2. Preliminaries

Throughout this work, for a vector $\textbf{x}\in\mathbb{R}^n$ , we denote the sum of its coordinates by $\textbf{1}\cdot \textbf{x}=\sum_i x_i$ . We work on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with a filtration $\{\mathcal{F}_{t}\}_{t\geq0}$ . We fix a finite time horizon $t\in[0,T]$ for some $T<\infty$ and assume $\{\mathcal{F}_{t}\}_{0\leq t \leq T}$ and all its sub-filtrations are right-continuous and complete. Unless stated otherwise, every stochastic process is càdlàg with a state-space that is a continuous subspace of $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))$ for some $n\in\mathbb{N}_+$ , where $\mathcal{B}(\mathbb{R}^n)$ is the Borel $\sigma$ -field.

Let $\{X_{t}\}_{t\geq 0}$ be a Lévy process taking values in $\mathbb{R}$ , such that the law of $X_{t}$ is absolutely continuous with respect to the Lebesgue measure for every $t\in[0,T]$ . In this case the density $f_t$ of $X_t$ exists and satisfies the Chapman–Kolmogorov convolution identity $f_{t}(x)=\int_{\mathbb{R}}f_{t-s}(x-y)f_{s}(y)\,\textrm{d} y$ , for $0<s<t\leq T$ and $x\in\mathbb{R}$ . Having independent and stationary increments, the finite-dimensional law of $\{X_{t}\}_{0\leq t \leq T}$ is given by

\[\mathbb{P}(X_{t_{1}}\in \,\textrm{d} x_{1},\ldots, X_{t_{m}}\in \,\textrm{d} x_{m})=\prod^{m}_{i=1}f_{t_{i}-t_{i-1}}(x_{i}-x_{i-1})\,\textrm{d} x_{i}\]

for $m\in\mathbb{N}_{+}$ , $0<t_{1}<\cdots <t_{m}<T$ and $x_{1},\ldots,x_n\in\mathbb{R}$ .

A Lévy bridge is a Lévy process conditioned to take some fixed value at a fixed future time. Since Lévy processes are homogeneous strong Markov processes, the definition of their bridges can be formalised in terms of Doob h-transformations. See [Reference Fitzsimmons, Pitman and Yor9] for further details on the bridges of Markov processes. Let $\{X^{(z)}_{t,T}\}_{0\leq t \leq T}$ be a bridge of $\{X_{t}\}_{0\leq t \leq T}$ to the value $z\in\mathbb{R}$ at time T, where $0<f_{T}(z)<\infty$ . The transition density of $\{X^{(z)}_{t,T}\}_{0\leq t < T}$ is given by the following Doob h-transform of the transition density of $\{X_{t}\}_{0\leq t < T}$ :

(2.1) \begin{equation} \mathbb{P}\Big(X^{(z)}_{t,T}\in \,\textrm{d} x \mid X^{(z)}_{s,T}=y\Big)=\dfrac{h_t(x)}{h_s(y)}f_{t-s}(x-y)\,\textrm{d} x\end{equation}

for $0\leq s <t <T$ , where $h_t(x)=f_{T-t}(z-x)$ . Note that $\{h_t\}_{0\leq t < T}$ defined as such is harmonic with respect to $\{X_{t}\}_{0\leq t < T}$ . Note also that $\mathbb{P}(0<h_t(X^{(z)}_{t,T})<\infty)=1$ for all $0\leq t < T$ , so the ratios of densities in (2.1) are almost surely well-defined (this is discussed in the remark following Proposition 1 of [Reference Fitzsimmons, Pitman and Yor9]). Similar ratios feature throughout this work and are likewise almost surely well-defined, and we may pass by them without further comment.

Lévy random bridges (LRBs) are an extension of Lévy bridges. Their interpretation in [Reference Hoyle, Hughston and Macrina15] is as a bridge to an arbitrary random variable at time T, rather than a fixed value. A process $\{L_{t}\}_{0\leq t \leq T}$ is an LRB with generating law $\nu$ if it satisfies: (i) $L_{T}$ has marginal law $\nu$ , (ii) there exists a Lévy process $\{X_{t}\}_{0\leq t \leq T}$ such that the density $f_t$ of $X_t$ exists for all $t\in(0,T]$ , (iii) $\nu$ concentrates mass where $f_{T}$ is positive and finite $\nu$ -a.s., (iv) for all $m\in\mathbb{N}_{+}$ , every $0<t_{1}<\cdots<t_{m}<T$ , every $(x_{1},\ldots,x_{m})\in\mathbb{R}^{m}$ , and $\nu$ -a.e. z,

\[\mathbb{P}(L_{t_{1}}\leq x_{1},\ldots, L_{t_{m}}\leq x_{m} \mid L_{T}=z)=\mathbb{P}(X_{t_{1}}\leq x_{1},\ldots, X_{t_{m}}\leq x_{m} \mid X_{T}=z).\]

The finite-dimensional distribution of $\{L_{t}\}_{0\leq t \leq T}$ is given by

(2.2) \begin{equation}\mathbb{P}(L_{t_{1}}\in \,\textrm{d} x_{1},\ldots, L_{t_{m}}\in \,\textrm{d} x_{m}, L_{T}\in \,\textrm{d} z)=\prod^{m}_{i=1}(f_{t_{i}-t_{i-1}}(x_{i}-x_{i-1})\,\textrm{d} x_{i})\vartheta_{t_{m}}(\textrm{d} z;\, x_{m}),\end{equation}

where $\vartheta_{0}(\textrm{d} z;\, y)=\nu(\textrm{d} z)$ and $\vartheta_{t}(\textrm{d} z;\, y)=\nu(\textrm{d} z)f_{T-t}(z-y)/f_{T}(z)$ for $t\in(0,T)$ . It follows that LRBs are Markov processes with stationary increments, where the transition law of $\{L_{t}\}_{0\leq t \leq T}$ is

(2.3) \begin{equation}\mathbb{P}(L_{T}\in \,\textrm{d} z \mid L_{s}=y)=\dfrac{\vartheta_{s}(\textrm{d} z;\, y)}{\vartheta_{s}(\mathbb{R};\, y)}, \quad\mathbb{P}(L_{t}\in \,\textrm{d} x \mid L_{s}=y)=\dfrac{\vartheta_{t}(\mathbb{R};\, x)}{\vartheta_{s}(\mathbb{R};\, y)}f_{t-s}(x-y)\,\textrm{d} x\end{equation}

for $0\leq s < t$ . We note that the finite-dimensional distributions of LRBs with discrete state-spaces have similar transition probabilities given in terms of probability mass functions (for details see [Reference Hoyle, Hughston and Macrina15]). The extension of many later results to discrete processes follows from this.

Let $X_1,\ldots,X_n$ be random variables taking values in $\mathbb{R}$ with a joint density of the form

(2.4) \begin{equation}p\biggl( \sum_{i=1}^n x_i \biggr)\prod_{i=1}^n \phi_{a_i}(x_i),\end{equation}

where $a_1,\ldots,a_n>0$ are parameters, and the set of functions $\{\phi_a\colon a>0\}$ satisfies the convolution property $\phi_{a}*\phi_{b}=\phi_{a+b}$ . In [Reference Gupta and Richards13], this is referred to as a ‘Liouville density function’. Indeed, according to the definition given in [Reference Gupta and Richards13], $(X_1,\ldots,X_n)$ then has a Liouville distribution, although we prefer to refer to this as the generalised Liouville distribution to distinguish it from the original and special case that $\{\phi_a\}$ are gamma densities (see [Reference Fang, Kotz and Ng8], [Reference Gupta and Richards10], [Reference Gupta and Richards11], and [Reference Gupta and Richards12]). The actual definition of the generalised Liouville distribution given in [Reference Gupta and Richards13] replaces the functions $\{\phi_{a}\}$ with measures, and so it includes examples where the joint density may not exist. For our purposes, it is convenient to relax (2.4) in a different way. We keep $\{\phi_a\}$ , but replace the function p with a measure $\nu$ .

