# Optimal Stopping and Utility in a Simple Modelof Unemployment Insurance

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimal Stopping Problem

#### 2.1. The Model of Unemployment Insurance

**Remark**

**1.**

#### 2.2. Setting the Optimal Stopping Problem

**Example**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Lemma**

**1.**

**Proof.**

#### 2.3. Allowing for Mortality

#### 2.4. A Priori Properties of the Value Function

**Lemma**

**2.**

- (i)
- $v(0)=0$ and, moreover, $v(x)\ge 0$ for all $x\ge 0$;
- (ii)
- $v(x)<\infty $ for all $x\ge 0$.

**Proof.**

#### 2.5. The Optimal Stopping Rule

#### 2.6. Deterministic Case

## 3. Solving the Optimal Stopping Problem

#### 3.1. Guessing the Solution

#### 3.2. Free-Boundary Problem

#### 3.3. Verification of the Found Solution

**Remark**

**4.**

- (i)
- Let us first show that $u(x)\ge v(x)$ $(x>0$). If the map $x\mapsto u(x)$ was a ${C}^{2}$-function (i.e., with continuous second derivative), then the classical Itô formula (Øksendal 2003, Theorem 4.1.2, p. 44) applied to ${\mathrm{e}}^{-\tilde{r}t}u({X}_{t})$ would yield, on account of (1) and (36),$${\mathrm{e}}^{-\tilde{r}t}u({X}_{t})=u(x)+{\int}_{0}^{t}{\mathrm{e}}^{-\tilde{r}s}\left(Lu({X}_{s})-\tilde{r}u({X}_{s})\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}s+{M}_{t}\phantom{\rule{1.em}{0ex}}({\mathrm{P}}_{x}-\mathrm{a}.\mathrm{s}.),$$$${M}_{t}:={\int}_{0}^{t}{\mathrm{e}}^{-\tilde{r}s}{u}^{\prime}({X}_{s})\phantom{\rule{0.166667em}{0ex}}\sigma {X}_{s}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{B}_{s}\phantom{\rule{1.em}{0ex}}(t\ge 0).$$However, for the function $u(x)$ given by (43), its ${C}^{2}$-smoothness breaks down at the point $x=b$, where it is only ${C}^{1}$. However, $u(x)$ is strictly convex on $(0,b)$ (i.e., ${u}^{\u2033}(x)>0$) and linear on $(b,\infty )$, and we can define the action $Lu(x)$ at $x=b$ by using the one-sided second derivative, say,$${u}^{\u2033}(b-)=P{q}_{*}{b}^{-2}.$$In this situation, a generalization of the Itô formula holds, known as the Itô–Meyer formula (see (Shiryaev 1999, chp. VIII, §2a, p. 757)), which ensures that the representation (45) is still valid.Recall that by construction (see the differential equation in (38)), we have$$Lu(x)-\tilde{r}u(x)=0,\phantom{\rule{1.em}{0ex}}x\in (0,b).$$Moreover, it is easy to check using (47) that the equality (48) also extends to $x=b$. On the other hand, on account of the condition (11) and the definition of b in (42), for $x>b$ we get$$\begin{array}{cc}\hfill Lu(x)-\tilde{r}u(x)& =\mu {\beta}_{1}x-\tilde{r}({\beta}_{1}x-P)\hfill \\ \hfill & ={\beta}_{1}x(\mu -\tilde{r})+\tilde{r}P\hfill \\ \hfill & <{\beta}_{1}b(\mu -\tilde{r})+\tilde{r}P\hfill \\ \hfill & =\frac{P(\mu {q}_{*}-\tilde{r})}{{q}_{*}-1}<0,\hfill \end{array}$$$$\mu {q}_{*}-\tilde{r}=-{\textstyle \frac{1}{2}}{\sigma}^{2}{q}_{*}({q}_{*}-1)<0.$$Thus, combining (48) and (49) we obtain$$Lu(x)-\tilde{r}u(x)\le 0\phantom{\rule{1.em}{0ex}}(x>0).$$Substituting the inequality (50) into formula (45), we conclude that, for any $x>0$ and all $t\ge 0$,$$u(x)+{M}_{t}\ge {\mathrm{e}}^{-\tilde{r}t}u({X}_{t})\phantom{\rule{1.em}{0ex}}({\mathrm{P}}_{x}-\mathrm{a}.\mathrm{s}.).