# Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates

## Abstract

**:**

## 1. Introduction

## 2. Basic Life Insurance Framework

- ${\left(r\left(t\right)\right)}_{t\ge 0}$ is a stochastic process with continuous paths,
- ${\left(m\left(t\right)\right)}_{t\ge 0}$ is a stochastic process with continuous paths,
- ${\left({\mathcal{F}}_{t}\right)}_{t\ge 0}$ is the natural filtration generated by the joint process, $(r,m)$.

## 3. Gaussian Diffusion Approximation

- (a)
- the column vector $W\left(t\right)$ is a d-dimensional standard Wiener process, $d\in \mathbb{\text{N}}$,
- (b)
- the mapping, $(x,t)\mapsto \alpha (x,t)\phantom{\rule{0.166667em}{0ex}}x+\beta \left(t\right)$, which is composed of the bounded and measurable mappings $\alpha :[0,\omega ]\to \mathbb{\text{R}}$ and $\beta :[0,\omega ]\to \mathbb{\text{R}}$, is jointly measurable on $\mathbb{\text{R}}\times [0,\omega ]$,
- (c)
- the mapping $\sigma :\mathbb{\text{R}}\times [0,\omega ]\to {\mathbb{\text{R}}}^{1\times d}$ is jointly measurable on $\mathbb{\text{R}}\times [0,\omega ]$,
- (d)
- there exists a constant $K>0$, such that$$\parallel \sigma (x,t)-\sigma (y,t)\parallel \le {K|x-y|,\phantom{\rule{1.em}{0ex}}\parallel \sigma (x,t)\parallel}^{2}\le {K}^{2}{(1+|x|}^{2})$$

**Proposition 3.1.**Let the process ${X}^{*}$ be defined by:

**Theorem 3.2.**Assume that the properties (a) to (d) hold. Then, for each $(k,l)\in {\mathbb{\text{N}}}_{0}^{2}$, there exists a constant ${C}_{kl}<\infty $ in such a way that:

**Example 3.3.**Let $\alpha \left(t\right)=1/2$ and $\sigma (t,x)=x$. Following the ideas of the proof of Theorem 3.2, we can show that:

**Corollary 3.4.**Under the assumptions of Theorem 3.2, for all $t\in [0,\omega ]$ and $\varsigma \in [0,1]$ we have

## 4. Affine Diffusion Approximation

**Theorem 4.1.**Under the assumptions (a) to (d) in Section 3 and given that ${\partial}_{xx}\sigma (x,t)$ exists and is bounded by K, for each $(k,l)\in {\mathbb{\text{N}}}_{0}^{2}$, there exists a constant, ${C}_{kl}<\infty $, in such a way that:

**Corollary 4.2.**Under the assumptions of Theorem 4.1, for all $t\in [0,\omega ]$ and $\varsigma \in [0,1]$, we have:

## 5. Approximation Error When Valuating Insurance Claims

**Theorem 5.1.**Assume that X satisfies (10) with properties (a) to (d) and that $X\left(t\right)$ is nonnegative for all $t\ge 0$. Then, there exists a constant, C, such that:

## 6. Numeric Illustration

**Figure 1.**$\sigma (t,x)$ in green, $\sigma (t,E[X\left(t\right)\left]\right)$ in red and ${\sigma}^{lin}(t,x)$ in blue for $t=0$ (

**left**) and $t=10$ (

**right**).

**Figure 2.**Expectation function $E\left[X\right(t\left)\right]$ (black) and simulated paths of $X\left(t\right)$ (green), $\tilde{X}\left(t\right)$ (red) and $\widehat{X}\left(t\right)$ (blue) for $\varsigma =1$ (

**top left**), $\varsigma =0.75$ (

**top right**), $\varsigma =0.5$ (

**bottom left**), $\varsigma =0.25$ (

**bottom right**).

**Figure 3.**Relative difference of $E\left[X\right(t\left)\right]$ (black), $\tilde{X}\left(t\right)$ (red) and $\widehat{X}\left(t\right)$ (blue) with respect to $X\left(t\right)$ for $\varsigma =1$ (

**top left**), $\varsigma =0.75$ (

**top right**), $\varsigma =0.5$ (

**bottom left**), $\varsigma =0.25$ (

**bottom right**).

**Table 1.**One-path simulation for (40) and its corresponding expectation approximation, Gaussian approximation and affine approximation.

