A Continuous-Time Optimal Insurance Design with Costly Monitoring

Abstract

We provide a theoretical and numerical framework to study optimal insurance design under asymmetric information. We consider a continuous-time model where neither the efforts nor the outcome of an insured firm are observable to an insurer. The insured may then cause two interconnected information problems: moral hazard and fraudulent claims. We show that, when costly monitoring is available, an optimal insurance contract distinguishes the one problem from the other. Furthermore, if the insured’s downward-risk aversion is weak and if the participation constraint is not too tight, then a higher level of the monitoring technology can mitigate both problems.

Appendices

Appendix A: Proof of Proposition 3.1

As in Eq. (2.1), for $u \in \mathcal{A}_1$, $X_t^u = W_t + \int _0^t u_s {{\mathrm{\mathrm{d} }}}s$. As it was shown above, $X^u$ is a driftless Brownian motion under the probability measure ${{\mathrm{\mathbb {P}}}}^u$ characterized by the Radon-Nikodym derivative:

$$\begin{aligned} \frac{{{\mathrm{\mathrm{d} }}}{{\mathrm{\mathbb {P}}}}^u}{{{\mathrm{\mathrm{d} }}}{{\mathrm{\mathbb {P}}}}} = \mathcal{E} \Big ( -\int \limits u {{\mathrm{\mathrm{d} }}}W \Big )_T. \end{aligned}$$

Define the martingale $M^u$ as

$$\begin{aligned} M^u_t:={{\mathrm{\mathbb {E}}}}^u [ \, e^{ f(X_T^u)} \,| \, \mathcal{F}_t^{X^u} ], \quad t\in [0,T]. \end{aligned}$$

By the Martingale Representation Theorem, there exists an $\mathbb {F} ^{X^u}$-adapted process $H^u$ such that

$$\begin{aligned} M^u_t=M^u_0+\int \limits _0^t H_s^u{{\mathrm{\mathrm{d} }}}X_s^u, \quad t\in [0,T]\,\text {a.s.}. \end{aligned}$$

We also define $\gamma ^u$ as

$$\begin{aligned} \gamma _t^u:=\frac{H_t^u}{M^u_t}, \quad t\in [0,T]. \end{aligned}$$

Since $\int \limits \gamma ^u {{\mathrm{\mathrm{d} }}}X^u= \int \limits \frac{{{\mathrm{\mathrm{d} }}}M^u}{M^u}$, it holds that

$$\begin{aligned} M^u=M^u_0\,\mathcal {E}\left( \int \limits \gamma ^u{{\mathrm{\mathrm{d} }}}X^u\right) \end{aligned}$$

and thus

$$\begin{aligned} e^{f(X_T^u)} ={{\mathrm{\mathbb {E}}}}^u [ e^{f(X_T^u)} ] \, \mathcal{E} \Big ( \int \limits \gamma ^u {{\mathrm{\mathrm{d} }}}X^u \Big )_T . \end{aligned}$$

(6.1)

Since the Clark–Ocone formula is applicable to $e^{f(\cdot )}$,¹⁴

$$\begin{aligned} H_t^u={{\mathrm{\mathbb {E}}}}^u [ \, e^{ f(X_T^u)} f'(X_T^u) \,| \, \mathcal{F}_t^{X^u} ]. \end{aligned}$$

(6.2)

Since $X^u$ is the driftless Brownian motion under ${{\mathrm{\mathbb {P}}}}^u$, $X_T^u \sim N(X_t^u, T-t)$ conditional on $\mathcal{F}_t^{X^u}$. Hence,

$$\begin{aligned} \gamma _t^u=\frac{{{\mathrm{\mathbb {E}}}}^u [ \, e^{ f(X_T^u)} f'(X_T^u) \,| \, \mathcal{F}_t^{X^u} ]}{ {{\mathrm{\mathbb {E}}}}^u [ \, e^{ f(X_T^u)} \,| \, \mathcal{F}_t^{X^u} ]} =h(X_t^u, t), \end{aligned}$$