Definition 2.1. Let $X_1,\ldots,X_n$ be random variables taking values in $\mathbb{R}$ , let $\nu\colon \mathcal{B}(\mathbb{R})\rightarrow\mathbb{R}_+$ be a probability law, and let $\mathcal{A}=\{\phi_a\colon 0<a\leq A < \infty)\}$ be a family of functions satisfying the convolution property: $\phi_{a}*\phi_{b}=\phi_{a+b}$ , for $a+b\leq A$ . Then $(X_1,\ldots,X_n)$ has a generalised multivariate Liouville distribution if its joint probability law is of the form

(2.5) \begin{equation}\mathbb{P}\biggl(X_1\in \textrm{d} x_1, \ldots, X_{n-1}\in \textrm{d} x_{n-1}, \sum_{i=1}^n X_{i}\in \textrm{d} z\biggr) = \dfrac{\phi_{a_n}\big(z-\sum_{i=1}^{n-1}x_i\big)\nu(\textrm{d} z )}{\phi_{\textbf{1}\cdot \textbf{a}}(z)} \prod_{i=2}^n \phi_{a_i}(x_i) \,\textrm{d} x_i\end{equation}

for $x_1,\ldots,x_n\in \mathbb{R}$ , $\phi_{a_1},\ldots,\phi_{a_n}\in\mathcal{A}$ , $\textbf{a}=(a_1,\ldots,a_n)^\top\in\mathbb{R}^n_+$ , $\textbf{1}\cdot \textbf{a}\leq A$ .

Remark 2.2. Writing $B+x=\{y\colon y-x\in B\}$ , for $B\subset\mathbb{R}$ and $x\in\mathbb{R}$ , then (2.5) is equivalent to

(2.6) \begin{align}&\mathbb{P}(X_1\in \textrm{d} x_1, \ldots, X_{n-1}\in \textrm{d} x_{n-1}, X_{n}\in B) \nonumber \\* &\quad =\prod_{i=2}^n (\phi_{a_i}(x_i) \,\textrm{d} x_i)\int_{z\in B+\sum_{i=1}^{n-1}x_i} \dfrac{\phi_{a_n}\big(z-\sum_{i=1}^{n-1}x_i\big)}{\phi_{\textbf{1}\cdot \textbf{a}}(z)} \nu(\textrm{d} z ) \nonumber \\* &\quad =\prod_{i=2}^n (\phi_{a_i}(x_i) \,\textrm{d} x_i)\int_{x_n\in B} \dfrac{\phi_{a_n}(x_n)}{\phi_{\textbf{1}\cdot\textbf{a}}(\!\sum_ix_i)}\nu\biggl(\sum_{i=1}^{n-1}x_i + \textrm{d} x_n \biggr).\end{align}

Furthermore, if $\nu$ admits a density p, then (2.6) can be written in the form of a Liouville density:

\begin{equation*}\mathbb{P}(X_1\in \textrm{d} x_1, \ldots, X_{n-1}\in \textrm{d} x_{n-1}, X_{n}\in \textrm{d} x_n)=\dfrac{p(\!\sum_{i}x_i)}{\phi_{\textbf{1}\cdot \textbf{a}}(\!\sum_ix_i)}\prod_{i=1}^n (\phi_{a_i}(x_i) \,\textrm{d} x_i).\end{equation*}

3. Generalised Liouville processes

To construct a GLP, we start with a ‘master’ LRB $\{L_t\}_{0\leq t \leq u_n}$ for $u_n\in\mathbb{R}_+$ and $n\geq2$ , where $L_{u_n}$ has marginal law $\nu$ . We assume that $\nu$ has no continuous singular part and split $\{L_t\}_{0\leq t \leq u_n}$ into n non-overlapping subprocesses.

Definition 3.1. For $m_1,\ldots, m_n >0$ ( $n\geq 2$ ), define the strictly increasing sequence $\{u_{i}\}^{n}_{i=1}$ by $u_{0}=0$ and $u_{i}=u_{i-1}+m_{i}$ for $i=1,\ldots,n$ . Then a process $\{{\xi}_{t}\}_{0\leq t \leq 1}$ is an n-dimensional generalised Liouville process (GLP) if

\begin{equation*}\{{\xi}_t\}_{0\leq t \leq 1} \overset{\textnormal{law}}{=} \{(L_{tm_1}-L_{0},\ldots,L_{tm_i+u_{i-1}}-L_{u_{i-1}},\ldots,L_{tm_n+u_{n-1}}-L_{u_{n-1}})^\top\}_{0\leq t \leq 1}\end{equation*}

for some LRB $\{L_t\}_{0\leq t \leq u_n}$ with generating law $\nu$ . We say that the generating law of $\{{\xi}_{t}\}_{0\leq t \leq 1}$ is $\nu$ and the activity parameter of $\{{\xi}_{t}\}_{0\leq t \leq 1}$ is $\textbf{m}=(m_1,\ldots,m_n)^\top$ .

We have restricted the definition of GLPs to the time horizon [0,1] for convenience. It is straightforward to generalise to an arbitrary closed time horizon. Each coordinate $\{\xi_t^{(i)}\}_{0\leq t \leq 1}$ of $\{{\xi}_t\}_{0\leq t \leq 1}$ is a subprocess of an LRB. Since subprocesses of LRBs are themselves LRBs (see [Reference Hoyle, Hughston and Macrina15]), GLPs form a multivariate generalisation of LRBs. For the rest of the paper, we let $\{{\xi}_t\}_{0\leq t \leq 1}$ be an n-dimensional GLP with generating law $\nu$ , and let $\{L_t\}_{0\leq t \leq u_n}$ be the master process of $\{{\xi}_t\}_{0\leq t \leq 1}$ . In addition, we denote the filtration generated by $\{{\xi}_t\}_{0\leq t \leq 1}$ by $\{\mathcal{F}_t^{{\xi}}\}_{0\leq t \leq 1} \subset \{\mathcal{F}_t\}_{0\leq t \leq 1}$ . Explicitly, we have $\mathcal{F}_t^{{\xi}}=\sigma(\{{\xi}_u\}\colon 0\leq u \leq t)$ .

