**6. Numerical Examples**

In this section, we confirm the obtained results by a sequence of numerical examples. Here, we assume that *X* is of the form

$$X\_t - X\_0 = t + 0.5B\_t - \sum\_{n=1}^{N\_t} Z\_{n\prime} \quad 0 \le t < \infty,$$

where *B* = {*Bt* : *t* ≥ <sup>0</sup>}, *N* = {*Nt* : *t* ≥ <sup>0</sup>}, and *Z* = {*Zn*}*n*≥<sup>1</sup> are a standard Brownian motion, a Poisson process with arrival rate *λ* = 0.4, and an i.i.d. sequence of random variables with distribution Gamma (1,2), respectively, which are assumed mutually independent. Since there is no closed form for the scale function *W*(*q*) associated with *X*, we use a numerical algorithm presented in Surya (2008) in order to approximate the inverse Laplace transform of Equation (4). Similarly, we approximate the derivatives of the scale functions and use the trapezoidal rule to calculate its integrals.

We first consider the case without transaction cost presented in Section 3.2. In Figure 1 (left), we plot the function *x* → *<sup>V</sup>*Λ(*x*) + Λ*K* for various values of Λ and a fixed value of *K*. For *x* ≥ *x*0, where *x*0 is such that *Kx*0 = *K*, its minimum over the considered values of Λ provides (an approximation of) *<sup>V</sup>*(*<sup>x</sup>*, *<sup>K</sup>*), indicated by the solid red line in the plot. Since the process has unbounded variation, then *K* = ∞. In Figure 1 (right), we plot, for *x* > *x*0, the Lagrange multiplier Λ∗ given in Theorem 2. We observe that Λ∗ goes to infinity as *x* ↓ *x*0 and remains always above 1.

**Figure 1.** (**Left**) Plots of *x* → *<sup>V</sup>*Λ(*x*) + Λ*K* for Λ = 1, 1.1, ... , 2, 3, ... , 10, 20, ... , 100, 200, ... , 1000, 2000, ... , 10, 000, 20, 000 (dotted) for the case *K* = 2.7. The minimum of *<sup>V</sup>*Λ(*x*) + Λ*K* over Λ is shown in solid bold-face red line. (**Right**) Plot of the Lagrange multiplier Λ∗ for *x* > *x*0, where *x*0 is such that *Kx*0= *K*.

In Figure 2, we show the values of *<sup>V</sup>*(*<sup>x</sup>*, *K*) and Lagrange multiplier Λ∗ as functions of (*<sup>x</sup>*, *<sup>K</sup>*). It is confirmed that *<sup>V</sup>*(*<sup>x</sup>*, *K*) increases as *x* and *K* increase, while Λ∗ increases as *x* and *K* decrease.

**Figure 2.** Plots of *<sup>V</sup>*(*<sup>x</sup>*, *K*) (**left**); and the Lagrange multiplier Λ∗ (**right**) as functions of *x* and *K*.

We now move to the case with transaction cost. First, we illustrate the results shown in Section 4. In Figure 3 (left), we plot the function *x* → *ζ*Λ(*x*) for the values of Λ = 1, ... , 9. We also plot its maximum value attained at *a*Λ and the value attained at the corresponding optimal values (*c*<sup>Λ</sup> 1 , *c*<sup>Λ</sup> 2 ) with transaction cost *δ* = 0.05. Note that, when Λ = 1, *a*Λ = *c*<sup>Λ</sup> 1 = 0 and for the other values of Λ, *ζ*Λ(*c*<sup>Λ</sup> 1 ) = *ζ*Λ(*c*<sup>Λ</sup> 2 ) < *ζ*Λ(*<sup>a</sup>*Λ). In Figure 3 (right), we plot the optimal thresholds *a*Λ, *c*<sup>Λ</sup> 1 and , *c*<sup>Λ</sup> 2 as function of Λ.

**Figure 3.** (**Left**) Plots of *x* → *ζ*Λ(*x*) for Λ = 1, ... , 9 and the corresponding values of *a*Λ, *c*<sup>Λ</sup> 1 and , *c*<sup>Λ</sup> 2 for *δ* = 0.05. (**Right**) Plots of the functions Λ → *a*Λ, *c*<sup>Λ</sup> 1 and , *c*<sup>Λ</sup> 2 .

In Figure 4, we illustrate the findings of Section 5.2. This figure is analogous to Figure 1 but with transaction cost *δ* as above. It can be seen that the change in the function *<sup>V</sup>δ*(*<sup>x</sup>*, *K*) is relatively very small, but the change in the optimal Lagrange multiplier Λ∗ is significant, being smaller in the case of transaction cost. A similar figure as Figure 2 in the case of transaction cost is omitted since both have the same shape.

**Figure 4.** (**Left**) Plots of *x* → *<sup>V</sup>*Λ(*x*) + Λ*K* for Λ = 1, 1.1, ... , 2, 3, ... , 10, 20, ... , 100, 200, ... , 1000, 2000, ... , 10, 000, 20, 000 (dotted) for the case *K* = 2.7. The minimum of *<sup>V</sup>δ*,<sup>Λ</sup>(*x*) + Λ*K* over Λ is plotted in solid bold-face red line. (**Right**) Plots of the Lagrange multipliers Λ∗ for *x* > *x*0, where *x*0 is such that *Kx*0= *K* with *δ* = 0 and *δ* = 0.05.
