## 1. Introduction

Over recent years, self-exciting processes, especially the Hawkes point processes, have been brought to bear in the modelling and analysis of phenomena as diverse as earthquakes, credit defaults and arrivals of orders in the limit-order books in financial markets. Numerous papers have looked at modelling finance and insurance risk based on them. The theoretical foundation can be traced back to a series of papers written by

Hawkes (

1971a,

1971b);

Hawkes and Oakes (

1974);

Brémaud and Massoulié (

1996,

2001,

2002); and more recently,

Dassios and Zhao (

2011);

Zhu (

2013a,

2015);

Jaisson and Rosenbaum (

2015) and

Boumezoued (

2016). Early applications concentrated on the fields such as seismology (see

Vere-Jones (

1975,

1978);

Adamopoulos (

1976);

Ozaki (

1979);

Vere-Jones and Ozaki (

1982) and

Ogata (

1988)).

In particular, these papers related to the aforementioned literature assume that interest rates equal zero, except for the work of

Jang and Dassios (

2013) where the interest rate is assumed to be constant. Previous works dealing with the effect of constant interest rates in terms of premium setting can be found in

Léveillé and Garrido (

2001);

Jang (

2004) and

Jang and Krvavych (

2004). Considering the claim inflation experienced cancels out interest earned, we can ignore the effect of the rate of interest. However, the interest rate might be more variable than the claims themselves. Hence, in this paper, we consider a stochastic interest rate to the aggregate claim amounts. Besides, to further accommodate the clustering effects of claims due to increases in the frequency of natural or man-made disasters, improved models are required to predict claims arising from catastrophic events. For these, we now study a generalised CIR process with externally-exciting and self-exciting jumps, which can be considered as a model extension of

Zhu (

2013a) or

Dassios and Zhao (

2017b,

2017a). It is also a generalisation of

Jang (

2007), where he studied a stochastic interest rate for the aggregate claim amounts using a jump-diffusion process without the self-exciting component.

Since the global financial crisis of 2007, interest rates have been lowered to avoid a great recession, and developed countries have delayed the rises of interest rates due to their fragile economies. However, this low interest rate regime will not continue forever. In addition, recent Greece’s “No” vote on the bailout conditions proposed by the relevant international institutions (EU, IMF and ECB) brought about the increases of the yields on the country’s government bonds as well as the yields on Italian and Spanish government bonds. Even though Greece and the rest of eurozone reached an agreement that could lead to a third bailout and keep the country in the eurozone, undoubtedly there would be sudden jumps in the yields of government bonds due to the clustering arrivals of shocks, such as news of failing to reign in their budget deficits and debts. There have also been sudden interest rate rises in the market in the past, for instance, when the UK crashed out of the ERM in 1992, the East Asian financial crisis of 1997, and the European sovereign debt crisis since 2009. We attempt to model the evolution of interest rates in a continuous-time setting by using a flexible stochastic process that includes a mean-reverting diffusion, externally-exciting and self-exciting jumps all together within a single framework. The arrivals of externally-exciting jumps are assumed to be distributed according to a simple Poisson process. We then calculate the prices of default-free zero-coupon bonds at time 0 paying $1 at time t.

This article is structured as follows. We define and characterise the generalised CIR process with externally-exciting and self-exciting jumps in

Section 2. It is then followed by

Section 3 analysing its theoretical distributional properties based on martingale methodology. Examining variations of this process in modelling the aggregate claim amounts with/without interest rate and also with/without a cluster of claims, we provide insurance premium calculations based on these moments in

Section 4.1. The comparisons between the moments of aggregate claims with/without self-exciting jumps and with/without a diffusion coefficient are also made. In

Section 4.2, we apply the results in

Section 2 to modelling interest rates and pricing government zero-coupon bonds. The comparisons between the bond prices with/without self-exciting component are also made. The sensitivities are also shown with respect to the underlying parameters in this section.

Section 5 contains some concluding remarks.

## 2. Mathematical Background

In this section, let us first provide a mathematical definition as below for this generalised CIR process.