Proof. See the Appendix. □

In what follows, we define a family of unnormalised measures $\{\theta_t\}_{0\leq t < 1}$ , such that

(3.1) \begin{equation}\theta_0(B;\, x)=\nu(B), \quad\theta_t(B;\, x)=\int_B \dfrac{f_{\textbf{1}\cdot \textbf{m}(1-t)}(z-x)}{f_{\textbf{1}\cdot \textbf{m}}(z)} \, \nu(\textrm{d} z)\end{equation}

for $t\in[0,1)$ , $x\in\mathbb{R}$ and $B\in\mathcal{B}(\mathbb{R})$ . We also write $\Theta_t(x)=\theta_t(\mathbb{R};\, x)$ . We define $R_t$ to be the sum of coordinates of ${\xi}_t$ :

\begin{equation*}R_t=\sum_{i=1}^n \xi^{(i)}_t = \textbf{1}\cdot {\xi}_t.\end{equation*}

Proposition 3.2. The GLP $\{{\xi}_t\}_{0\leq t \leq 1}$ is a Markov process with the transition law given by

\begin{align*} & \mathbb{P}\big(\xi_1^{(1)}\in\textrm{d} z_1,\ldots, \xi_1^{(n-1)}\in\textrm{d} z_{n-1},\xi_1^{(n)}\in B \mid {\xi}_s=\textbf{x} \big)\\* &\quad = \dfrac{\theta_{\tau(s)}\big(B+\sum_{i=1}^{n-1}z_i;\, x_n+\sum_{i=1}^{n-1}z_i\big)}{\Theta_s(\textbf{1}\cdot \textbf{x})}\prod_{i=1}^{n-1}f_{(1-s)m_i}(z_i-x_i)\,\textrm{d} z_i,\end{align*}

and

(3.2) \begin{equation}\mathbb{P}( {\xi}_t\in \,\textrm{d}\textbf{y} \mid {\xi}_s=\textbf{x} )=\dfrac{\Theta_{t}(\textbf{1}\cdot \textbf{y})}{\Theta_s(\textbf{1}\cdot \textbf{x})}\prod_{i=1}^{n}f_{(t-s)m_i}(y_i-x_i) \,\textrm{d} y_i,\end{equation}

where $\textbf{x}, \textbf{y}\in\mathbb{R}^n$ , $\tau(t)=1-m_n(1-t)/(\textbf{1}\cdot \textbf{m})$ , $0\leq s<t<1$ , and $B\in\mathcal{B}(\mathbb{R})$ .

Proof. See the Appendix. □

Remark 3.3. From Proposition 3.2, if the generating law $\nu$ admits a density p, we get a neater transition law to the terminal value, given by

\begin{equation*}\mathbb{P}( {\xi}_1 \in \,\textrm{d}\textbf{z} \mid {\xi}_s=\textbf{x} )=\dfrac{p(\textbf{1}\cdot \textbf{z})}{\Theta_s(\textbf{1}\cdot \textbf{x}) f_{\textbf{1}\cdot \textbf{m}}(\textbf{1}\cdot \textbf{z})}\prod_{i=1}^n f_{(1-s)m_i}(z_i-x_i) \,\textrm{d} z_i.\end{equation*}

Remark 3.4. Our definition of GLPs is somewhat heuristic. A formal definition is possible through a Doob h-transform, since $\{\Theta_{t}\}_{0\leq t < 1}$ is harmonic to a Lévy process $\{\textbf{X}_t\}_{t\geq 0}$ taking values in $\mathbb{R}^n$ with marginal density $g_t(\textbf{x}) = \prod_{i=1}^{n}f_{m_i t}(x_i)$ . To see this, note that we can alternatively write (3.2) as

\[\mathbb{P}( {\xi}_t\in \,\textrm{d}\textbf{y} \mid {\xi}_s=\textbf{x} )=\dfrac{\tilde{\Theta}_{t}(\textbf{y})}{\tilde{\Theta}_s(\textbf{x})}g_{t-s}(\textbf{y}-\textbf{x}) \,\textrm{d} \textbf{y},\]

where $\tilde{\Theta}_{t}(\textbf{x})=\Theta_t(\textbf{1}\cdot \textbf{x})$ , for $0\leq t < 1$ . To see that $\{\tilde{\Theta}_{t}\}_{0\leq t < 1}$ is harmonic to $\{\textbf{X}\}_{0\leq t <1}$ , note that

(3.3) \begin{align}\int_{\mathbb{R}^n}g_{t-s}(\textbf{y}-\textbf{x})\tilde{\Theta}_{t}(\textbf{y}) \,\textrm{d}\textbf{y}&=\int_{\mathbb{R}^n}\prod_{i=1}^{n}f_{(t-s)m_i}(y_i-x_i)\tilde{\Theta}_{t}(\textbf{y}) \,\textrm{d}\textbf{y} \notag \\[2pt]&=\int_{\mathbb{R}}\int_{\mathbb{R}^n}f_{\textbf{1}\cdot \textbf{m} (1-t)}(z-\textbf{1}\cdot\textbf{y})\prod_{i=1}^{n}f_{(t-s)m_i}(y_i-x_i) \,\textrm{d}\textbf{y} \dfrac{\textrm{d}\nu(z)}{f_{\textbf{1}\cdot \textbf{m}}(z)} \notag \\[2pt] &=\int_{\mathbb{R}}f_{\textbf{1}\cdot \textbf{m} (1-s)}(z-\textbf{1}\cdot\textbf{x}) \dfrac{\textrm{d}\nu(z)}{f_{\textbf{1}\cdot\textbf{m}}(z)}\\ &=\tilde{\Theta}_{s}(\textbf{x}) \notag \end{align}

for $0\leq s < t < 1$ , where (3.3) follows from repeated use of the convolution property of $\{f_t\}_{0\leq t\leq \textbf{1}\cdot \textbf{m}}$ .

Remark 3.4 demonstrates that the laws of $\{{\xi}_t\}_{0\leq t <1}$ and $\{\textbf{X}_t\}_{0\leq t <1}$ are equivalent, which we formalise by the corollary below.

Corollary 3.1. Suppose that $\{{\xi}_{t}\}_{t\geq0}$ is a Lévy process under measure $\widetilde{\mathbb{P}}$ with $\widetilde{\mathbb{P}}({\xi}_{t}\in \,\textrm{d} \textbf{x})=g_t(\textbf{x})\,\textrm{d}\textbf{x}$ . Then $\{\Theta_t(R_t)^{-1}\}_{0\leq t <1}$ is a Radon–Nikodým density process that defines the measure change

(3.4) \begin{equation}\dfrac{\textrm{d} \widetilde{\mathbb{P}}}{\textrm{d} \mathbb{P}}\biggr|_{\mathcal{F}_t^{{\xi}}}=\Theta_t(R_t)^{-1} \quad (0\leq t <1),\end{equation}

and $\{{\xi}_t\}_{0\leq t < 1}$ is a $\mathbb{P}$ -GLP with generating law $\nu$ and activity parameter $\textbf{m}$ .

Proof. See the Appendix. □

Proof. For all $s\in[0,1)$ the transition probabilities to ${\xi}_t$ ( $s<t<1$ ) can be computed from (3.2) by first substituting $\nu$ with the Dirac measure $\delta_{\textbf{1}\cdot \textbf{z}}$ in (3.1), yielding

\begin{equation*}\mathbb{P}({\xi}_t \in \textrm{d} \textbf{y} \mid {\xi}_1=\textbf{z}, \mathcal{F}_s^{{\xi}}) = \prod_{i=1}^n \dfrac{f_{m_i(t-s)}\big(y_i-\xi^{(i)}_s\big)f_{m_i(1-t)}(z_i-y_i)}{f_{m_i(1-s)}\big(z_i-\xi^{(i)}_s\big)} \,\textrm{d} y_i\end{equation*}

for almost every $\textbf{z}\in\mathbb{R}^n$ . Conditional on ${\xi}_1=\textbf{z}$ , we see that the transition laws of the coordinates of $\{{\xi}_t\}_{0\leq t \leq 1}$ are independent, and that each is the transition law of a Lévy bridge. □

Using the Markov property of $\{{\xi}_t\}_{0\leq t \leq 1}$ , we can also provide the conditional laws of the coordinates $\xi^{(i)}_t$ given $\mathcal{F}_s^{\xi^{(i)}}=\sigma(\{\xi^{(i)}_u\}\colon 0\leq u \leq s)$ or given $\mathcal{F}_s^{{\xi}}$ , for $s<t$ .

