# A Generalised CIR Process with Externally-Exciting and Self-Exciting Jumps and Its Applications in Insurance and Finance

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

**Definition**

**1**(Generalised CIR Process with Externally-exciting and Self-exciting Jumps)

**.**

- ${S}_{0}>0$ is the initial value at time $t=0$;
- $a\ge 0$ is the constant mean-reversion level;
- $\delta >0$ is the constant mean-reversion rate;
- $\sigma >0$ is the constant that governs the volatility;
- ${\left\{{W}_{t}\right\}}_{t\ge 0}$ is a standard Brownian motion;
- ${\left\{{X}_{i}\right\}}_{i=1,2,...}$ are the sizes of externally-exciting jumps, a sequence of i.i.d. positive r.v.s with distribution function $H\left(x\right),x>0$, occurring at the corresponding random times ${\left\{{T}_{i}^{\left(X\right)}\right\}}_{i=1,2,...}$ following a Poisson process ${N}_{t}^{\left(X\right)}$ of constant rate $\varrho >0$;
- ${\left\{{Y}_{j}\right\}}_{j=1,2,...}$ are the sizes of self-exciting jumps, a sequence of i.i.d. positive r.v.s with distribution function $G\left(y\right),y>0$, occurring at the corresponding random times $N\equiv {\left\{{T}_{j}^{\left(Y\right)}\right\}}_{j=1,2,...}$, and this point process ${N}_{t}$ has a stochastic intensity linearly dependent on ${S}_{t}$, i.e.,$${\lambda}_{t}=b+c{S}_{t},\phantom{\rule{2.em}{0ex}}b,c\ge 0;$$
- the sequences ${\left\{{X}_{i}\right\}}_{i=1,2,...}$, ${\left\{{Y}_{j}\right\}}_{j=1,2,...}$, ${\left\{{T}_{i}^{\left(X\right)}\right\}}_{i=1,2,...}$ and ${\left\{{W}_{t}\right\}}_{t\ge 0}$ are assumed to be independent of each other.

- The first two terms correspond to the classical square-root process (Feller 1951) or CIR process (Cox et al. 1985).
- The third term corresponds to the impact of exogenous shocks.
- The last term corresponds to the impact of past exogenous shocks acting on the future intensity, and this term corresponds to the self-exciting component in a generalised Hawkes framework.