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

**.** Generalised CIR process with externally-exciting and self-exciting jumps is a jump-diffusion processwhere ${S}_{0}>0$ 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.,and 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.

Equivalently, Equation (

1) can be expressed by the stochastic differential equation (SDE)

where

$J}_{t}^{\left(X\right)}:=\sum _{i=1}^{{N}_{t}^{\left(X\right)}}{X}_{i$ and

$J}_{t}^{\left(Y\right)}:=\sum _{j=1}^{{N}_{t}}{Y}_{j$. Basically, this stochastic process

${S}_{t}$ has four terms:

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.

The resulting process can be considered either as a natural generalisation of a CIR process or a Markovian Hawkes process

1. Hence, it can be considered as the extensions of some recent models proposed by

Zhu (

2013a) and

Dassios and Zhao (

2017b,

2017a). This process presents some unique features which might be suitable for mimicking the dynamics of some financial quantities, such as the aggregate losses for insurance companies and interest rates in the fixed-income markets. In particular, a crucial relationship between the process level and the jump arrivals is specified by Equation (

2)

2, and it essentially controls the degree of “contagion” effects: when the level of process is high, more jump arrivals are expected to follow afterwards, hence, contagion spreads accordingly. To illustrate how this new process looks, by setting parameters by

$(a,{\lambda}_{0},\delta ,\sigma )=(2,2,1,1)$ and assuming jump sizes follow the exponential distribution of rate

$1.5$, simulated sample paths of

${S}_{t}$ as defined in Equation (

1) within the time horizons of

$[0,0.1]$,

$[0,1]$ and

$[0,10]$ are presented in

Figure 1.

For notational simplification, we denote the moments and Laplace transforms by

and the

aggregated process by

${Z}_{t}:=\underset{0}{\overset{t}{\int}}{S}_{u}\mathrm{d}u$. For the well-posedness of the process,

$\delta >{\mu}_{{1}_{G}}$ is the

stationary condition for the original Hawkes process, and we also need it in some parts of this paper. However, the conventional

Feller’s condition $2\delta a\ge {\sigma}^{2}$ for the original CIR process is not required throughout this paper as we allow the process to reach the zero level flexibly.

## 3. Distributional Properties

Note that this model as defined in Equation (

1) is still within the classical

affine framework Duffie et al. (

2000,

2003). Without losing generality, in this paper, we only consider the canonical case when

$b=0$ and

$c=1$ for the intensity process in Equation Equation (

2), as indeed it is mathematically trivial to derive all associated results below for a general setup based on

$b,c\ge 0$ (see also in

Zhu (

2014)). Let us first provide the joint Laplace transform of the distribution of

$\left({S}_{t},{Z}_{t}\right)$:

**Proposition** **1.** For constants $\nu ,\xi \ge 0$, we have the conditional joint Laplace transformwhere $C\left(t\right)$ is determined by the non-linear ordinary differential equation (ODE)with the boundary condition $C\left(T\right)=\nu $; and $D\left(T\right)-D\left(t\right)$ is determined by **Theorem** **1.** Under the condition $\delta >{\mu}_{{1}_{G}}$, for any $\nu \in [0,{a}^{+})$ and $\xi >0$, the joint Laplace transform of $\left({S}_{T},{Z}_{T}\right)$ conditional on ${S}_{0}$ is given bywhereand ${a}^{+}$ is the unique positive solution to the equation $1+\xi -\delta u-\widehat{g}\left(u\right)-\frac{1}{2}{\sigma}^{2}{u}^{2}=0$. **Proof.** By setting

$t=0$ in Equation (

4), we have

where

${\mathcal{F}}_{t}$ is the sigma-algebra generated by

${S}_{t}$, and

$C\left(0\right)$ is uniquely determined by the non-linear ODE

with the boundary condition

$C\left(T\right)=\nu $. Under the condition

$\delta >{\mu}_{{1}_{G}}$, it can be solved by the following steps:

Set

$C\left(t\right)=A(T-t)$ and

$\tau =T-t$. Then, Equation Equation (

9) becomes

with the initial condition

$A\left(0\right)=\nu \ge 0;$ we define the right-hand side of Equation (

10) as the function

${f}_{1}\left(A\right)$, i.e.,

Under the condition of

$\delta >{\mu}_{{1}_{G}}$, we have

then,

${f}_{1}\left(A\right)$ is a strictly decreasing function of

$A\ge 0$. Thus, we have

${f}_{1}\left(A\right)<\xi $ for

$A>0$, since

${f}_{1}\left(0\right)=\xi >0$; there is one unique positive solution

${a}^{+}$ to

${f}_{1}\left(A\right)=0$ for

$A\ge 0$, and

${f}_{1}\left(A\right)>0$ for

$A\in [0,{a}^{+})$.

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

Integrate both sides from time 0 to

$\tau $ with the initial condition

$A\left(0\right)=\nu \ge 0$, then we have

Define the function on the left-hand side as

then, we have

${\mathcal{G}}_{\nu ,\xi}\left(A\right)=\tau $; it is obvious that

$A\to \nu $ when

$\tau \to 0$.

By convergence test, we have

Obviously,

$\underset{v}{\overset{{a}^{+}}{\int}}\frac{1}{{a}^{+}-u}\mathrm{d}u=\infty$, then,

thus

$A\to {a}^{+}$ when

$\tau \to \infty $. Therefore,

${\mathcal{G}}_{\nu ,\xi}\left(A\right)=\tau :[\nu ,{a}^{+})\to [0,\infty )$ is a well defined (strictly increasing) function and its inverse function

${\mathcal{G}}_{\nu ,\xi}^{-1}\left(\tau \right)=A:[0,\infty )\to [\nu ,{a}^{+})$ exists.

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

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

Finally, substitute

$C\left(0\right)$ and

$D\left(T\right)-D\left(0\right)$ into Equation (

8) and the result follows.

□

**Corollary** **1.** The Laplace transform of aggregated process ${Z}_{T}$ conditional on ${S}_{0}$ is given bywhere **Proof.** Setting

$\nu =0$ in Equation (

7), the result follows immediately. □

If we set $T\to \infty $, then $\mathbb{E}\left[{e}^{-\xi {Z}_{T}}\mid {S}_{0}\right]\to 0$, which means that ${Z}_{T}\to \infty $ almost surely when $T\to \infty $.

Note that to derive the Laplace transform of

${S}_{T}$, we cannot trivially set

$\xi =0$ in Equation (

7), since

${a}^{+}$ does not exist when

$\xi =0$.

Dassios and Zhao (

2017a) derived the Laplace transform of

${S}_{T}$ and its moments, for which we state the means and variances directly from their results as follows:

**Proposition** **2.** The expectation of ${S}_{t}$ conditional on ${S}_{0}$ is given bywhere $\iota :=\delta -{\mu}_{{1}_{G}}$. **Proposition** **3.** The variance of ${S}_{t}$ conditional on ${S}_{0}$ is given by Similar results for some special cases could also be found in

Zhu (

2014).

## 5. Conclusions

We studied a generalised CIR process with externally-exciting and self-exciting jumps, and examined the distributional properties. The joint Laplace transform of the process and its integrated process was derived by applying the standard martingale theory. Using the first and second moments of the process, we provided insurance premium calculations and their comparisons with/without self-exciting jumps, and with/without a diffusion coefficient. As a financial application, we present how this Laplace transform can be used for pricing default-free zero-coupon bonds. Numerical calculations for bond prices are illustrated with/without self-exciting jumps, and with/without a diffusion coefficient. Changing the relevant parameters of the process, their sensitivities are also presented for both applications. The estimation exercise for the parameters of this model is important, which is left as future research.

Dassios and Zhao (

2017a) derived the Laplace transform of

${S}_{T}$ and its moments. Maximum likelihood estimation requires the inversion of the Laplace transform for

${S}_{T}$, which is a complicated numerical problem. An alternative is moment-based estimation, where moments can be obtained by successively differentiating the Laplace transform for

${S}_{T}$ and indeed the first two are given by Propositions 2 and 3.