(6.3)

as in Eq. (3.1). Therefore, $\gamma ^u$ is bounded. Taking logarithms on both sides of Eq. (6.1),

$$\begin{aligned} f(X_T^u)&= \log {{\mathrm{\mathbb {E}}}}^u [ e^{f(X_T^u)} ] + \int \limits _0^T \gamma _s^u {{\mathrm{\mathrm{d} }}}X_s^u - \frac{1}{2} \int \limits _0^T (\gamma _s^u)^2 {{\mathrm{\mathrm{d} }}}s \\&= \log {{\mathrm{\mathbb {E}}}}^u [ e^{f(X_T^u)} ] + \int \limits _0^T \gamma _s^u {{\mathrm{\mathrm{d} }}}W_s -\frac{1}{2} \int \limits _0^T \Big \{ (\gamma _s^u)^2 - 2 u_s \gamma _s^u \Big \} {{\mathrm{\mathrm{d} }}}s. \end{aligned}$$

Subtracting $\frac{1}{2} \int _0^T (u_s)^2 {{\mathrm{\mathrm{d} }}}s$ from both sides and taking expectations under the probability measure ${{\mathrm{\mathbb {P}}}}$,

$$\begin{aligned}&{{\mathrm{\mathbb {E}}}}\left[ f\Big ( W_T + \int \limits _0^T u_s \, {{\mathrm{\mathrm{d} }}}s \Big ) - \frac{1}{2} \int \limits _0^T (u_s)^2 \, {{\mathrm{\mathrm{d} }}}s \right] = \log {{\mathrm{\mathbb {E}}}}^u [ e^{f(X_T^u)} ]\nonumber \\&-\frac{1}{2}{{\mathrm{\mathbb {E}}}}\left[ \int \limits _0^T (\gamma _s^u - u_s)^2 {{\mathrm{\mathrm{d} }}}s \right] \nonumber \\&\le \log {{\mathrm{\mathbb {E}}}}^u [ e^{f(X_T^u)} ]. \end{aligned}$$

(6.4)

Note that $\log {{\mathrm{\mathbb {E}}}}^u [ e^{f(X_T^u)} ]$ on the right-hand side of Eq. (6.4) is independent of $u$. If $\gamma ^u = u$, i.e., if there exists some $u \in \mathcal{A}_1$ that satisfies the implementability condition (3.3), it is an optimal solution. Moreover,

$$\begin{aligned} \gamma ^u = u&\Longleftrightarrow&u_\cdot = h(X_\cdot ^u, \cdot ) \qquad \because \hbox {Eq}.(6.3) \nonumber \\&\Longleftrightarrow&X_\cdot ^u = W_\cdot + \int \limits _0^\cdot h(X_s^u, s) {{\mathrm{\mathrm{d} }}}s. \end{aligned}$$

(6.5)

It is well known that, by Zvonkin (1974)(see also Karatzas and Shreve (1997), Chapter 5, Notes, p.396), the last Markovian stochastic differential equation (6.5):

$$\begin{aligned} {{\mathrm{\mathrm{d} }}}X_t^u = {{\mathrm{\mathrm{d} }}}W_t + h(X^u_t, t) {{\mathrm{\mathrm{d} }}}t, \qquad X^u_0=0 \end{aligned}$$

has a unique strong solution $X^*$. Since it is the strong solution, $\mathbb {F} ^{X^*} = \mathbb {F} ^W$. $\square$

Remark 5.1

The remark right after the statement of Proposition 3.1 is proved as follows. For every adapted process $u$ satisfying $\mathbb {E} [\int \limits _0^T(u_t)^2 {{\mathrm{\mathrm{d} }}}t ]<\infty$, it follows from Lebesgue’s dominated convergence theorem that

$$\begin{aligned} J(u)=\lim _{n \rightarrow \infty } J(u^{(n)}) \end{aligned}$$

where $u^{(n)}_t:=u_t \mathbf {1}_{\{|u_t|\le n\}}$. Since $u^{(n)}$ is a bounded process, it satisfies Eq. (6.4). By taking the limit $n \rightarrow \infty$, the process $u$ also satisfies Eq. (6.4).