Proof. See the Appendix. □

Proposition 3.5. The process $\{R_t\}_{0\leq t \leq 1}$ is an LRB with generating law $\nu$ and the transition law

(3.5) \begin{align}\mathbb{P}(R_1\in \textrm{d} r\mid {\xi}_s=\textbf{x}) &= \dfrac{\theta_{s}(\textrm{d} r;\, \textbf{1}\cdot \textbf{x})}{\Theta_{s}(\textbf{1}\cdot \textbf{x})}, \end{align}

(3.6) \begin{align}\mathbb{P}(R_t\in \textrm{d} r\mid {\xi}_s=\textbf{x})&= \dfrac{\Theta_{t}(r)}{\Theta_{s}(\textbf{1}\cdot \textbf{x})}f_{(t-s)\textbf{1}\cdot \textbf{m}}(r-\textbf{1}\cdot \textbf{x})\,\textrm{d} r. \end{align}

Proof. See the Appendix. □

The next statement is a key result for defining stochastic integrals of integrable LRBs, and hence integrable GLPs.

Proposition 3.6. If $\mathbb{E}(|R_t|)<\infty$ for all $t\in[0,1]$ , then $\{R_t\}_{0\leq t < 1}$ admits the canonical semimartingale representation

(3.7) \begin{equation}R_t = \int_{0}^{t}\dfrac{\mathbb{E}( R_{1} \mid \mathcal{F}^{{\xi}}_{s}) - R_{s}}{1-s}\,\textrm{d} s + M_t\end{equation}

for $0\leq t < 1$ , where $\{M_t\}_{0\leq t < 1}$ is an $(\mathcal{F}_t^{{\xi}},\mathbb{P})$ -martingale with initial state $M_0=0$ .

Proof. From Proposition 3.5, $\{R_t\}_{0\leq t \leq 1}$ is an LRB. Hence, if $\mathbb{E}(|R_t|)<\infty$ for all $t\in(0,1]$ , then

(3.8) \begin{equation}\mathbb{E}(R_t \mid {\xi}_{s} = \textbf{x}) = \dfrac{1-t}{1-s}\textbf{1}\cdot \textbf{x} + \dfrac{t-s}{1-s}\mathbb{E}(R_1 \mid {\xi}_{s} = \textbf{x} ), \quad s\in[0,t).\end{equation}

We shall use (3.8) to prove that $\{M_t\}_{0\leq t < 1}$ given in (3.7) is an $\mathcal{F}_t^{{\xi}}$ -martingale. Since $\{{\xi}_t\}_{0\leq t \leq 1}$ is Markov,

\begin{align*}\mathbb{E}(M_t - M_s \mid \mathcal{F}_s^{{\xi}}) &= \mathbb{E}(R_t - R_s \mid \mathcal{F}_s^{{\xi}}) - \int_s^t \dfrac{\mathbb{E}(R_1 \mid {\xi}_s) - \mathbb{E}(R_u \mid {\xi}_s)}{1-u} \,\textrm{d} u \notag \\[2pt]&= \dfrac{1-t}{1-s}\textbf{1}\cdot {\xi}_{s} + \dfrac{t-s}{1-s}\mathbb{E}(R_1 \mid {\xi}_{s}) - R_{s} - \int_s^t \dfrac{\mathbb{E}(R_1 \mid {\xi}_s)}{1-u} \,\textrm{d} u \notag \\[2pt]&\quad\, + \int_s^t \dfrac{1}{1-u}\biggl(\dfrac{1-u}{1-s}\textbf{1}\cdot {\xi}_{s} + \dfrac{u-s}{1-s}\mathbb{E}(R_1 \mid {\xi}_s)\biggr) \,\textrm{d} u \notag \\[2pt]&=0\end{align*}

for $0\leq s <t \leq 1$ . Given $\mathbb{E}(|R_t|)<\infty$ ,

\[\mathbb{E}\biggl(\int_0^t\frac{|\mathbb{E}(R_1 \mid {\xi}_s) - R_s|}{1-s} \,\textrm{d} s\biggr) <\infty\quad \text{for $0\leq t < 1$}\]

remains to be shown:

\begin{align*}\mathbb{E}\biggl(\int_0^t \dfrac{|\mathbb{E}(R_1 \mid {\xi}_s) - R_s|}{1-s} \,\textrm{d} s\biggr) &\leq \mathbb{E}\biggl(\int_0^t \dfrac{|\mathbb{E}(R_1 \mid {\xi}_s)|}{1-s} \,\textrm{d} s\biggr) + \mathbb{E}\biggl(\int_0^t \dfrac{|R_s|}{1-s} \,\textrm{d} s\biggr) \notag \\[2pt]&= \int_0^t \mathbb{E}\biggl(\dfrac{|\mathbb{E}(R_1 \mid {\xi}_s)|}{1-s}\biggr) \,\textrm{d} s + \int_0^t \mathbb{E}\biggl(\dfrac{|R_s|}{1-s} \biggr)\,\textrm{d} s \notag \\[2pt]& < \infty,\end{align*}

since $\{\mathbb{E}(R_1 \mid \mathcal{F}^{{\xi}}_t)\}_{0\leq t < 1}$ is a martingale. Hence $\mathbb{E}(|M_t|)<\infty$ for $0\leq t < 1$ . Finally, $M_0=0$ since $R_0=0$ . □

Remark 3.6. Let $\alpha_t = (1-t)^{-1}$ and $\beta_t = \mathbb{E}(R_{1} \mid {\xi}_{t})$ . Then

\begin{equation*}\textrm{d} R_t = \alpha_t(\beta_t - R_t )\,\textrm{d} t + \textrm{d} M_t\end{equation*}

for $0\leq t < 1$ . In this form, the dynamics of LRBs resemble those of an Ornstein–Uhlenbeck process, with an increasing mean-reversion rate $\{\alpha_t\}_{0\leq t < 1}$ and a state-dependent reversion level $\{\beta_t\}_{0\leq t < 1}$ . We can write

\begin{equation*}R_t = \int_{0}^{t}\dfrac{1-t}{(1-s)^2}\mathbb{E}( R_{1} \mid {\xi}_{s})\,\textrm{d} s + \int_{0}^{t} \dfrac{1-t}{1-s}\,\textrm{d} M_s \quad \text{for $0\leq s < t < 1$}.\end{equation*}

The following two propositions are motivated by [Reference Mansuy and Yor18]. We first recall that a measurable process $\{H_t\}_{t\geq0}$ is called a harness if, for all $t\geq 0$ , $\mathbb{E}(|H_t|)<\infty$ and for all $0\leq a<b<c<d$ ,

\begin{equation*}\mathbb{E}\biggl( \dfrac{H_c - H_b}{c-b} \mid \mathcal{H}_{a,d} \biggr) = \dfrac{H_d - H_a}{d-a},\end{equation*}

where $\mathcal{H}_{a,d}= \sigma(\{H_t\}_{t\leq a}, \{H_t\}_{t\geq d})$ .

Proof. See the Appendix. □

Proposition 3.8. Let $\varphi$ be a $C^1$ -function. If $\mathbb{E}(|R_t|)<\infty$ for all $t\in(0,1]$ , then the stochastic process $\{Z_t\}_{0\leq t < 1}$ defined by

\begin{equation*}Z_{t} = \dfrac{\mathbb{E}( R_{1} \mid {\xi}_t) - R_t}{1-t}\int_t^1 \varphi(u)\,\textrm{d} u + \int_0^t \varphi(u)\,\textrm{d} R_u \quad (0\leq t <1)\end{equation*}

is an $(\mathcal{F}_{t}^{{\xi}},\mathbb{P})$ -martingale.

