# Merton Investment Problems in Finance and Insurance for the Hawkes-Based Models

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. General Compound Hawkes process

#### 3.1. Hawkes Process

**Definition**

**1.**

**(One-dimensionalHawkes Process) (Hawkes 1971a, 1971b).**The one-dimensional Hawkes process is a point process $N\left(t\right)$ which is characterized by its intensity $\lambda \left(t\right)$ with respect to its natural filtration:

**LLN for HP**(Bacry et al. 2013). Let $0<\widehat{\mu}:={\int}_{0}^{+\infty}\mu \left(s\right)ds<1.$ Then

**Remark**

**1.**

**FCLT for HP**(Bacry et al. 2013). Under LLN and ${\int}_{0}^{+\infty}s\mu \left(s\right)ds<+\infty $ conditions

**Remark**

**2.**

#### 3.2. General Compound Hawkes Process

**Definition**

**2.**

**(General Compound Hawkes Process).**General compound Hawkes Process is defined as (Swishchuk 2020; Swishchuk and Huffman 2020; Swishchuk 2017b)

**-in finance**:

**-in insurance**:

#### 3.3. LLN and FCLT for GCHP

**Lemma 1.**

**(LLN for GCHP)**

**(Swishchuk 2020; Swishchuk and Huffman 2020; Swishchuk 2017b)**. Let $\widehat{\mu}:={\int}_{0}^{+\infty}\mu \left(s\right)ds<1,$ and Markov chain ${X}_{i}$ is ergodic with stationary probabilities ${\pi}_{i}^{*}.$ Then the GCHP ${S}_{nt}$ satisfies the following weak convergence in the Skorokhod topology:

**Theorem 1.**

**(FCLT (or Jump-Diffusion Limit) for GCHP)**

**(Swishchuk 2020; Swishchuk and Huffman 2020; Swishchuk 2017b).**Let ${X}_{k}$ be an ergodic Markov chain and with ergodic probabilities $({\pi}_{1}^{*},{\pi}_{2}^{*},...,{\pi}_{n}^{*}).$ Let also ${S}_{t}$ be LGCHP, and $0<\widehat{\mu}:={\int}_{0}^{+\infty}\mu \left(s\right)ds<1\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\int}_{0}^{+\infty}\mu \left(s\right)sds<+\infty .$

**Remark**

**3.**

**Theorem 2.**

**(FCLT (or Pure Diffusion Limit) for GCHP**

**(Swishchuk 2020; Swishchuk and Huffman 2020; Swishchuk 2017b; Swishchuk et al. 2020).**Let ${X}_{k}$ be an ergodic Markov chain and with ergodic probabilities $({\pi}_{1}^{*},{\pi}_{2}^{*},...,{\pi}_{n}^{*}).$ Let also ${S}_{t}$ be LGCHP, and $0<\widehat{\mu}:={\int}_{0}^{+\infty}\mu \left(s\right)ds<1\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\int}_{0}^{+\infty}\mu \left(s\right)sds<+\infty .$

**Remark**

**4.**

**Remark**

**5.**

## 4. Merton Investment Problem in Finance for the Hawkes-Based Model

**Proposition**

**1.**

**Remark**

**6.**

## 5. Merton Investment Problem in Insurance for the Hawkes-Based Risk Model

**Proposition**

**2.**

**Remark**

**7.**

**Corollary**

**1.**

## 6. Discussion

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LLN | Law of Large Numbers |

FCLT | Functional Central Limit Theorem |

GCHP | General Compound Hawkes Process |

HJB | Hamilton–Jaocobi–Bellman Equation |

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Swishchuk, A.
Merton Investment Problems in Finance and Insurance for the Hawkes-Based Models. *Risks* **2021**, *9*, 108.
https://doi.org/10.3390/risks9060108

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Swishchuk A.
Merton Investment Problems in Finance and Insurance for the Hawkes-Based Models. *Risks*. 2021; 9(6):108.
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2021. "Merton Investment Problems in Finance and Insurance for the Hawkes-Based Models" *Risks* 9, no. 6: 108.
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