*4.2. Choice of Optimal Thresholds*

To choose the optimal thresholds among reflected policies, we maximize the function *G*Λ.

**Proposition 1.** *The function G*Λ*, defined in Equation* (19)*, attains its maximum on* A*.*

**Proof.** Let *c*1 ≥ 0 be fixed. The first derivative of *G*Λ with respect to *c*2 is given by

$$\partial\_{c\_2} G\_\Lambda(c\_1, c\_2) = \frac{q F\_\Lambda(c\_1, c\_2) W^{(q)}(c\_2)}{(Z^{(q)}(c\_2) - Z^{(q)}(c\_1))^2},\tag{20}$$

where

$$\begin{split} F\_{\Lambda}(\boldsymbol{c}\_{1},\boldsymbol{c}\_{2}) &:= \frac{(Z^{(q)}(\boldsymbol{c}\_{2}) - Z^{(q)}(\boldsymbol{c}\_{1}))}{qW^{(q)}(\boldsymbol{c}\_{2})} (1 - \Lambda Z^{(q)}(\boldsymbol{c}\_{2})) - \left(\boldsymbol{c}\_{2} - \boldsymbol{c}\_{1} - \delta - \Lambda \left(\overline{Z}^{(q)}(\boldsymbol{c}\_{2}) - \overline{Z}^{(q)}(\boldsymbol{c}\_{1})\right)\right) \\ &= -\Lambda \left[\frac{(Z^{(q)}(\boldsymbol{c}\_{2}))^{2}}{qW^{(q)}(\boldsymbol{c}\_{2})} - \overline{Z}^{(q)}(\boldsymbol{c}\_{2}) - \left(\frac{(Z^{(q)}(\boldsymbol{c}\_{2}))}{qW^{(q)}(\boldsymbol{c}\_{2})} Z^{(q)}(\boldsymbol{c}\_{1}) - \overline{Z}^{(q)}(\boldsymbol{c}\_{1})\right)\right] \\ &+ \frac{Z^{(q)}(\boldsymbol{c}\_{2}) - Z^{(q)}(\boldsymbol{c}\_{1})}{qW^{(q)}(\boldsymbol{c}\_{2})} - (\boldsymbol{c}\_{2} - \boldsymbol{c}\_{1} - \delta). \end{split} \tag{21}$$

On the other hand, taking *a* = *c*2 in Equation (12), we see

$$\frac{[Z^{(q)}(\mathfrak{c}\_2)]^2}{q\mathcal{W}^{(q)}(\mathfrak{c}\_2)} - k^{(q)}(\mathfrak{c}\_2) \ge 0 \quad \text{and} \quad \frac{Z^{(q)}(\mathfrak{c}\_2)}{q\mathcal{W}^{(q)}(\mathfrak{c}\_2)} Z^{(q)}(\mathfrak{c}\_1) - k^{(q)}(\mathfrak{c}\_1) \ge 0.$$

Then, using Equation (10), we have

$$F\_{\Lambda}(c\_1, c\_2) < \frac{Z^{(q)}(c\_2)}{q \mathcal{W}^{(q)}(c\_2)} + \Lambda \left[ \frac{Z^{(q)}(c\_2)}{q \mathcal{W}^{(q)}(c\_2)} Z^{(q)}(c\_1) - k^{(q)}(c\_1) \right] - (c\_2 - c\_1 - \delta). \tag{22}$$

Therefore, since lim *<sup>c</sup>*2→<sup>∞</sup> *<sup>Z</sup>*(*q*)(*<sup>c</sup>*2) *<sup>q</sup>W*(*q*)(*<sup>c</sup>*2) = 1 <sup>Φ</sup>(*q*) (see Remark 3), the right-hand side of the aforementioned inequality goes to −∞ as *c*2 goes to <sup>∞</sup>, which implies

$$
\partial\_{c\_2} G\_\Lambda(c\_1, c\_2) < 0,\quad \text{for } c\_2 \text{ large enough.}\tag{23}
$$