## 3. Distributional Properties

**Proposition**

**1.**

**Theorem**

**1.**

**Proof.**

- Set $C\left(t\right)=A(T-t)$ and $\tau =T-t$. Then, Equation Equation (9) becomes$$\frac{\mathrm{d}A\left(\tau \right)}{\mathrm{d}\tau}=1-\delta A\left(\tau \right)-\widehat{g}\left(A\left(\tau \right)\right)-\frac{1}{2}{\sigma}^{2}{A}^{2}\left(\tau \right)+\xi ,$$$${f}_{1}\left(A\right):=1+\xi -\delta A-\widehat{g}\left(A\right)-\frac{1}{2}{\sigma}^{2}{A}^{2}.$$
- Under the condition of $\delta >{\mu}_{{1}_{G}}$, we have$$\frac{\partial {f}_{1}\left(A\right)}{\partial A}=\underset{0}{\overset{\infty}{\int}}y{e}^{-Ay}\mathrm{d}G\left(y\right)-\delta -{\sigma}^{2}A\le \underset{0}{\overset{\infty}{\int}}y\mathrm{d}G\left(y\right)-\delta ={\mu}_{{1}_{G}}-\delta <0,\phantom{\rule{2.em}{0ex}}A\ge 0,$$
- As $\nu $ should be approachable to zero, we assume $A\left(0\right)=\nu \in [0,{a}^{+})$, we have $A\left(\tau \right)\in [v,{a}^{+})$ and ${f}_{1}\left(A\left(\tau \right)\right)>0$, then, Equation Equation (10) can be written as$$\frac{\mathrm{d}A\left(\tau \right)}{1+\xi -\delta A\left(\tau \right)-\widehat{g}\left(A\left(\tau \right)\right)-\frac{1}{2}{\sigma}^{2}{A}^{2}\left(\tau \right)}=\mathrm{d}\tau .$$Integrate both sides from time 0 to $\tau $ with the initial condition $A\left(0\right)=\nu \ge 0$, then we have$$\underset{\nu}{\overset{A}{\int}}\frac{1}{1+\xi -\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u=\tau ,\phantom{\rule{2.em}{0ex}}A\in [v,{a}^{+}).$$Define the function on the left-hand side as$${\mathcal{G}}_{\nu ,\xi}\left(A\right):=\underset{\nu}{\overset{A}{\int}}\frac{1}{1+\xi -\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u,\phantom{\rule{2.em}{0ex}}A\in [v,{a}^{+}),$$
- By convergence test, we have$$\begin{array}{ccc}\hfill \underset{u\uparrow {a}^{+}}{lim}\frac{\frac{1}{{a}^{+}-u}}{\frac{1}{1+\xi -\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}}& =& \underset{u\uparrow {a}^{+}}{lim}\frac{1+\xi -\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}{{a}^{+}-u}\hfill \\ & =& \underset{v\downarrow 0}{lim}\frac{1+\xi -\delta ({a}^{+}-v)-\widehat{g}\left({a}^{+}-v\right)-\frac{1}{2}{\sigma}^{2}{({a}^{+}-v)}^{2}}{v}\hfill \\ & =& \delta -\underset{0}{\overset{\infty}{\int}}y{e}^{-{a}^{+}y}\mathrm{d}G\left(y\right)+{\sigma}^{2}{a}^{+}>\delta -{\mu}_{{1}_{G}}+{\sigma}^{2}{a}^{+}>0.\hfill \end{array}$$Obviously, $\underset{v}{\overset{{a}^{+}}{\int}}\frac{1}{{a}^{+}-u}\mathrm{d}u=\infty$, then,$$\underset{v}{\overset{{a}^{+}}{\int}}\frac{1}{1+\xi -\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u=\infty ,$$
- The unique solution is found by $A\left(\tau \right)={\mathcal{G}}_{\nu ,\xi}^{-1}\left(\tau \right)={\mathcal{G}}_{\nu ,\xi}^{-1}(T-t)$. Hence, $C\left(0\right)=A\left(T\right)={\mathcal{G}}_{\nu ,\xi}^{-1}\left(T\right)$.
- Now, $D\left(T\right)-D\left(0\right)$ is determined by$$D\left(T\right)-D\left(0\right)=\varrho \underset{0}{\overset{T}{\int}}\left[1-\widehat{h}\left({\mathcal{G}}_{\nu ,\xi}^{-1}\left(\tau \right)\right)\right]\mathrm{d}\tau +a\delta \underset{0}{\overset{T}{\int}}{\mathcal{G}}_{\nu ,\xi}^{-1}\left(\tau \right)\mathrm{d}\tau .$$By the change of variable ${\mathcal{G}}_{\nu ,\xi}^{-1}\left(\tau \right)=u,$ we have $\tau ={\mathcal{G}}_{\nu ,\xi}\left(u\right)$, and$$\begin{array}{ccc}\hfill \underset{0}{\overset{T}{\int}}\left[1-\widehat{h}\left({\mathcal{G}}_{\nu ,\xi}^{-1}\left(\tau \right)\right)\right]\mathrm{d}\tau & =& \underset{{\mathcal{G}}_{\nu ,\xi}^{-1}\left(0\right)}{\overset{{\mathcal{G}}_{\nu ,\xi}^{-1}\left(T\right)}{\int}}\left[1-\widehat{h}\left(u\right)\right]\frac{\partial \tau}{\partial u}\mathrm{d}u=\underset{\nu}{\overset{{\mathcal{G}}_{\nu ,\xi}^{-1}\left(T\right)}{\int}}\frac{1-\widehat{h}\left(u\right)}{1+\xi -\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u,\hfill \\ \hfill \underset{0}{\overset{T}{\int}}{\mathcal{G}}_{\nu ,\xi}^{-1}\left(\tau \right)\mathrm{d}\tau & =& \underset{{\mathcal{G}}_{\nu ,\xi}^{-1}\left(0\right)}{\overset{{\mathcal{G}}_{\nu ,\xi}^{-1}\left(T\right)}{\int}}u\frac{\partial \tau}{\partial u}\mathrm{d}u=\underset{\nu}{\overset{{\mathcal{G}}_{\nu ,\xi}^{-1}\left(T\right)}{\int}}\frac{u}{1+\xi -\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u.\hfill \end{array}$$
- Finally, substitute $C\left(0\right)$ and $D\left(T\right)-D\left(0\right)$ into Equation (8) and the result follows.