Valuation w.r.t. the Single Simulated Path | $\varsigma =1$ | $\varsigma =0.75$ | $\varsigma =0.5$ | $\varsigma =0.25$ |
---|---|---|---|---|

true value | 0.37988 | 0.43749 | 0.50274 | 0.57575 |

expectation approximation | 0.65644 | 0.65644 | 0.65644 | 0.65644 |

(relative error) | 0.72804 | 0.50046 | 0.30573 | 0.14016 |

Gaussian approximation | 0.39773 | 0.44968 | 0.50926 | 0.57768 |

(relative error) | 0.04700 | 0.02785 | 0.01296 | 0.00336 |

affine approximation | 0.41127 | 0.45185 | 0.50731 | 0.57635 |

(relative error) | 0.08264 | 0.03283 | 0.00910 | 0.00105 |

Valuation w.r.t. 10,000 Simulations | $\varsigma =1$ | $\varsigma =0.75$ | $\varsigma =0.5$ | $\varsigma =0.25$ |
---|---|---|---|---|

true value | 0.68733 | 0.67571 | 0.66712 | 0.65827 |

expectation approximation | 0.65644 | 0.65644 | 0.65644 | 0.65644 |

(mean absolute deviation) | 0.16151 | 0.13076 | 0.09186 | 0.04839 |

Gaussian approximation | 0.70483 | 0.68155 | 0.66873 | 0.65835 |

(mean absolute deviation) | 0.10065 | 0.05486 | 0.02272 | 0.00500 |

linear noise approximation | 0.70280 | 0.68152 | 0.66875 | 0.65839 |

(mean absolute deviation) | 0.04802 | 0.02486 | 0.00863 | 0.00113 |

## 7. Proofs

**Proof of Theorem 3.2.**For a shorter notation, define:

- we have $-1\le {g}_{\epsilon}^{\prime}\left(x\right)\le 1$ and $0\le x\phantom{\rule{0.166667em}{0ex}}{g}_{\epsilon}^{\prime \prime}\left(x\right)\le 3/2$ for all $x\in \mathbb{\text{R}}$,
- the continuous function, α, has a finite bound, $|\alpha \left(t\right)|\le {\alpha}_{max}$, on $[0,\omega ]$,
- the mapping, $\sigma (x,t)$, is Lipschitz-continuous in x with Lipschitz constant K,
- $\parallel \sigma (X\left(s\right),s)\parallel \le \parallel \sigma (X\left(s\right),s)-\sigma (\overline{X}\left(s\right),s)\parallel +\parallel \sigma (\overline{X}\left(s\right),s)\parallel \le K|X\left(s\right)-\overline{X}\left(s\right)|+\parallel \sigma (\overline{X}\left(s\right),s)\parallel $ for all s,
- ${\parallel \sigma (X\left(s\right),s)\parallel}^{2}\le 2{K}^{2}|X\left(s\right)-\overline{X}{\left(s\right)|}^{2}+2{\parallel \sigma (\overline{X}\left(s\right),s)\parallel}^{2}$ for all s,
- the continuous function $t\mapsto \parallel \sigma (\overline{X}\left(t\right),t)\parallel $ has a finite bound ${\sigma}^{max}$ on $[0,\omega ]$,

**Proof of Theorem 4.1.**The proof is completely analogous to the proof of Theorem 3.2, apart from the fact that the terms:

**Proof of Theorem 5.1.**Taylor’s theorem applied on the function $x\mapsto exp\{-x\}$ yields:

## 8. Conclusions

## Acknowledgments

## Conflicts of Interest

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Christiansen, M.C.
Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates. *Risks* **2013**, *1*, 81-100.
https://doi.org/10.3390/risks1030081

**AMA Style**

Christiansen MC.
Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates. *Risks*. 2013; 1(3):81-100.
https://doi.org/10.3390/risks1030081

**Chicago/Turabian Style**

Christiansen, Marcus C.
2013. "Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates" *Risks* 1, no. 3: 81-100.
https://doi.org/10.3390/risks1030081