Proof. We have

\begin{align*}\mathbb{E}\biggl( \int_t^1 \varphi(u)\,\textrm{d} R_u \mid \mathcal{F}_t^{{\xi}} \biggr) &=\varphi(1)\,\mathbb{E}( R_{1} \mid \mathcal{F}_t^{{\xi}}) - \varphi(t)R_t - \int_t^1\mathbb{E}(R_u \mid \mathcal{F}_t^{{\xi}})\,\textrm{d} \varphi(u) \notag \\*&= \dfrac{\mathbb{E}( R_{1} \mid {\xi}_t) - R_t}{1-t}\int_t^1 \varphi(u)\,\textrm{d} u,\end{align*}

from the integration-by-parts formula, (3.8), and the Markov property of $\{{\xi}_t\}_{0\leq t \leq 1}$ . Hence $Z_t = \mathbb{E}(\int_0^1 \varphi(u)\,\textrm{d} R_u \mid \mathcal{F}_t^{{\xi}})$ , which is an $(\mathcal{F}_{t}^{{\xi}},\mathbb{P})$ -martingale. □

Similar to Proposition 3.6, we have the following result (we omit the proof to avoid repetition).

Proposition 3.9. If $\mathbb{E}(|{\xi}_t|)<\infty$ for all $t\in(0,1]$ , then $\{{\xi}_t\}_{0\leq t < 1}$ admits the canonical semimartingale representation

(3.9) \begin{equation}{\xi}_t = \int_{0}^{t}\dfrac{\mathbb{E}( {\xi}_{1} \mid \mathcal{F}^{{\xi}}_{s}) - {\xi}_{s}}{1-s}\,\textrm{d} s + \textbf{M}_t \quad (0\leq t < 1),\end{equation}

where $\{\textbf{M}_t\}_{0\leq t < 1}$ is an $(\mathcal{F}_t^{{\xi}},\mathbb{P})$ -martingale.

In [Reference Jakubowski and Pytel17] it is shown that Archimedean survival processes satisfy the weak Markov consistency condition, but not necessarily the strong Markov consistency condition. Motivated by this, Proposition 3.10 below provides a generalised version of this result for GLPs. First we recall the weak and strong Markov consistency conditions. Let $\{\textbf{X}_t\}_{t\geq0}$ be an n-dimensional real-valued Markov process and let $\mathcal{F}_t^{\textbf{X}}=\sigma(\{\textbf{X}_u\}\colon 0\leq u \leq t)$ . Also, for each coordinate process $\{X^{(i)}_t\}_{t\geq 0}$ , $i=1,\ldots,n$ , write $\mathcal{F}_t^{X^{(i)}}=\sigma(\{X^{(i)}_u\}\colon 0\leq u \leq t)\subset \mathcal{F}_t^{\textbf{X}}$ . The process $\{\textbf{X}_t\}_{t\geq0}$ satisfies the weak Markov consistency condition if

(3.10) \begin{equation}\mathbb{P}\big(X^{(i)}_t \in B \mid \mathcal{F}_s^{X^{(i)}} \big) = \mathbb{P}\big( X^{(i)}_t \in B \mid X^{(i)}_s \big)\end{equation}

for every $i=1,\ldots,n$ and every $B\in\mathcal{B}(\mathbb{R})$ . Further, $\{\textbf{X}_t\}_{t\geq 0}$ satisfies the strong Markov consistency condition if

(3.11) \begin{equation}\mathbb{P}\big( X^{(i)}_t \in B \mid \mathcal{F}_s^{\textbf{X}} \big) = \mathbb{P}\big( X^{(i)}_t \in B \mid X^{(i)}_s \big)\end{equation}

for every $i=1,\ldots,n$ and every $B\in\mathcal{B}(\mathbb{R})$ .

Proof. Each coordinate $\{\xi^{(i)}_t\}_{0\leq t \leq 1}$ of the GLP $\{{\xi}_t\}_{0\leq t \leq 1}$ is an LRB, since every subprocess of an LRB is an LRB (see [Reference Hoyle, Hughston and Macrina15]). Thus (3.10) is satisfied for every $i=1,\ldots,n$ , every $B\in\mathcal{B}(\mathbb{R})$ , and all $0\leq s < t \leq 1$ . However, (3.11) does not necessarily hold since

\begin{align*}\mathbb{P}\big( \xi^{(i)}_t -\xi^{(i)}_s \in \textrm{d} y \mid \mathcal{F}_s^{{\xi}} \big) = \mathbb{P}\biggl( \xi^{(i)}_t -\xi^{(i)}_s \in \textrm{d} y \mid \sum_j \xi^{(j)}_s \biggr)\\[-15pt]\end{align*}

is only equal to $\mathbb{P}( \xi^{(i)}_t -\xi^{(i)}_s \in \textrm{d} y \mid \xi^{(i)}_s )$ if both $\sum_j \xi^{(j)}_s$ and $\xi^{(i)}_s$ are independent from the increment $\xi^{(i)}_t -\xi^{(i)}_s$ for all $0\leq s < t \leq 1$ . In such a case the coordinates of $\{{\xi}_t\}_{0\leq t \leq 1}$ are independent Lévy processes. □

In the same spirit we shall introduce weak and strong semimartingale consistency conditions. Definition 3.2 below goes beyond a Markov setting, but in the context of GLPs it offers links to Markov consistency.

Proof. Let $\mathbb{E}(|{\xi}_t|)<\infty$ for all $t\in(0,1]$ and define $\alpha_t=(1-t)^{-1}$ for $t<1$ . Following similar steps to the proof of Proposition 3.6, each coordinate of $\{{\xi}_{t}\}_{0\leq t < 1}$ admits a decomposition $\xi^{(i)}_t = a^{(i)}_t + m^{(i)}_t$ , where

\begin{equation*}a^{(i)}_t = \int_{0}^{t}\alpha_s(\mathbb{E}( \xi^{(i)}_1 \mid \mathcal{F}^{\xi^{(i)}}_{s}) - \xi^{(i)}_s)\,\textrm{d} s,\end{equation*}

which is $\{\mathcal{F}_t^{\xi^{(i)}}\}$ -adapted, and $\{m^{(i)}_t\}_{0\leq t < 1}$ is an $(\mathcal{F}_t^{\xi^{(i)}},\mathbb{P})$ -martingale, for $i=1,\ldots,n$ . Hence $\{{\xi}_{t}\}_{0\leq t < 1}$ is weak semimartingale consistent. From Proposition 3.9, we also know that $\xi^{(i)}_t=A^{(i)}_t+M^{(i)}_t$ , where

\begin{equation*}A^{(i)}_t = \int_{0}^{t}\alpha_s(\mathbb{E}( \xi^{(i)}_1 \mid \mathcal{F}^{{\xi}}_{s}) - \xi^{(i)}_s)\,\textrm{d} s,\end{equation*}

which is $\{\mathcal{F}_t^{{\xi}}\}$ -adapted, and $\{M_t^{(i)}\}_{0\leq t < 1}$ is an $(\mathcal{F}_t^{{\xi}},\mathbb{P})$ -martingale. Since $\{{\xi}_{t}\}_{0\leq t \leq 1}$ is Markov and using Proposition 3.10, we know that $\mathbb{E}( \xi^{(i)}_1 \mid {\xi}_{t})$ is not necessarily equal to $\mathbb{E}( \xi^{(i)}_1 \mid \xi^{(i)}_{t})$ . Hence $\{A^{(i)}_t\}_{0\leq t < 1}$ is not necessarily $\{\mathcal{F}_t^{\xi^{(i)}}\}$ -adapted. Also, $\{M_t^{(i)}\}_{0\leq t < 1}$ is not necessarily an $(\mathcal{F}_t^{\xi^{(i)}},\mathbb{P})$ -martingale. □