**Corollary**

**1.**

**Proof.**

**Proposition**

**2.**

**Proposition**

**3.**

## 4. Applications

#### 4.1. An Application in Insurance: Insurance Premium Calculation

- If there are no self-exciting jumps and no diffusion in Equation (12), it becomes a simple Poisson shot-noise process, denoted by ${L}_{t}$, i.e.,$$\mathrm{d}{L}_{t}=-\delta {L}_{t}\mathrm{d}t+\mathrm{d}{J}_{t}^{\left(X\right)}.$$This process has been used for actuarial applications as a discounted aggregate loss process, see Jang (2004, 2007) and Jang and Krvavych (2004). If we assume (often implicitly) that interest rate is zero, i.e., $\delta =0$, it becomes a simple compound Poisson process ${L}_{t}={\sum}_{i=1}^{{N}_{t}^{\left(X\right)}}{X}_{i}$.
- If we replace $-\delta $ by $\delta $ and set $\sigma =0$ in Equation (12) and ${S}_{0}=0$, then we have a process of$${M}_{t}:=\sum _{0\le {T}_{i}^{\left(X\right)}<t}{X}_{i}{e}^{\delta \left(t-{T}_{i}^{\left(X\right)}\right)}+\sum _{0\le {T}_{j}^{\left(Y\right)}<t}{Y}_{j}{e}^{\delta \left(t-{T}_{j}^{\left(Y\right)}\right)},$$$$\mathrm{d}{M}_{t}=\delta {M}_{t}\mathrm{d}t+\mathrm{d}{J}_{t}^{\left(X\right)}+\mathrm{d}{J}_{t}^{\left(Y\right)}.$$
**Remark****1.**This shot-noise self-exciting jump process in Equation (15) may be interpreted in the context of non-life insurance. A single event (e.g., natural catastrophe) may induce losses for a line of business. Each loss may produce a cluster of losses according to a branching structure (Dassios and Zhao 2011). Both losses are accumulated on a constant risk-free force of interest rate δ.If there are no self-exciting jumps, from Equation (14), we have$${\Gamma}_{t}:=\sum _{i=1}^{{N}_{t}^{\left(X\right)}}{X}_{i}{e}^{\delta \left(t-{T}_{i}\right)},$$$$\mathrm{d}{\Gamma}_{t}=\delta {\Gamma}_{t}\mathrm{d}t+\mathrm{d}{J}_{t}^{\left(X\right)},$$ - In contrast, now let us consider a stochastic interest rate to the aggregate loss amounts up to time t, denoted by ${L}_{t}$, as it is not deterministic in practice. Thus, if we replace $-\delta $ by $\eta >0$ in Equation (12), then we can extend our study from Equations (15) and (16) to$$\mathrm{d}{L}_{t}=\eta {L}_{t}\mathrm{d}t+\sigma \sqrt{{L}_{t}}\mathrm{d}{W}_{t}+\mathrm{d}{J}_{t}^{\left(X\right)}+\mathrm{d}{J}_{t}^{\left(Y\right)}.$$
**Remark****2.**This shot-noise self-exciting jump-diffusion process in Equation (17) may be also interpreted in the context of non-life insurance. Similarly, a single event (e.g., a natural catastrophe) may induce losses for a line of business. Compared with Equation (16), both losses are accumulated on a stochastic force of interest rate. The proposed model captures the effect of sudden intensity increases due to external events, together with the accumulation of losses on a stochastic interest rate. Hence, it does have a potential interest in the insurance field.

#### 4.1.1. Expectation of Loss Process ${L}_{t}$

- If there are no self-exciting jumps, from Equation (18), we have$$\mathbb{E}\left[{L}_{t}\mid {L}_{0}\right]={L}_{0}{e}^{\eta t}+\frac{{\mu}_{{1}_{H}}\varrho}{\eta}\left({e}^{\eta t}-1\right).$$
- If we only consider self-exciting jumps, i.e., set ${\mu}_{{1}_{H}}=0$ in Equation (18), we have$$\mathbb{E}\left[{L}_{t}\mid {L}_{0}\right]={L}_{0}{e}^{\zeta t}.$$