$$According to formula (46), $({M}_{t})$ is a continuous local martingale (Shiryaev 1999, chp. II, §1c, p. 101). Let $({\tau}_{n})$ be a localizing sequence of bounded stopping times, so that ${\tau}_{n}\uparrow \infty $ (${\mathrm{P}}_{x}$-a.s.) and the stopped process $({M}_{{\tau}_{n}\wedge t})$ is a martingale, for each $n\in \mathbb{N}$.Now, let $\tau $ be an arbitrary stopping time of (${X}_{t})$. From (51) we get$$\begin{array}{cc}\hfill u(x)+{M}_{{\tau}_{n}\wedge \tau}& \ge {\mathrm{e}}^{-\tilde{r}({\tau}_{n}\wedge \tau )}u({X}_{{\tau}_{n}\wedge \tau})\hfill \\ \hfill & \ge {\mathrm{e}}^{-\tilde{r}({\tau}_{n}\wedge \tau )}g({X}_{{\tau}_{n}\wedge \tau})\phantom{\rule{1.em}{0ex}}({\mathrm{P}}_{x}-\mathrm{a}.\mathrm{s}.),\hfill \end{array}$$$$u(x)\ge {\mathrm{E}}_{x}\left[{\mathrm{e}}^{-\tilde{r}({\tau}_{n}\wedge \tau )}g({X}_{{\tau}_{n}\wedge \tau})\right],$$$${\mathrm{E}}_{x}({M}_{{\tau}_{n}\wedge \tau})={\mathrm{E}}_{x}({M}_{0})=0.$$By Fatou’s lemma (Shiryaev 1996, §II.6, Theorem 2(a), p. 187), from (53) it follows$$u(x)\ge {\mathrm{E}}_{x}\left(\underset{n\to \infty}{lim\; inf}{\mathrm{e}}^{-\tilde{r}({\tau}_{n}\wedge \tau )}g({X}_{{\tau}_{n}\wedge \tau})\right)={\mathrm{E}}_{x}\left({\mathrm{e}}^{-\tilde{r}\tau}g({X}_{\tau})\right).$$Finally, taking in (54) the supremum over all stopping times $\tau $, we obtain$$u(x)\ge \underset{\tau}{sup}{\mathrm{E}}_{x}\left({\mathrm{e}}^{-\tilde{r}\tau}g({X}_{\tau})\right)=v(x)\phantom{\rule{1.em}{0ex}}(x>0),$$
- (ii)
- Let us now prove the opposite inequality, $u(x)\le v(x)$ ($x>0$). According to (30) and (44), we readily have $u(x)=g(x)\le v(x)$ for $x\in [\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}b,+\infty )$. Next, fix $x\in (0,b)$ and consider the representation (45) with t replaced by ${\tau}_{n}\wedge {\tau}_{b}$, where $({\tau}_{n})$ is the localizing sequence of stopping times for (${M}_{t}$) as before. Then, by virtue of the identity (48) (which, as has been explained, is also true for $x=b$), it follows that$$u(x)+{M}_{{\tau}_{n}\wedge \tau}={\mathrm{e}}^{-\tilde{r}({\tau}_{n}\wedge {\tau}_{b})}u({X}_{{\tau}_{n}\wedge {\tau}_{b}})\phantom{\rule{1.em}{0ex}}({\mathrm{P}}_{x}-\mathrm{a}.\mathrm{s}.).$$Similarly as above, taking expectation on both sides of the equality (55) and again applying Doob’s optional sampling theorem to the martingale $({M}_{{\tau}_{n}\wedge t})$, we obtain$$u(x)={\mathrm{E}}_{x}\left[{\mathrm{e}}^{-\tilde{r}({\tau}_{n}\wedge {\tau}_{b})}u({X}_{{\tau}_{n}\wedge {\tau}_{b}})\right].$$Note that, for $0<x<b$, we have $0\le u(x)\le u(b)$ and $0\le {X}_{{\tau}_{n}\wedge {\tau}_{b}}\le b$ (${\mathrm{P}}_{x}$-a.s.), hence$$0\le {\mathrm{e}}^{-\tilde{r}({\tau}_{n}\wedge {\tau}_{b})}u({X}_{{\tau}_{n}\wedge {\tau}_{b}})\le u(b)\phantom{\rule{1.em}{0ex}}({\mathrm{P}}_{x}-\mathrm{a}.\mathrm{s}.).$$Using that ${\tau}_{n}\uparrow \infty $, observe that, ${\mathrm{P}}_{x}$-a.s.,$$\begin{array}{cc}\hfill \underset{n\to \infty}{lim}{\mathrm{e}}^{-\tilde{r}({\tau}_{n}\wedge {\tau}_{b})}u({X}_{{\tau}_{n}\wedge {\tau}_{b}})& ={\mathrm{e}}^{-\tilde{r}{\tau}_{b}}u({X}_{{\tau}_{b}}){\mathrm{\U0001d7d9}}_{\{{\tau}_{b}<\infty \}}+\underset{n\to \infty}{lim}{\mathrm{e}}^{-\tilde{r}{\tau}_{n}}u({X}_{{\tau}_{n}}){\mathrm{\U0001d7d9}}_{\{{\tau}_{b}=\infty \}}\hfill \\ \hfill & ={\mathrm{e}}^{-\tilde{r}{\tau}_{b}}u(b){\mathrm{\U0001d7d9}}_{\{{\tau}_{b}<\infty \}},\hfill \end{array}$$$$\begin{array}{cc}\hfill u(x)& ={\mathrm{E}}_{x}\left({\mathrm{e}}^{-\tilde{r}{\tau}_{b}}u(b){\mathrm{\U0001d7d9}}_{\{{\tau}_{b}<\infty \}}\right)\hfill \\ \hfill & ={\mathrm{E}}_{x}\left({\mathrm{e}}^{-\tilde{r}{\tau}_{b}}g(b){\mathrm{\U0001d7d9}}_{\{{\tau}_{b}<\infty \}}\right)\hfill \\ \hfill & ={\mathrm{E}}_{x}\left({\mathrm{e}}^{-\tilde{r}{\tau}_{b}}g({X}_{{\tau}_{b}}){\mathrm{\U0001d7d9}}_{\{{\tau}_{b}<\infty \}}\right)\hfill \\ \hfill & \le v(x),\hfill \end{array}$$