From here and Remark 8, we obtain that there exists *c*<sup>∗</sup> ∈ (*<sup>c</sup>*1, ∞) (that depends on *c*1) such that

$$G\_{\Lambda}(c\_1, c\_2) \le G\_{\Lambda}(c\_1, c^\*), \quad \text{for all } c\_2 > c\_1.$$

Taking *d*∗(*<sup>c</sup>*1) := sup{*c*<sup>∗</sup> > *c*1 : *<sup>G</sup>*Λ(*<sup>c</sup>*1, *<sup>c</sup>*2) ≤ *<sup>G</sup>*Λ(*<sup>c</sup>*1, *c*<sup>∗</sup>) for all *c*2 > *<sup>c</sup>*1}, with *c*1 ≥ 0, we see *d*∗(*<sup>c</sup>*1) < ∞ for each *c*1 ≥ 0, since Equation (23) holds. From Equation (20) and the fact that *<sup>∂</sup><sup>c</sup>*2*G*Λ(*<sup>c</sup>*1, *d*∗(*<sup>c</sup>*1)) = 0, it follows that *<sup>F</sup>*Λ(*<sup>c</sup>*1, *d*∗(*<sup>c</sup>*1)) = 0 for *c*1 ≥ 0. Then, by the definitions of *F*Λ and *ζ*Λ—see Equations (21) and (14), respectively—we ge<sup>t</sup>

$$G\_{\Lambda}(\boldsymbol{c}\_{1}, \boldsymbol{d}^{\*}(\boldsymbol{c}\_{1})) = \frac{d^{\*}(\boldsymbol{c}\_{1}) - \boldsymbol{c}\_{1} - \delta - \Lambda(\mathbb{Z}^{(q)}(\boldsymbol{d}^{\*}(\boldsymbol{c}\_{1})) - \mathbb{Z}^{(q)}(\boldsymbol{c}\_{1}))}{Z^{(q)}(\boldsymbol{d}^{\*}(\boldsymbol{c}\_{1})) - Z^{(q)}(\boldsymbol{c}\_{1})} = \mathbb{J}\_{\Lambda}(\boldsymbol{d}^{\*}(\boldsymbol{c}\_{1})), \quad \text{for each } \boldsymbol{c}\_{1} \ge 0.$$

Now, let us take *c*¯1 > *a*Λ (where *a*Λ is defined in Equation (14)). Then, using the fact that *ζ*Λ is strictly decreasing in (*<sup>a</sup>*Λ, ∞) (see Remark 6), we have that for any *c*2 > *c*1 > *d*∗(*c*¯1) it holds that *d*∗(*c*¯1) < *d*∗(*<sup>c</sup>*1) and

$$G\_{\Lambda}(\mathfrak{c}\_1, \mathfrak{c}\_2) \le G\_{\Lambda}(\mathfrak{c}\_1, d^\*(\mathfrak{c}\_1)) = \mathbb{Z}\_{\Lambda}(d^\*(\mathfrak{c}\_1)) < \mathbb{Z}\_{\Lambda}(d^\*(\mathfrak{c}\_1)) = G\_{\Lambda}(\mathfrak{c}\_1, d^\*(\mathfrak{c}\_1)).$$

This implies that the maximum of the function *G*Λ has to be achieved on the set

$$\{(c\_1, c\_2) \in \mathbb{R}\_+^2 : c\_1 < c\_2 \text{ and } c\_1 \in [0, \overline{c}\_1] \}\dots$$