#### 4.1.2. Variance of Loss Process ${L}_{t}$

- If there are no self-exciting jumps, from Equation (21), we have$$\begin{array}{ccc}\phantom{\rule{6.0pt}{0ex}}\mathrm{Var}\left[{L}_{t}\mid {L}_{0}\right]\phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{{\sigma}^{2}}{2\eta}\left[\frac{{\mu}_{{1}_{H}}\varrho}{\eta}+{\mu}_{{2}_{H}}\varrho +2{L}_{0}\right]{e}^{2\eta t}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}\hfill & -\frac{{\sigma}^{2}}{\eta}\left({L}_{0}+\frac{{\mu}_{{1}_{H}}\varrho}{\eta}\right){e}^{\eta t}-\frac{\varrho}{2\eta}\left({\mu}_{{2}_{H}}-\frac{{\sigma}^{2}{\mu}_{{1}_{H}}}{\eta}\right).\hfill \end{array}$$
- If we only consider self-exciting jumps, i.e., set ${\mu}_{{1}_{H}}={\mu}_{{2}_{H}}=0$ in Equation (21), we have$$\mathrm{Var}\left[{L}_{t}\mid {L}_{0}\right]=\frac{\left({\mu}_{{2}_{G}}+{\sigma}^{2}\right){L}_{0}}{\zeta}\left({e}^{2\zeta t}-{e}^{\zeta t}\right).$$
- If we set $\sigma =0$ in Equation (21) and denote the special case of ${L}_{t}$ by ${V}_{t}$, then it is given by$$\begin{array}{ccc}\phantom{\rule{6.0pt}{0ex}}\mathrm{Var}\left[{V}_{t}\mid {V}_{0}\right]\phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{1}{2\zeta}\left(\frac{{\mu}_{{2}_{G}}{\mu}_{{1}_{H}}\varrho}{\zeta}+{\mu}_{{2}_{H}}\varrho +2{\mu}_{{2}_{G}}{V}_{0}\right){e}^{2\zeta t}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \phantom{\rule{1.em}{0ex}}\hfill & -\frac{{\mu}_{{2}_{G}}}{\zeta}\left({V}_{0}+\frac{{\mu}_{{1}_{H}}\varrho}{\zeta}\right){e}^{\zeta t}-\frac{\varrho}{2\zeta}\left({\mu}_{{2}_{H}}-\frac{{\mu}_{{2}_{G}}{\mu}_{{1}_{H}}}{\zeta}\right).\hfill \end{array}$$

#### 4.1.3. Numerical Examples

**Remark**

**3.**

**Remark**

**4.**

#### 4.2. An Application in Finance: Default-Free Bond Pricing

- If there are no self-exciting jumps, from Equation (26), we have$$\mathbb{E}\left[{e}^{-{Z}_{T}}\mid {r}_{0}\right]={e}^{-{\mathcal{G}}_{0,1}^{-1}\left(T\right){r}_{0}}\times exp\left(-\underset{0}{\overset{{\mathcal{G}}_{0,1}^{-1}\left(T\right)}{\int}}\frac{a\delta u+\varrho \left[1-\widehat{h}\left(u\right)\right]}{2-\delta u-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u\right),$$$${\mathcal{G}}_{0,1}\left(A\right):=\underset{0}{\overset{A}{\int}}\frac{1}{2-\delta u-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u.$$
- If we only consider the self-exciting jumps, i.e., $\varrho =0$ in Equation (26), we have$$\mathbb{E}\left[{e}^{-{Z}_{T}}\mid {r}_{0}\right]={e}^{-{\mathcal{G}}_{0,1}^{-1}\left(T\right){r}_{0}}\times exp\left(-\underset{0}{\overset{{\mathcal{G}}_{0,1}^{-1}\left(T\right)}{\int}}\frac{a\delta u}{2-\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u\right),$$$${\mathcal{G}}_{0,1}\left(A\right):=\underset{0}{\overset{A}{\int}}\frac{1}{2-\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}}\mathrm{d}u.$$

#### Numerical Examples

**Remark**

**5.**

**Remark**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof for Proposition 1

**Proof.**

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1 | A Markovian Hawkes process is the one with exponential fertility rate. |

2 | A similar setup as Equation (2) for constructing dependency between the interest rate and default rate was presented in Lando (2004, p. 123). |

$\mathbb{E}\left[{L}_{t}\mid {L}_{0}\right]$ | Equation (18) | 24.28 |

$\mathbb{E}\left[{L}_{t}\mid {L}_{0}\right]$, if there are no self-exciting jumps | Equation (19) | 6.18 |

$\mathbb{E}\left[{L}_{t}\mid {L}_{0}\right]$, if we only consider self-exciting jumps | Equation (20) | 7.77 |

$\mathrm{Var}\left[{L}_{t}\mid {L}_{0}\right]$ | Equation (21) | 620.77 |

$\mathrm{Var}\left[{L}_{t}\mid {L}_{0}\right]$, if there are no self-exciting jumps | Equation (22) | 14.22 |

$\mathrm{Var}\left[{L}_{t}\mid {L}_{0}\right]$, if we only consider self-exciting jumps | Equation (23) | 230.81 |