## 4. Elementary Solution of the Reduced Problem

#### 4.1. Distribution of the Hitting Time

**Proposition**

**1.**

**Remark**

**5.**

#### 4.2. Alternative Derivation

**Remark**

**6.**

#### 4.3. Direct Maximization

## 5. Statistical Issues and Numerical Illustration

#### 5.1. Specifying the Model Parameters

- The loss of job rate ${\lambda}_{0}$ can be extracted from the publicly available data about the mean length at work, which is theoretically given by $\mathrm{E}({\tau}_{0})=1/{\lambda}_{0}$.
- Likewise, the inflation rate r is also in the public domain.
- To specify the wage growth rate $\mu $, a simple approach is just to set $\mu =r$ as a crude version of a “tracking” rule. However, it may be possible that the individual’s wage growth rate $\mu $ is, to some extent, stipulated by the job contract—for example, that it must not exceed the inflation rate r by more than 1% per annum (applicable, e.g., to civil servants) or, by contrast, that it must be no less than r minus 0.5% per annum (more realistic in the private sector). In practical terms, this would often mean that the actual growth rate $\mu $ is kept on the lowest predefined level.
- More generally, the wage growth rate $\mu $ can be estimated by observing the wage process ${X}_{t}$. This can be implemented by first using regression analysis on ${Y}_{t}=ln{X}_{t}$ and estimating the regression line slope $\mu -\frac{1}{2}{\sigma}^{2}$ (see (2)). In addition, the volatility ${\sigma}^{2}$ can be estimated by using a suitable quadratic functional of the sample path $({Y}_{t})$.
- Finally, knowing the benefit schedule (which should be available through the insurance policy’s terms and conditions), it is in principle possible to calculate, or at least estimate the value $\beta $.