We used Proposition 3.10 to prove Proposition 3.11; we shall note another link between semimartingale consistency and Markov consistency. From [Reference Bielecki, Jakubowski and Nieweglowski1], if a Markov process $\{\textbf{X}_t\}_{t\geq0}$ satisfies the weak Markov consistency with respect to its marginal $\{X^{(i)}_t\}_{t\geq0}$ , then $\{\textbf{X}_t\}_{t\geq0}$ is also strongly Markov consistent with respect to $\{X^{(i)}_t\}_{t\geq0}$ if and only if $\{\mathcal{F}_t^{X^{(i)}}\}_{t\geq0}$ is $\mathbb{P}$ -immersed in $\{\mathcal{F}_t^{\textbf{X}}\}_{t\geq0}$ . Here $\mathbb{P}$ -immersion means that if $\{X^{(i)}_t\}_{t\geq0}$ is an $(\mathcal{F}_t^{X^{(i)}},\mathbb{P})$ -local martingale, then it is an $(\mathcal{F}_t^{\textbf{X}},\mathbb{P})$ -local martingale. As an opposite direction to $\mathbb{P}$ -immersion, we prove a result that links strong martingale consistency and strong Markov consistency.

Proof. Since $\{\boldsymbol{S}_t\}_{t\geq 0}$ is an $(\mathcal{F}_t^{\boldsymbol{S}},\mathbb{P})$ -martingale, we have $\mathbb{E}( S^{(i)}_t \mid \mathcal{F}^{\boldsymbol{S}}_{u})=S^{(i)}_u$ for $0\leq u < t$ . Then, if $\{\boldsymbol{S}_t\}_{t\geq0}$ is strong martingale consistent, we have $\mathbb{E}( S^{(i)}_t \mid \mathcal{F}^{\boldsymbol{S}}_{u})=\mathbb{E}( S^{(i)}_t \mid \mathcal{F}_u^{S^{(i)}})=S^{(i)}_u$ . Thus, given that $\{\boldsymbol{S}_t\}_{t\geq 0}$ is Markovian satisfying weak Markov consistency,

\begin{align*} \mathbb{E}( S^{(i)}_t \mid \mathcal{F}^{\boldsymbol{S}}_{u}) & = \int_{\mathbb{R}}x\mathbb{P}( S^{(i)}_t \in \textrm{d} x \mid \mathcal{F}^{\boldsymbol{S}}_{u} )\\[3pt] &=\int_{\mathbb{R}}x\mathbb{P}( S^{(i)}_t \in \textrm{d} x \mid \boldsymbol{S}_u ) \\[3pt] & =\mathbb{E}( S^{(i)}_t \mid \mathcal{F}_u^{S^{(i)}})\\[3pt] &= \int_{\mathbb{R}}x\mathbb{P}( S^{(i)}_t \in \textrm{d} x \mid \mathcal{F}_u^{S^{(i)}} )\\[3pt] &=\int_{\mathbb{R}}x\mathbb{P}( S^{(i)}_t \in \textrm{d} x \mid S^{(i)}_u ).\end{align*}

For the opposite direction, if $\{\boldsymbol{S}_t\}_{t\geq0}$ is strong Markov consistent, then since $\{\boldsymbol{S}_t\}_{t\geq 0}$ is an $(\mathcal{F}_t^{\boldsymbol{S}},\mathbb{P})$ -martingale satisfying weak Markov consistency,

\begin{align*}\int_{\mathbb{R}}x\mathbb{P}( S^{(i)}_t \in \textrm{d} x \mid \mathcal{F}^{\boldsymbol{S}}_{u} ) & =\mathbb{E}( S^{(i)}_t \mid \mathcal{F}^{\boldsymbol{S}}_{u}) =S^{(i)}_u \\[3pt] &=\int_{\mathbb{R}}x\mathbb{P}( S^{(i)}_t \in \textrm{d} x \mid S^{(i)}_u ) \\[3pt] &= \int_{\mathbb{R}}x\mathbb{P}( S^{(i)}_t \in \textrm{d} x \mid \mathcal{F}_u^{S^{(i)}} ) \\[3pt] &= \mathbb{E}( S^{(i)}_t \mid \mathcal{F}_u^{S^{(i)}}).\end{align*}

Hence $\{S_t^{(i)}\}_{t\geq0}$ is also an $(\mathcal{F}_t^{S^{(i)}},\mathbb{P})$ -martingale, and the statement follows. □

4. Examples

We shall now study two examples of GLPs in more detail: Brownian Liouville processes and Poisson Liouville processes.

4.1. Brownian Liouville processes

As a subclass of GLPs, let us consider what we call Brownian Liouville processes (BLPs). In Definition 3.1, we let $\{L_t\}_{0\leq t \leq u_n}$ be a Brownian random bridge given by

(4.1) \begin{equation}L_t = \dfrac{t}{\textbf{1}\cdot \textbf{m}}L_{\textbf{1}\cdot \textbf{m}}+\sigma \biggl(W_t-\dfrac{t}{\textbf{1}\cdot \textbf{m}}W_{\textbf{1}\cdot \textbf{m}}\biggr),\end{equation}

where $\sigma>0$ and $\{W_t\}_{0\leq t \leq u_n}$ is a standard Brownian motion independent of the random variable $L_{\textbf{1}\cdot \textbf{m}}$ . For a background of the anticipative orthogonal representation given in (4.1) for a Brownian random bridge, we refer the reader to [Reference Brody and Hughston2] and [Reference Brody, Hughston and Macrina6]. We also note that the Gaussian process $\{W_t-({{t}/{\textbf{1}}\cdot \textbf{m}})W_{\textbf{1}\cdot \textbf{m}}\}_{0\leq t \leq \textbf{1}\cdot \textbf{m}}$ in (4.1) is a Brownian bridge starting and ending at zero. The following proposition is analogous to [Reference Hoyle and Menguturk14, Proposition 3.10] for Archimedean survival processes. We denote the Hadamard (i.e. element-wise) product of vectors $\textbf{x}, \textbf{y} \in \mathbb{R}^n$ by $\textbf{x} \circ \textbf{y}$ . We say $\{\beta_t\}_{0\leq t \leq 1}$ is a standard Brownian bridge if (a) it is a Brownian bridge, (b) $\beta_0=\beta_1=0$ , and (c) $\textrm{Var}(\beta_t)=(1-t)^2$ .