Finally, from Equation (22), we obtain

$$F\_{\Lambda}(\boldsymbol{c}\_{1},\boldsymbol{c}\_{2}) < \frac{Z^{(q)}(\boldsymbol{c}\_{2})}{q\mathcal{W}^{(q)}(\boldsymbol{c}\_{2})} + \Lambda \sup\_{\boldsymbol{c}\_{1} \in [0,\boldsymbol{\varepsilon}\_{1}]} \left[ \frac{Z^{(q)}(\boldsymbol{c}\_{2})}{q\mathcal{W}^{(q)}(\boldsymbol{c}\_{2})} Z^{(q)}(\boldsymbol{c}\_{1}) - k^{(q)}(\boldsymbol{c}\_{1}) \right] - (\boldsymbol{c}\_{2} - \boldsymbol{c}\_{1} - \boldsymbol{\delta}), \quad \text{for } \boldsymbol{c}\_{1} \in [0,\boldsymbol{\varepsilon}\_{1}].$$

Hence, for any *c*1 ∈ [0, *<sup>c</sup>*¯1], we can find *c*¯2 > *c*¯1 such that

$$
\partial\_{\mathcal{C}\_2} G\_{\Lambda}(\mathcal{c}\_1, \mathcal{c}\_2)(\mathcal{c}\_1, \mathcal{c}\_2) \preccurlyeq 0,\quad \text{for any } 0 \le \mathcal{c}\_1 \le \mathcal{c}\_1 \text{ and } 0 \le \mathcal{c}\_2 \le \mathcal{c}\_2.
$$

Therefore, the function *G*Λ attains its maximum on the set

$$\{(c\_1, c\_2) \in [0, \mathfrak{c}\_1] \times [0, \mathfrak{c}\_2] : c\_1 < c\_2\} \subset \mathcal{A}. \quad \Box$$

Note that by Proposition 1 the set B⊂A defined as

$$\mathcal{B} := \{ (c\_1^\*, c\_2^\*) \in \mathcal{A} : G\_\Lambda(c\_1^\*, c\_2^\*) \ge G\_\Lambda(c\_1, c\_2) \text{ for all } (c\_1, c\_2) \in \mathcal{A} \},$$

is not empty. Moreover, since *G*Λ ∈ C<sup>1</sup>(A) and using Equation (14), it follows that

$$\partial\_{\mathbb{G}\_1} G\_{\Lambda}(\mathbf{c}\_1^\*, \mathbf{c}\_2^\*) = \frac{q \mathcal{W}^{(q)}(\mathbf{c}\_1^\*)}{Z^{(q)}(\mathbf{c}\_2^\*) - Z^{(q)}(\mathbf{c}\_1^\*)} \left( G\_{\Lambda}(\mathbf{c}\_1^\*, \mathbf{c}\_2^\*) - \zeta\_{\Lambda}(\mathbf{c}\_1^\*) \right) \le 0, \text{ for } (\mathbf{c}\_1^\*, \mathbf{c}\_2^\*) \in \mathcal{B}, \tag{24}$$

with equality if *c*1 > 0, and

$$\partial\_{c\_2^\*} G\_\Lambda(c\_1^\*, c\_2^\*) = -\frac{q \mathcal{W}^{(q)}(c\_2^\*)}{Z^{(q)}(c\_2^\*) - Z^{(q)}(c\_1^\*)} \left( G\_\Lambda(c\_1^\*, c\_2^\*) - \zeta\_\Lambda(c\_2^\*) \right) = 0, \quad \text{for } (c\_1^\*, c\_2^\*) \in \mathcal{B}. \tag{25}$$

**Proposition 2.** *There exists a unique pair* (*c*Λ1 , *c*Λ2 ) *in* B*. Furthermore,* 0 ≤ *c*Λ1 ≤ *a*Λ < *c*Λ2 < <sup>∞</sup>*, with a*Λ *defined in Equation* (16)*, and the value function associated with the* (*c*Λ1 , *c*Λ2 )*-policy is*

$$w^{c^\Lambda\_1, c^\Lambda\_2}\_{\delta, \Lambda}(\mathbf{x}) = \begin{cases} Z^{(q)}(\mathbf{x})\zeta\_\Lambda(c^\Lambda\_2) + \Lambda k^{(q)}(\mathbf{x}), & \text{if } \mathbf{x} \le c^\Lambda\_2, \\\mathbf{x} - c^\Lambda\_2 + v^{c^\Lambda\_1, c^\Lambda\_2}\_{\delta, \Lambda}(c^\Lambda\_2), & \text{if } \mathbf{x} > c^\Lambda\_2. \end{cases} \tag{26}$$