#### 5.2. Estimating the Drift and Volatility

#### 5.3. Hypothesis Testing

#### 5.4. Numerical Examples

**Example**

**2.**

**Example**

**3.**

## 6. Parametric Dependencies

**Remark**

**7.**

#### 6.1. Monotonicity

#### 6.2. Limiting Values

#### 6.3. Comparative Statics and Sensitivity Analysis

#### 6.4. Economic Interpretation

## 7. Including Utility Considerations

#### 7.1. Perpetual American Call Option

#### 7.2. Heuristic Optimal Stopping Models with Utility

**Proposition**

**2.**

**Proof.**

#### 7.3. Suboptimal Solutions

**Remark**

**8.**

#### 7.4. Connections to Expected Utility Theory

**Remark**

**9.**

## 8. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | For technical convenience, we choose to work with continuous-time models, but our ideas can also be adapted to discrete time (which may be somewhat more natural, since the wage process is observed by the individual on a weekly time scale). |

2 | Impact of individualistic (not always rational) perception in economics and financial markets is the subject of the modern behavioral economics (see, e.g., a recent monograph by Dhami 2016). |

3 | More specifically, according to the French UI system back in the 1990s (Kerr 1996, p. 8), a worker aged 50 or more, with eight months of insurable employment in the last twelve months, was entitled to full benefits equal to 57.4% of the final wage payable for the first eight months, thereafter declining by 15% every four months; however, the payments continued for no longer than 21 months overall. This leads to choosing the following numerical values in (6): ${h}_{0}=0.574$, ${s}_{0}=8\left(52/12\right)\doteq 34.7$ (weeks) and $\delta =-(3/52)ln\left(1-0.15\right)\doteq 0.0094=0.94\%$ (per week). The restriction of the benefit term by $21\left(52/12\right)=91$ weeks can be taken into account in our model by adjusting the parameter ${\lambda}_{1}$ from the condition $\mathrm{E}({\tau}_{1})=91$, giving ${\lambda}_{1}\doteq 0.0110$. A more conservative choice is to use a tail probability condition, for example, $\mathrm{P}({\tau}_{1}>91)=0.10$, yielding ${\lambda}_{1}=-ln\left(0.10\right)/91\doteq 0.0253$ (with $\mathrm{E}({\tau}_{1})\doteq 39.5$). |

4 | This conclusion is in accord with the general optimal stopping theory (Peskir and Shiryaev 2006, §2.2). |

5 | The equivalence of the problems (105) and (106), which we have established directly, is not a coincidence: it is known (Villeneuve 2007, Proposition 3.1, p.185) that, under mild assumptions, the solution of the general optimal stopping problem $v(x)={sup}_{\tau}{\mathrm{E}}_{x}\left({\mathrm{e}}^{-r\tau}g({X}_{\tau})\right)$ does not change with the positive truncation of $g(\xb7)$. |

**Figure 1.**A time chart of the unemployment insurance scheme. The horizontal axis shows (continuous) time; the vertical axis indicates the pay rate (i.e., income receivable per unit time). The origin $t=0$ indicates the start of employment. Two pieces of a random path ${X}_{t}$ depict the dynamics of the individual’s wage whilst in employment. The individual joins the UI scheme at entry time $\tau $ (by paying a premium P). When the current job ends (at time ${\tau}_{0}>\tau $), a benefit proportional to the final wage ${X}_{{\tau}_{0}}$ is payable according to a predefined schedule (e.g., see Example 1), until a new job is found after the unemployment spell of duration ${\tau}_{1}$.

**Figure 2.**Schematic diagram of possible transitions in the unemployment insurance scheme. Here, ${\tau}_{0}$ and ${\tau}_{1}$ are the (exponential) holding times in states 0 and 1, with parameters ${\lambda}_{0}$ and ${\lambda}_{1}$, respectively, whereas $\tau $ is the entry time (i.e., from state 0– to state 0+), which is subject to optimal control based on observations over the wage process $({X}_{t})$.

