*4.3. Verification*

Let us denote by *<sup>v</sup>δ*,<sup>Λ</sup> the function given in Equation (26), which is the optimal value function among reflected policies. We now prove some properties of this function.

**Lemma 2.** *The function <sup>v</sup>δ*,<sup>Λ</sup> *is C*<sup>2</sup>((0, <sup>∞</sup>)\{*c*Λ2 }) *and C*<sup>1</sup>(0, <sup>∞</sup>)*.*

**Proof.** By Assumption 1, we have that, for each *q* ≥ 0, the function *W*(*q*) is continuously differentiable on (0, <sup>∞</sup>). This implies, by Equation (26), that *<sup>v</sup>δ*,<sup>Λ</sup> is *C*<sup>2</sup>((0, <sup>∞</sup>)\{*c*Λ2 }). On the other hand, using Equation (26), we have that for *x* ≤ *c*Λ2 ,

$$
\psi\_{\delta,\Lambda}'(\mathbf{x}) = q\mathcal{W}^{(q)}(\mathbf{x})\zeta\_{\Lambda}(c\_2^{\Lambda}) + \Lambda Z^{(q)}(\mathbf{x}) \\
= q\mathcal{W}^{(q)}(\mathbf{x})\left(\frac{1-\Lambda Z^{(q)}(c\_2^{\Lambda})}{q\mathcal{W}^{(q)}(c\_2^{\Lambda})}\right) + \Lambda Z^{(q)}(\mathbf{x}).
$$

This implies that *<sup>v</sup>δ*,<sup>Λ</sup>(*c*Λ<sup>2</sup> −) = 1. For *x* > *c*Λ2 , we obtain by Equation (26) that

$$v\_{\delta,\Lambda}'(c\_2^{\Lambda}+)=1=v\_{\delta,\Lambda}'(c\_2^{\Lambda}-),$$

which implies the result. Let L be the operator defined as follows,

$$\mathcal{L}F(\mathbf{x}) := \gamma F'(\mathbf{x}) + \frac{\sigma^2}{2} F''(\mathbf{x}) + \int\_{(0,\infty)} (F(\mathbf{x} - \mathbf{z}) - F(\mathbf{x}) + F'(\mathbf{x})\mathbf{z}\mathbf{1}\_{\{0 < z \le 1\}}) \Pi(d\mathbf{z}), \quad \mathbf{x} > 0, \mathbf{z} \in \Omega$$

where *x* ∈ R and *F* is a function on R such that L*F*(*x*) is well defined.