**Proof.** Let *M* be the maximum value of *G*Λ in B; therefore, for any (*c*<sup>∗</sup>1, *c*∗2 ) ∈ B, we have that *ζ*Λ(*c*<sup>∗</sup>2 ) = *M* by Equation (25). From Remark 6, we know that *ζ*Λ is strictly increasing on (0, *<sup>a</sup>*Λ) and strictly decreasing on (*<sup>a</sup>*Λ, <sup>∞</sup>). If *ζ*Λ(0) ≥ *M*, *ζ*Λ attains *M* at a unique *c*Λ2 > *a*Λ and therefore (0, *c*Λ2 ) is the only point that satisfies Equation (24). On the other hand, if *ζ*Λ(0) < *M*, *ζ*Λ can only attain the value *M* at a unique *c*Λ1 < *a*Λ and a unique *c*Λ2 > *a*Λ. Hence, (*c*Λ1 , *c*Λ2 ) is the only point that satisfies Equations (24) and (25), that is, the only existing point in B. Now, from Lemma 1 and using that *<sup>G</sup>*Λ(*c*Λ1 , *c*Λ2 ) = *ζ*Λ(*c*Λ2 ), we obtain the first part of Equation (26). For the second part, let *x* > *c*Λ2 , then

$$v\_{\delta,\Lambda}^{\varepsilon\_1^{\Lambda},\varepsilon\_2^{\Lambda}}(\mathbf{x}) = \mathbf{x} - c\_1^{\Lambda} - \delta + v\_{\delta,\Lambda}^{\varepsilon\_1^{\Lambda},\varepsilon\_2^{\Lambda}}(c\_1^{\Lambda}) = \mathbf{x} - c\_2^{\Lambda} + c\_2^{\Lambda} - c\_1^{\Lambda} - \delta + v\_{\delta,\Lambda}^{\varepsilon\_1^{\Lambda},\varepsilon\_2^{\Lambda}}(c\_1^{\Lambda}) = \mathbf{x} - c\_2^{\Lambda} + v\_{\delta,\Lambda}^{\varepsilon\_1^{\Lambda},\varepsilon\_2^{\Lambda}}(c\_2^{\Lambda}). \quad \Box$$

The following properties of *vc*Λ1 ,*c*Λ<sup>2</sup> *δ*,Λ are used below in the verification theorem.

**Remark 9.** *From Equations* (10) *and* (26)*, we note*

$$v\_{\delta,\Lambda}^{c\_1^{\Lambda},c\_2^{\Lambda}}(\mathbf{x}) \ge \frac{\Lambda \psi'(0+)}{q} + Z^{(q)}(c\_2^{\Lambda})\zeta(c\_2^{\Lambda}), \quad \text{for } \mathbf{x} > 0.$$

**Remark 10** (Continuity/smoothness at zero)**.** *Note that for x* < 0*, vc*Λ1 ,*c*Λ<sup>2</sup> *δ*,Λ (*x*) = *vc*Λ1 ,*c*Λ<sup>2</sup> *δ*,Λ (0) + Λ*x. Therefore,*

*(i) vc*Λ1 ,*c*Λ<sup>2</sup> *δ*,Λ*is continuous at zero.*

*(ii) For the case of unbounded variation, we have that*

$$v\_{\delta,\Lambda}^{\varepsilon\_1^{\Lambda},\varepsilon\_2^{\Lambda}\prime}(0+) = q\mathcal{W}^{(q)}(0+)\zeta\_{\Lambda}(\varepsilon\_2^{\Lambda}) + \Lambda = \Lambda = v\_{\delta,\Lambda}^{\varepsilon\_1^{\Lambda},\varepsilon\_2^{\Lambda}\prime}(0-).$$