Appendix B: Supplementary Note on the Clark–Ocone Formula

The Clark–Ocone formula for general functionals is stated in terms of the Malliavin-Fréchet derivative (see e.g. Ocone and Karatzas 1991; Revuz and Yor 1999; Rogers and Williams 2000), but, in order to show our formula (A.2), the following simple argument is sufficient.

Let us define the function $g: \mathbf {R} \times [0,T) \rightarrow \mathbf {R}$ by

$$\begin{aligned} g(x,t) := \int \limits _{-\infty }^\infty \frac{1}{\sqrt{2 \pi (T-t)}} e^{-\frac{y^2}{2(T-t)}} e^{f(x+y)} dy, \end{aligned}$$

which is a $C^2$ function since it is the convolution of $e^{f(x)}$ and $\frac{1}{\sqrt{2 \pi (T-t)}} e^{-\frac{x^2}{2(T-t)}}.$ Also, we have

$$\begin{aligned} \mathbb {E} \big [ e^{f(W_T)} \big | \mathcal{F}_t \big ] = g(W_t, t) \quad \text{ a.s. }, \; 0 \le t < T \end{aligned}$$

and thus the process $\{ g(W_t, t) \}_{0 \le t < T}$ is a martingale. It then follows from Itô’s formula that

$$\begin{aligned} g(W_t, t) = g(0,0) + \int \limits _0^t \frac{\partial g}{\partial x}(W_s ,s) dW_s, \quad 0 \le t < T, \;\; \text{ a.s. } \end{aligned}$$

It remains to prove that

$$\begin{aligned} \frac{\partial g(x,t)}{\partial x}&= \lim _{\epsilon \rightarrow 0} \int \limits _{-\infty }^\infty \frac{1}{\sqrt{2 \pi (T-t)}} e^{-\frac{y^2}{2(T-t)}} \frac{e^{f(x + y+\epsilon )} - e^{f(x+y)} }{\epsilon } {{\mathrm{\mathrm{d} }}}y \\&= \int \limits _{-\infty }^\infty \frac{1}{\sqrt{2 \pi (T-t)}} e^{-\frac{y^2}{2(T-t)}} f'(x+y) e^{f(x+y)} {{\mathrm{\mathrm{d} }}}y. \end{aligned}$$

This can be shown by the following two properties combined with Lebesgue’s dominated convergence theorem:

• For almost every $x \in \mathbf {R} ,$ $\displaystyle \lim _{\epsilon \rightarrow 0} \frac{e^{f(x +\epsilon )} - e^{f(x)}}{\epsilon } = f'(x) e^{f(x)} ;$

• Since the Lipschitz constant of $f$ is $K:=\sup _{x\in \mathbf {R} \backslash \{x_1,x_2,\cdots ,x_n \}} \left| f^\prime (x)\right| <\infty ,$ we have, for every $x \in \mathbf {R} ,$

  $$\begin{aligned} \sup _{0<|\epsilon | < 1} \Big | \frac{e^{f(x + \epsilon )} - e^{f(x)} }{\epsilon } \Big |&= e^{f(x)} \sup _{0<|\epsilon | < 1} \Big | \frac{e^{f(x+\epsilon )-f(x)} - 1 }{\epsilon } \Big | \\&\le e^{f(x)} \sup _{0<|\epsilon | < 1} \frac{e^{|f(x+\epsilon )-f(x)|} - 1 }{|\epsilon |} \\&\le e^{f(x)} \sup _{0<|\epsilon | < 1} \frac{e^{K|\epsilon |} - 1 }{|\epsilon |} \\&= (e^K - 1 ) e^{f(x)}. \end{aligned}$$