. □

Appendix B: Some special functions

We define the scaled generalized Laguerre polynomial as

$$ \ell_{n}^{(\nu)}(x):=\sqrt{\frac{n!}{\Gamma(\nu+n+1)}}L_{n}^{(\nu )}(x), n=0,1,\dots, $$

(B.1)

where $L_{n}^{(\nu)}(x)$ is the generalized Laguerre polynomial. We compute $\ell_{n}^{(\nu)}(x)$ in our implementation instead of $L_{n}^{(\nu)}(x)$. Based on the classical recursion for $L_{n}^{(\nu )}(x)$ (see e.g. [42, Eq. (9.12.3)]), $\ell_{n}^{(\nu )}(x)$ can be computed recursively as

$$\begin{aligned} \ell_{0}^{(\nu)}(x) = &\frac{1}{\sqrt{\Gamma(\nu+1)}} , \\ \ell_{1}^{(\nu)}(x) =& \frac{1+\nu-x}{\sqrt{\Gamma(\nu +2)}}, \\ \ell_{n}^{(\nu)}(x) = &\frac{\nu+2n-1-x}{\sqrt{n(\nu+n)}} \ell _{n-1}^{(\nu)}(x) - \sqrt{ \frac{(\nu+n-1)(n-1)}{(\nu+n)n}}\ell _{n-2}^{(\nu)}(x),\quad \,\, n\geq2. \qquad \quad \end{aligned}$$

(B.2)

Let $M(a,c;x)$ denote the Kummer confluent hypergeometric function which is defined as [5, Eq. (6.1.2)]

$$ M(a,c;z)= \sum_{n=0}^{\infty}\frac{(a)_{n}}{(c)_{n} n!}z^{n}= 1+\frac {a}{c}z+\frac{a(a+1)}{c(c+1)2!}z^{2}+\cdots $$

(B.3)

for $c\neq0,-1,-2,\dots$ and $(a)_{n}$ is the Pochhammer symbol which is defined as $(a)_{n}=a(a+1)\cdots(a+n-1)$. We define the scaled confluent hypergeometric Kummer function as

$$ \mathbf{M}(a,c;z):=M(a,c;z)/\Gamma(c),\qquad c>0. $$

(B.4)

Below we give several useful identities:

(1) The scaled confluent hypergeometric Kummer function and the generalized Laguerre polynomials are related as [65, Eq. (13.6.19)]

$$ \mathbf{M}(-n,a+1;z) = \frac{n!}{\Gamma(n+a+1)}L_{n}^{(a)}(z). $$

(B.5)

Using this relation, the eigenfunction of the ASubCIR process (see (6.2)) can be rewritten as

$$ \varphi_{n}(x) = \sqrt{\kappa} \alpha^{\frac{\beta-1}{2}} \sqrt {\frac{\Gamma(n+\beta)}{n!} } \mathbf{M}(-n,\beta; \alpha x). $$

(B.6)

(2) Kummer’s transformation identity [65, Eq. (13.2.39)]:

$$ \mathbf{M}(a,b;z)= e^{z} \mathbf{M}(b-a,b,-z). $$

(B.7)

(3) Integral representation for $\mathbf{M}(a,b,z)$ [65, Eq. (13.4.2)]: for $\Re(b)>\Re(c)>0$,

$$ \mathbf{M}(a,b,z) = \frac{1}{\Gamma(b-c)}\int_{0}^{1} \mathbf {M}(a,c,zt)t^{c-1}(1-t)^{b-c-1}dt, $$

(B.8)

where $\Re(x)$ denotes the real part of a complex number $x$.

Finally, Tricomi’s confluent hypergeometric function $U(a,c;x)$ and the Gauss hypergeometric function ${}_{2}F_{1}$ are defined as [5, Eq. (6.2.1) and Eq. (8.2.2)]

$$\begin{aligned} U(a,c;x) & := \frac{1}{\Gamma(a)}\int_{0}^{\infty}e^{-xt}t^{a-1}(1+t)^{c-a-1}dt, \quad\Re(a)>0, \end{aligned}$$

(B.9)

$$\begin{aligned} {}_{2}F_{1}\Big( \textstyle\begin{array}{c} a, b \\ c \end{array}\displaystyle ; x \Big) & := \sum_{n=0}^{\infty}\frac{(a)_{n} (b)_{n}}{(c)_{n} n!} x^{n}. \end{aligned}$$

(B.10)