Exact Simulation of the Jump Times of a Class of Piecewise Deterministic Markov Processes

Abstract

In this paper, we are interested in the exact simulation of a class of piecewise deterministic Markov processes. We show how to perform an efficient thinning algorithm depending on the jump rate bound. For different types of bounds, we compare theoretically the efficiency of the algorithm (measured by the mean ratio between the total number of jump times generated by thinning and the number of selected ones) and we compare numerically the computation times. We use the thinning algorithm on Hodgkin–Huxley models with Markovian ion channels dynamics to illustrate our results.

Appendix

In this section we compute the rate of acceptance for the thinning of Poisson processes. Let N and $\tilde{N}$ be two Poisson processes with jump rate $\lambda$ and $\tilde{\lambda }$ respectively and jump times $(T_{n})_{n\ge 1}$ and $(\tilde{T}_{n})_{n\ge 1}$ respectively. Assume that N is the thinning of $\tilde{N}$. Since $\mathbb {P}(\tilde{N}_{t}=0)=e^{-\int _{0}^{t}\tilde{\lambda }(s)ds}$, we define the rate of acceptance by $\mathbb {E}[N_{t}/\tilde{N}_{t}|\tilde{N}_{t}\ge 1]$. In the case of Poisson processes this indicator takes the following form

$$\begin{aligned} \mathbb {E}[\frac{N_{t}}{\tilde{N}_{t}}|\tilde{N}_{t}\ge 1]=\frac{\int _{0}^{t}\lambda (s)ds}{\int _{0}^{t}\tilde{\lambda }(s)ds}. \end{aligned}$$

(21)

To get (21), we use the following result which is similar to the n-uplet of non-ordering uniform variables in the Poisson homogeneous case

$$\begin{aligned} f_{(\tilde{T}_{1},\ldots ,\tilde{T}_{n}|\tilde{N}_{t}=n)}(t_{1},\ldots ,t_{n})=\frac{\tilde{\lambda }(t_{1})\ldots \tilde{\lambda }(t_{n})}{\left( \int _{0}^{t}\tilde{\lambda }(s)ds\right) ^n}{} \mathbf 1 _{(t_{1},\ldots ,t_{n})\in {[0,t]^{n}}}. \end{aligned}$$

(22)

Equation (22) gives an explicit formula of the conditional density of the vector $(\tilde{T}_{1},\ldots ,\tilde{T}_{n}|\tilde{N}_{t}=n)$. Note that we do not consider any ordering in points $(\tilde{T}_{k})_{0\le {k}\le {n}}$ and that conditionally on $\{\tilde{N}_{t}=n\}$, the points $\tilde{T}_{1},\ldots ,\tilde{T}_{n}$ are independent with density $\left( \tilde{\lambda }(s)/\int _{0}^{t}\tilde{\lambda }(u)du\right) \mathbf 1 _{s\in {[0,t]}}$. By noting that, for $k\le {n}$,

$$\begin{aligned}&\{N_{t}=k|\tilde{N}_{t}=n\}\\&\quad =\bigcup _{1\le {i_{1}}<\cdots <i_{k}\le {n}}\left[ \bigcap _{i\in {\{i_{1},\ldots ,i_{k}\}}}\{U_{i}\le {\frac{\lambda }{\tilde{\lambda }}(\tilde{T}_{i})|\tilde{N}_{t}=n}\} \bigcap _{i\in {\{i_{1},\ldots ,i_{k}\}}^{c}}\{U_{i}>\frac{\lambda }{\tilde{\lambda }}(\tilde{T}_{i})|\tilde{N}_{t}=n\}\right] , \end{aligned}$$

where $(U_{i})$ are independent variables uniformly distributed in [0, 1], independent of $(\tilde{T}_{i})$, we deduce that

$$\begin{aligned} \mathbb {P}(N_{t}=k|\tilde{N}_{t}=n)=\left( \begin{array}{c}n\\ k\end{array}\right) \mathbb {P}\left( U_{i}\le {\frac{\lambda }{\tilde{\lambda }}(\tilde{T}_{i})}|\tilde{N}_{t}=n\right) ^{k}\mathbb {P}\left( U_{i}>\frac{\lambda }{\tilde{\lambda }}(\tilde{T}_{i})|\tilde{N}_{t}=n\right) ^{n-k}. \end{aligned}$$

(23)

Thus, the law of the number of selected points is binomial conditionally on the number of generated points. With (22) and (23), one is able to determine that

$$\begin{aligned} \mathcal {L}(N_{t}|\tilde{N}_{t}=n)=\mathcal {B}(n,p), \end{aligned}$$

with $p=\int _{0}^{t}\lambda (s)ds/\int _{0}^{t}\tilde{\lambda }(s)ds$. Then, we find (21) by using that

$$\begin{aligned} \mathbb {E}[\frac{N_{t}}{\tilde{N}_{t}}|\tilde{N}_{t}\ge 1]=\frac{1}{\mathbb {P}(\tilde{N}_{t}\ge 1)}\sum _{n\ge 1}\frac{1}{n}\mathbb {E}[N_{t}|\tilde{N}_{t}=n]\mathbb {P}(\tilde{N}_{t}=n). \end{aligned}$$