**Figure 3.**Simulated wage process ${X}_{t}$ (left) and ${Y}_{t}=ln{X}_{t}$ (right) according to the geometric Brownian motion model (2), with ${X}_{0}=346$ (euros) and parameters $\mu =0.0004$ and $\sigma =0.02$ (see Example 3). The dashed horizontal line in the left plot indicates the optimal threshold ${b}^{*}\doteq 352.37$ (euros) first attained in this simulation at ${\tau}^{*}\phantom{\rule{-0.166667em}{0ex}}=54$ (weeks). The dashed line in the right plot shows the estimated drift of the log-transformed data (see Section 5.2).

**Figure 4.**Graphs illustrating parametric dependencies of the optimal threshold (23): (

**a**) on the wage drift $\mu <\tilde{r}$ and (

**b**) on the unemployment rate ${\lambda}_{0}>0\vee (\mu -r)$, for selected values of ${\lambda}_{0}$ and $\mu $, respectively. The values of other model parameters used throughout are as in Example 3: $r=0.0004$, $P=9\phantom{\rule{0.166667em}{0ex}}000$, $\beta =30$, and $\sigma =0.02$. The dashed horizontal line in both plots indicates the initial wage $x=346$. The dashed vertical line in (

**a**) indicates $\mu =r$. The lower dashed horizontal line in (

**b**) shows the asymptote $P/\beta =300$ (see (99)).

**Figure 5.**Graphs illustrating parametric dependencies of the value function (25): (

**a**) on the wage drift $\mu <\tilde{r}$ and (

**b**) on the unemployment rate ${\lambda}_{0}>0\vee (\mu -r)$, for selected values of ${\lambda}_{0}$ and $\mu $, respectively. The values of other model parameters used throughout are as in Example 3: $r=0.0004$, $P=9000$, $\beta =30$, $\sigma =0.02$, and $x=346$. The dashed horizontal lines in both plots correspond to the value ${v}_{*}:=\beta x-P=1380$. The dashed vertical line in (

**a**) indicates $\mu =r$; in this case, shown as curve II in plot (

**b**), $v(x)\equiv {v}_{*}$ for all ${\lambda}_{0}\ge {\lambda}_{*}\doteq 0.012420$ (see (92)). That is why curves III, IV and V in plot (

**a**) all intersect at $\mu =r$.

**Figure 6.**Isolines (level curves) of the optimal stopping problem solution on the $({\lambda}_{0},\mu )$-plane: (

**a**) ${b}^{*}({\lambda}_{0},\mu )=\mathrm{const}$ (optimal threshold (23)); (

**b**) $v({\lambda}_{0},\mu )=\mathrm{const}$ (value function (25)). The values of other parameters used throughout are as in Example 3: $r=0.0004$, $P=9\phantom{\rule{0.166667em}{0ex}}000$, $\beta =30$, $\sigma =0.02$, and $x=346$. The slanted dashed lines in both plots show the boundary $\mu ={\lambda}_{0}+r$ (see (11)). In plot b, the horizontal dashed line indicates $\mu =r$ and the vertical dashed line shows ${\lambda}_{*}\doteq 0.012420$ (cf. Figure 5b).

**Figure 7.**Functional dependence on the preference weight $\kappa $ in the reduced optimal stopping problem (115): (

**a**) the optimal threshold ${b}^{\u2020}$ (see (117)) and (

**b**) the value function ${u}^{\u2020}(x)$ (see (119)). Numerical values of the parameters used are as in Example 3: $r=\mu =0.0004$, $P=9\phantom{\rule{0.166667em}{0ex}}000$, $\beta =30$, $\sigma =0.02$, and $x=346$. In particular, if $\kappa =0$ then ${b}^{\u2020}$ coincides with ${b}^{*}\doteq 352.3705$ and ${u}^{\u2020}(x)$ coincides with $v(x)\doteq 1389.6190$. The dashed vertical lines on both plots indicate the value ${\kappa}^{\u2020}\doteq 162.7108$ (see (120)) separating different regimes for ${u}^{\u2020}(x)$ according to (119). When $\kappa ={\kappa}^{\u2020}$, we have ${b}^{\u2020}=x=346$, shown as a dashed horizontal line in plot a; the corresponding value function is given by ${u}^{\u2020}(x)={\beta}_{1}x+{\kappa}^{\u2020}-P\doteq 1542.7110$ (see (118)), shown as a dashed horizontal line in plot b. Note that the graph of ${u}^{\u2020}(x)$ in plot b looks almost linear for $\kappa \in [0,{\kappa}^{\u2020}]$, because the ratio $\kappa /P$ is quite small, $0\le \kappa /P\le {\kappa}^{\u2020}/P\doteq 0.01808$; the slope here is approximately $v(x)({q}_{*}-1)/P\doteq 0.88448$, as compared to slope 1 of the linear graph for $\kappa \ge {\kappa}^{\u2020}$.