Proposition 4.1. If $\{{\xi}_t\}_{0\leq t \leq 1}$ is a BLP with the master process (4.1), then it admits the independent Brownian bridge representation

(4.2) \begin{equation}{\xi}_t = t\biggl(\dfrac{\textbf{m}}{\textbf{1}\cdot \textbf{m}} R_1 + \sigma \textbf{Z} \biggr) + \sigma \sqrt{\textbf{m}} \circ \boldsymbol{\beta}_t,\end{equation}

where $\sqrt{\textbf{m}}=(\sqrt{m_1},\ldots,\sqrt{m_n})^\top$ , $\{\boldsymbol{\beta}_t\}$ is a vector of independent standard Brownian bridges, and the random vector $\textbf{Z} = (Z_1,\ldots,Z_n)^{\top}$ is multivariate Gaussian with

\begin{equation*} \textrm{Cov}(Z_i, Z_j) = \delta_{ij}m_i - \dfrac{m_im_j}{\textbf{1}\cdot \textbf{m}}.\end{equation*}

Proof. For the proof we use $W_t$ and W(t) interchangeably. We have

\begin{align*}\xi^{(i)}_t &= L(m_it-u_{i-1}) - L(u_{i-1}) \notag \\*&=\dfrac{m_it}{\textbf{1}\cdot \textbf{m}}L(\textbf{1}\cdot \textbf{m}) + \sigma\biggl( W(m_it+u_{i-1}) - W(u_{i-1})-\dfrac{m_it}{\textbf{1}\cdot \textbf{m}}W(\textbf{1}\cdot \textbf{m}) \biggr) \notag \\*&=\dfrac{m_it}{\textbf{1}\cdot \textbf{m}}R_1 + \sigma\sqrt{m_i}\beta^{(i)}_t + \sigma t\biggl( W(u_{i}) - W(u_{i-1})-\dfrac{m_i}{\textbf{1}\cdot \textbf{m}}W(\textbf{1}\cdot \textbf{m}) \biggr)\end{align*}

since $R_1=L_{\textbf{1}\cdot \textbf{m}}$ , and where

\begin{equation*}\sqrt{m_i}\beta^{(i)}_t = W(m_it + u_{i-1}) - W(u_{i-1}) - t(W(u_i) - W(u_{i-1})).\end{equation*}

So $\{\beta^{(i)}_t\}_{0\leq t \leq 1}$ is a standard Brownian bridge, and is independent of $W(u_i) - W(u_{i-1})$ and $W(\textbf{1}\cdot \textbf{m})$ . It is straightforward to verify the independence by noting that they are jointly Gaussian with nil covariation. Thus we can write

\begin{equation*}\xi^{(i)}_t = t \biggl(\dfrac{m_i}{\textbf{1}\cdot \textbf{m}}R_1 + \sigma Z_i\biggr)+ \sigma\sqrt{m_i}\beta^{(i)}_t,\end{equation*}

where $Z_i$ is given by

(4.3) \begin{equation}Z_i = W(u_{i}) - W(u_{i-1})-\dfrac{m_i}{\textbf{1}\cdot \textbf{m}}W(\textbf{1}\cdot \textbf{m}),\end{equation}

and $R_1$ , $Z_i$ and $\{\beta^{(i)}_t\}_{0\leq t \leq 1}$ are mutually independent. Further, $\{\beta^{(1)}_t\}_{0\leq t \leq 1},\, {\ldots}\,, \{\beta^{(n)}_t\}_{0\leq t \leq 1}$ are mutually independent, since they are jointly Gaussian with nil covariation. □

Proposition 4.1 provides an anticipative orthogonal representation for BLPs, whereas (3.9) provides a non-anticipative semimartingale representation when $\{{\xi}_t\}_{0\leq t \leq 1}$ is a BLP.

Remark 4.1. Note that $\textbf{1}\cdot \textbf{Z}=0$ from (4.3), and so its covariance matrix is singular. Also, using (4.2) from Proposition 4.1, $\{R_t\}_{0\leq t \leq 1}$ admits the anticipative representation

\begin{equation*} R_t = \textbf{1}\cdot {\xi}_t = t\textbf{1}\cdot\biggl(\dfrac{\textbf{m}}{\textbf{1}\cdot \textbf{m}} R_1 + \sigma \textbf{Z}\biggr) + \sigma\textbf{1}\cdot(\sqrt{\textbf{m}} \circ \boldsymbol{\beta}_t)= tR_1 + \sigma\sqrt{\textbf{1}\cdot \textbf{m}}\tilde{\beta}_t,\end{equation*}

where $\{\tilde{\beta}_t\}_{0\leq t \leq 1}$ is a standard Brownian bridge.

Proposition 4.2. Let $\pi_0(\textrm{d} \textbf{x}) = \mathbb{P}({\xi}_{1} \in \textrm{d} \textbf{x})$ and $\pi_t(\textrm{d} \textbf{x}) = \mathbb{P}({\xi}_{1} \in \textrm{d} \textbf{x} \mid \mathcal{F}_t^{{\xi}})$ . Also, let $\{\textbf{B}_t\}_{0\leq t < 1}$ be a vector of standard $(\mathcal{F}_t^{{\xi}},\mathbb{P})$ -Brownian motions. Then, if $\mathbb{E}(|{\xi}_{1}|)<\infty$ , the multivariate measure-valued process $\{\pi_t\}_{0\leq t < 1}$ satisfies

\begin{equation*}\pi_t(\textrm{d} \textbf{x}) = \pi_0(\textrm{d} \textbf{x}) + \int_{0}^t \pi_s(\textrm{d} \textbf{x}) \boldsymbol{\sigma}_s^\top(\sigma\sqrt{\textbf{m}}\circ\,\textrm{d} \textbf{B}_s)\end{equation*}

for $0\leq t < 1$ , where each coordinate of $\{\boldsymbol{\sigma}_t\}_{0\leq t < 1}$ is given by

\begin{equation*}\sigma_t^{(i)} = \dfrac{x^{(i)}-\mathbb{E}(\xi_1^{(i)}\mid {\xi}_t)}{\sigma^2 m_i (1-t)}.\end{equation*}

Proof. See the Appendix. □

Remark 4.2. Note that $\textbf{1}\cdot\tilde{\textbf{B}}_t=\textbf{1}\cdot(\sigma \sqrt{\textbf{m}}\circ\textbf{B}_t)$ gives the non-anticipative semimartingale representation for $\{R_t\}_{0\leq t < 1}$ , which is

\begin{equation*}R_t = \int_{0}^{t}\dfrac{\mathbb{E}( R_{1} \mid {\xi}_{s}) - R_{s}}{1-s}\,\textrm{d} s + \sigma\sum_{i=1}^n \sqrt{m_i}B_t^{(i)},\end{equation*}

which provides the explicit example of the $(\mathcal{F}_t^{{\xi}},\mathbb{P})$ -martingale in (3.7).

4.2. Poisson Liouville processes

Our second example is that of Poisson Liouville processes (PLPs), which are counting processes. Accordingly, in Definition 3.1, we let $\{L_t\}_{0\leq t \leq u_n}$ be a Poisson random bridge with $\mathbb{P}(L_{u_n}=i)=\nu(\{i\})$ , for $i\in\mathbb{N}_0$ .