$\mathit{\sigma}$ | $\mathbf{Var}\left[{\mathit{L}}_{\mathit{t}}\mid {\mathit{L}}_{0}\right]$ | $\mathbf{Var}\left[{\mathit{V}}_{\mathit{t}}\mid {\mathit{V}}_{0}\right]$ | $\mathbf{Var}\left[{\mathit{L}}_{\mathit{t}}\mid {\mathit{L}}_{0}\right]-\mathbf{Var}\left[{\mathit{V}}_{\mathit{t}}\mid {\mathit{V}}_{0}\right]$ |
---|---|---|---|

(21) | (24) | ||

0.0 | 567.88 | 567.88 | 0.00 |

0.5 | 581.10 | 567.88 | 13.22 |

0.6 | 586.92 | 567.88 | 19.04 |

0.7 | 593.80 | 567.88 | 25.92 |

0.8 | 601.73 | 567.88 | 33.85 |

0.9 | 610.72 | 567.88 | 42.84 |

1.0 | 620.77 | 567.88 | 52.89 |

$\mathit{\beta}$ | $\mathbb{E}\left[{\mathit{L}}_{\mathit{t}}\mid {\mathit{L}}_{0}\right]$ | $\mathbf{Var}\left[{\mathit{L}}_{\mathit{t}}\mid {\mathit{L}}_{0}\right]$ | $\mathbb{E}\left[{\mathit{L}}_{\mathit{t}}\mid {\mathit{L}}_{0}\right]$ | $\mathbf{Var}\left[{\mathit{L}}_{\mathit{t}}\mid {\mathit{L}}_{0}\right]$ | $\mathbb{E}\left[{\mathit{L}}_{\mathit{t}}\mid {\mathit{L}}_{0}\right]$ | $\mathbf{Var}\left[{\mathit{V}}_{\mathit{t}}\mid {\mathit{V}}_{0}\right]$ |
---|---|---|---|---|---|---|

(19) | (22) | (20) | (23) | (21) | (24) | |

10.00 | 6.18 | 14.22 | 1.16 | 1.28 | 15.91 | 11.75 |

5.00 | 6.18 | 14.22 | 1.28 | 1.58 | 18.03 | 13.35 |

1.00 | 6.18 | 14.22 | 2.86 | 15.17 | 72.77 | 59.89 |

0.50 | 6.18 | 14.22 | 7.77 | 230.81 | 620.77 | 567.88 |

0.25 | 6.18 | 14.22 | 57.40 | 26,376.00 | 46,440.00 | 45,156.00 |

$B(0,1)$ | Equation (26) | 0.9419 |

$B(0,1)$, if there are no self-exciting jumps | Equation (27) | 0.9423 |

$B(0,1)$, if we only consider self-exciting jumps | Equation (28) | 0.9552 |

$\mathit{\sigma}$ | $\mathit{B}(0,1)$ |
---|---|

0.01 | 0.9368 |

0.1 | 0.9369 |

0.5 | 0.9389 |

0.8 | 0.9419 |

10 | 0.9889 |

∞ | 1 |

$\mathit{\alpha}$ | $\mathit{B}(0,1)$ (26) | $\mathit{B}(0,1)$ (27) |
---|---|---|

∞ | 0.955201 | 0.955585 |

100 | 0.941880 | 0.942340 |

90 | 0.940422 | 0.940889 |

70 | 0.936278 | 0.936768 |

50 | 0.928904 | 0.929434 |

30 | 0.912734 | 0.912116 |

5 | 0.742420 | 0.743715 |

1 | 0.391674 | 0.393072 |

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**MDPI and ACS Style**

Dassios, A.; Jang, J.; Zhao, H. A Generalised CIR Process with Externally-Exciting and Self-Exciting Jumps and Its Applications in Insurance and Finance. *Risks* **2019**, *7*, 103.
https://doi.org/10.3390/risks7040103

**AMA Style**

Dassios A, Jang J, Zhao H. A Generalised CIR Process with Externally-Exciting and Self-Exciting Jumps and Its Applications in Insurance and Finance. *Risks*. 2019; 7(4):103.
https://doi.org/10.3390/risks7040103

**Chicago/Turabian Style**

Dassios, Angelos, Jiwook Jang, and Hongbiao Zhao. 2019. "A Generalised CIR Process with Externally-Exciting and Self-Exciting Jumps and Its Applications in Insurance and Finance" *Risks* 7, no. 4: 103.
https://doi.org/10.3390/risks7040103