**Figure 8.**Theoretical graphs for functionals of the hitting time ${\tau}_{b}$ versus threshold $b\ge 0$. Upper row: (

**a**) the hitting probability ${\mathrm{P}}_{x}({\tau}_{b}<\infty )$ (see (65)) and (

**b**) the mean hitting time ${\mathrm{E}}_{x}({\tau}_{b})$ (see (66)). Bottom row: the expected net present value $\mathrm{e}\phantom{\rule{0.24005pt}{0ex}}\mathrm{NPV}(x;{\tau}_{b})$ (see (72)) with $\mu <\frac{1}{2}{\sigma}^{2}$ (

**c**) or $\mu >\frac{1}{2}{\sigma}^{2}$ (

**d**). The values of parameters used throughout are as in Section 5.4: $x=346$, $P=9\phantom{\rule{0.166667em}{0ex}}000$, ${\beta}_{1}=30$, $\mu =0.0004$, and $\sigma =0.04$ (left) or $\sigma =0.02$ (right). The dashed vertical lines in each plot indicate x and ${b}^{*}\doteq 404.7410$ (left) (see Example 2) or ${b}^{*}\doteq 352.3705$ (right) (see Example 3).

**Table 1.**Sensitivity check of numerical results for the functions ${b}^{*}$ and v in Examples 2 and 3: (

**a**) parametric derivatives and (

**b**) increments in response to a $1\%$-change in the background parameters.

(a) Derivatives | ||

Derivative | Example 2 | Example 3 |

$\mathrm{d}{b}^{*}/\mathrm{d}\mu $ | $-16\phantom{\rule{0.166667em}{0ex}}037.57$ | $-13\phantom{\rule{0.166667em}{0ex}}962.43$ |

$\mathrm{d}v/\mathrm{d}\mu $ | $842\phantom{\rule{0.166667em}{0ex}}062.30$ | $993\phantom{\rule{0.166667em}{0ex}}991.20$ |

$\mathrm{d}{b}^{*}/\mathrm{d}{\lambda}_{0}$ | $-6\phantom{\rule{0.166667em}{0ex}}323.813$ | $-3\phantom{\rule{0.166667em}{0ex}}161.906$ |

$\mathrm{d}v/\mathrm{d}{\lambda}_{0}$ | $-46\phantom{\rule{0.166667em}{0ex}}485.530$ | $-8\phantom{\rule{0.166667em}{0ex}}768.435$ |

(b) Increments (euro) | ||

Increment | Example 2 | Example 3 |

$\Delta {b}^{*}$ ($\mu $) | $-0.06415$ | $-0.05585$ |

$\Delta v$ ($\mu $) | $3.36825$ | $3.97597$ |

$\Delta {b}^{*}$ (${\lambda}_{0}$) | $-0.63238$ | $-0.31619$ |

$\Delta v$ (${\lambda}_{0}$) | $-4.64855$ | $-0.87684$ |

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**MDPI and ACS Style**

Anquandah, J.S.; Bogachev, L.V.
Optimal Stopping and Utility in a Simple Modelof Unemployment Insurance. *Risks* **2019**, *7*, 94.
https://doi.org/10.3390/risks7030094

**AMA Style**

Anquandah JS, Bogachev LV.
Optimal Stopping and Utility in a Simple Modelof Unemployment Insurance. *Risks*. 2019; 7(3):94.
https://doi.org/10.3390/risks7030094

**Chicago/Turabian Style**

Anquandah, Jason S., and Leonid V. Bogachev.
2019. "Optimal Stopping and Utility in a Simple Modelof Unemployment Insurance" *Risks* 7, no. 3: 94.
https://doi.org/10.3390/risks7030094