Proposition 4.3. Let $\{\lambda_t^R\}_{0\leq t \leq 1}$ be the intensity process of the $L_1$ -norm process $\{R_t\}_{0\leq t \leq 1}$ . If $\mathbb{E}(R_1)<\infty$ , then

(4.4) \begin{equation}\lambda^R_t = \dfrac{\mathbb{E}(R_1 \mid {\xi}_{t}) - R_t}{1-t} \quad for\ \text{$0\leq t < 1$}.\end{equation}

Proof. Since $\{R_t\}_{0\leq t \leq 1}$ is a counting process, we have $\lambda^R_t = \lim_{h\rightarrow 0}\mathbb{E}(R_{t+h} - R_{t} \mid \mathcal{F}_t^{{\xi}})/h$ . Since $\{R_t\}_{0\leq t \leq 1}$ is a Markov process with respect to $\{\mathcal{F}_t^{{\xi}}\}_{0\leq t \leq 1}$ , using (3.8), we have

\begin{align*}\lambda^R_t &= \lim_{h\rightarrow 0} \biggl(\dfrac{\mathbb{E}(R_{t+h} \mid {\xi}_t)}{h} -\dfrac{R_t}{h}\biggr) \notag \\*&= \lim_{h\rightarrow 0} \biggl(\dfrac{1-t-h}{(1-t)h}\textbf{1}\cdot {\xi}_t + \dfrac{h\mathbb{E}(R_{1} \mid {\xi}_t)}{(1-t)h} -\dfrac{R_t}{h}\biggr) \notag\\&= \dfrac{\mathbb{E}(R_{1} \mid {\xi}_t)}{(1-t)} + \lim_{h\rightarrow 0} \biggl(R_t\biggl(\dfrac{1-t-h}{(1-t)h} -\dfrac{1}{h}\biggr)\biggr) \notag \\*&= \dfrac{\mathbb{E}(R_{1} \mid {\xi}_t)}{1-t} - \lim_{h\rightarrow 0} \biggl(R_t\dfrac{h}{(1-t)h}\biggr),\end{align*}

which yields the result. □

Remark 4.3. When $\{{\xi}_t\}_{0\leq t \leq 1}$ is a PLP, Proposition 4.3 provides an alternative proof for Proposition 3.6, since $\{R_t\}_{0\leq t \leq 1}$ is a counting process.

Proposition 4.4. Let $\{\lambda_t^{(i)}\}_{0\leq t \leq 1}$ be the intensity process of the coordinate $\{\xi^{(i)}_t\}_{0\leq t \leq 1}$ . If $\mathbb{E}(R_1)<\infty$ , then

\begin{equation*}\lambda^{(i)}_t = \dfrac{m_i}{\textbf{1}\cdot \textbf{m}}\lambda^R_t \quad \text{for $0\leq t < 1$.}\end{equation*}

Proof. Fix $0\leq s < t < 1$ . From Remark 3.5, we know that, given ${\xi}_s$ , the increment ${\xi}_t - {\xi}_s$ has a generalised multivariate Liouville distribution with generating law

\[ \nu^*(\{i\})=\nu_{st}(\{i+R_s\}) \]

for $i\in\mathbb{N}_0$ and parameter vector $\textbf{m}(t-s)$ . We define

\begin{equation*} \mu_{st} = \sum_{i=0}^\infty i\, \nu^*(\{i\}) = \sum_{i=R_s}^{\infty} i \, \nu_{st}(\{i\}) - R_s = \mathbb{E}(R_t \mid {\xi}_{s}) - R_s= \dfrac{t-s}{1-s}(\mathbb{E}(R_1 \mid R_{s}) - R_s),\end{equation*}

where the last equality comes from (3.8). Then, from [Reference Fang, Kotz and Ng8, Theorem 6.3] and [Reference Gupta and Richards13], we have

\begin{equation*}\mathbb{E}(\xi^{(i)}_t \mid {\xi}_{s}) = \dfrac{m_i}{\textbf{1}\cdot \textbf{m}}\mu_{st} + \xi^{(i)}_s.\end{equation*}

Since $\{\xi^{(i)}_t\}_{0\leq t \leq 1}$ is a counting process, we have

\[\lambda^{(i)}_t = \lim_{h\rightarrow 0}\mathbb{E}[\xi^{(i)}_{t+h} - \xi^{(i)}_{t} \mid \mathcal{F}_t^{{\xi}}]/h.\]

Thus

\begin{equation*}\lambda^{(i)}_t = \dfrac{m_i}{\textbf{1}\cdot \textbf{m}}\lim_{h\rightarrow 0}\dfrac{\mu_{t,t+h}}{h} = \dfrac{m_i}{\textbf{1}\cdot \textbf{m}}\dfrac{\mathbb{E}(R_1 \mid R_{t}) - R_t}{1-t}.\end{equation*}

The result then follows from (4.4). □

Remark 4.4. Note that $\lambda^{R}_t = \sum_i\lambda^{(i)}_t$ .

If we let $\{P_t\}_{0\leq t \leq 1}$ denote a Poisson process, and define $\boldsymbol{\Delta}$ by $\Delta_i = P_{t_i}-P_{t_{i-1}}$ for some partition $0=t_0<t_1<\cdots<t_n$ , we have

\begin{align*}\mathbb{P}(\boldsymbol{\Delta}=\textbf{x}\mid P_{t_n}=k) = \begin{cases} k! \prod_{i=1}^n {{p_i^{x_i}}/{x_i!}} &\text{for $\textbf{x}\in\mathbb{N}_0^n$ with $\textbf{1}\cdot \textbf{x}=k$,}\\* 0 &\text{otherwise,}\end{cases}\end{align*}

where $p_i=(t_i-t_{i-1})/t_n$ . In other words, given $P_{t_n}$ , $\boldsymbol{\Delta}$ has a multinomial distribution. Let $\{{\xi}_t\}_{0\leq t \leq 1}$ be a Poisson Liouville process with generating law $\nu(\{k\})=A(k)$ , $k\in\mathbb{N}_0$ . Then

\begin{equation*}\mathbb{P}({\xi}_1=\textbf{x}) = A(\textbf{1}\cdot \textbf{x}) (\textbf{1}\cdot \textbf{x})! \prod_{i=1}^n \dfrac{p_i^{x_i}}{x_i!}.\end{equation*}

Write $G_\nu$ for the probability generating function of $\nu$ :

\begin{equation*}G_{\nu}(z)=\sum_{k=0}^{\infty}z^kA(k).\end{equation*}

Let $T^{(i)}$ be the time of the first jump of the ith marginal process. If $\xi^{(i)}_1\,{<}\,1$ , then we set $T^{(i)}\,{=}\,\infty$ .

Proposition 4.5. The random times $\{T^{(i)};\, i=1,\ldots,n\}$ satisfy the following:

\begin{align*}\mathbb{P}(T^{(i)}>s) &= G_{\nu}(1-s p_i), \notag \\*\mathbb{P}(T^{(i)}=\infty) &= G_{\nu}(1-p_i), \notag \\\mathbb{P}(T^{(i)}> s_i;\, i=1,\ldots,n)&=G_{\nu}\biggl( 1-\sum_{i=1}^n p_i s_i \biggr), \notag \\*\mathbb{P}(T^{(i)}=\infty;\, i=1,\ldots,n) &= A(0)\end{align*}

for $s\in[0,1]$ and $\textbf{s}\in [0,1]^n$ .

Proof. See the Appendix. □

Here $G_\nu$ is increasing (and invertible) on [0,1]. Write $\psi(x)=G_\nu(1-x)$ , and note that $\psi$ is invertible on [0,1]. If $u_i\in[0,\psi(p_i)]$ , then we have

\begin{equation*}\mathbb{P}\biggl(T^{(i)}> \dfrac{\psi^{-1}(u_i)}{p_i}\biggr) = u_i.\end{equation*}

It follows that the conditioned random variable $\psi(p_i T^{(i)}) \mid \{T^{(i)} < 1\}$ is uniformly distributed. Furthermore

\begin{equation*}\mathbb{P}\biggl(T^{(i)}> \dfrac{\psi^{-1}(u_i)}{p_i};\, i=1,\ldots,n\biggr) = \psi\biggl(\sum_i \psi^{-1}(u_i)\biggr).\end{equation*}

The form of the joint survival function of $\textbf{T}=\{T^{(i)};\, i=1,\ldots,n\}$ resembles that of an Archimedean copula. However, the fact that $\mathbb{P}(T^{(i)}\geq 1)>0$ means that it is not an Archimedean copula.