**3. Solutions to the Boundary-Value Problem**

In this section, we obtain closed-form solutions to the boundary-value problem in (18)–(22) under various relations on the parameters of the model.

### *3.1. The General Solution of the Ordinary Differential Equation*

We first observe that the general solution of the equation in (18) has the form

$$\mathcal{V}(\mathbf{x}, \mathbf{s}, q) = \mathbb{C}\_1(\mathbf{s}, q) \,\,\Psi\_1(\mathbf{x}, \mathbf{s}, q) + \mathbb{C}\_2(\mathbf{s}, q) \,\,\Psi\_2(\mathbf{x}, \mathbf{s}, q) + \frac{\mathfrak{a}}{\mathfrak{a} + \lambda} \,\, e^{-\theta \mathfrak{s} - \kappa \eta} \tag{23}$$

where *Ci*(*<sup>s</sup>*, *q*), *i* = 1, 2, are some arbitrary continuously differentiable functions, and <sup>Ψ</sup>*i*(*<sup>x</sup>*,*s*, *q*), *i* = 1, 2, are the two fundamental positive solutions (i.e., nontrivial linearly independent particular solutions) of the homogeneous version of the second-order ordinary differential equation in (18). Without loss of generality, we may assume that <sup>Ψ</sup>1(*<sup>x</sup>*,*s*, *q*) and <sup>Ψ</sup>2(*<sup>x</sup>*,*s*, *q*) are the (strictly) increasing and decreasing (convex) functions, respectively. Note that these solutions should satisfy the properties <sup>Ψ</sup>1(*<sup>r</sup>*,*r*,*<sup>r</sup>*) ↑ ∞ and <sup>Ψ</sup>2(*<sup>r</sup>*,*r*,*<sup>r</sup>*) ↓ 0 as *r* ↑ ∞ and <sup>Ψ</sup>1(*<sup>r</sup>*,*r*,*<sup>r</sup>*) ↓ 0 and <sup>Ψ</sup>2(*<sup>r</sup>*,*r*,*<sup>r</sup>*) ↑ ∞ as *r* ↓ −∞ on the state space *E*<sup>3</sup> of the process (*<sup>X</sup>*, *S*, *Q*). These functions can be represented as the functionals

$$\Psi\_1(\mathbf{x}, \mathbf{s}, q) = \begin{cases} E\_{\mathbf{x}, \mathbf{s}, q} [e^{-\lambda\_5^{\mathbf{x}\prime}} I(\mathbf{\xi}^{\prime} < \infty)], & \text{if} \quad \mathbf{x} \le \mathbf{x}^{\prime} \\ 1/E\_{\mathbf{x}^{\prime}, \mathbf{s}, q} [e^{-\lambda\_5^{\mathbf{x}}} I(\mathbf{\xi}^{\prime} < \infty)], & \text{if} \quad \mathbf{x} \ge \mathbf{x}^{\prime} \end{cases} \tag{24}$$

and

$$\Psi\_2(\mathbf{x}, \mathbf{s}, q) = \begin{cases} 1/E\_{\mathbf{x}', \mathbf{s}, q} [e^{-\lambda\_\theta^x} I(\xi < \infty)], & \text{if} \quad \mathbf{x} \le \mathbf{x}' \\ E\_{\mathbf{x}, \mathbf{s}, q} [e^{-\lambda\_\theta^y} I(\xi' < \infty)], & \text{if} \quad \mathbf{x} \ge \mathbf{x}' \end{cases} \tag{25}$$

of the first hitting times *ξ* = inf{*t* ≥ 0 | *Xt* = *x*} and *ξ* = inf{*t* ≥ 0 | *Xt* = *x*} of the process *X* solving the stochastic differential equation in (1) and started at *x* and *x* such that (*<sup>x</sup>*,*s*, *q*),(*x*,*s*, *q*) ∈ *E*3, respectively (see, e.g., Rogers and Williams 1987, chp. V, sct. 50 for further details).

Hence, by applying the conditions of (19)–(22) to the function in (23), we obtain the equalities

$$\mathbb{C}\_{1}(s,q)\,\Psi\_{1}(s-a,s,q) + \mathbb{C}\_{2}(s,q)\,\Psi\_{2}(s-a,s,q) = \frac{\lambda}{a+\lambda}\,e^{-\theta s - \kappa q} \tag{26}$$

for *s* − *q* ≥ *a*,

$$\mathcal{C}\_1(s,q)\,\Psi\_1(q+b,s,q) + \mathcal{C}\_2(s,q)\,\Psi\_2(q+b,s,q) = -\frac{a}{a+\lambda}\,e^{-\theta\varepsilon-\kappa q} \tag{27}$$

for *s* − *q* ≥ *b*,

$$\sum\_{i=1}^{2} \left( \partial\_{\boldsymbol{q}} \mathbb{C}\_{i}(\mathbf{s}, \boldsymbol{q}) \, \middle| \, \mathbb{V}\_{i}(\boldsymbol{q}, \boldsymbol{s}, \boldsymbol{q}) + \mathbb{C}\_{i}(\mathbf{s}, \boldsymbol{q}) \, \partial\_{\boldsymbol{q}} \mathbb{V}\_{i}(\mathbf{x}, \boldsymbol{s}, \boldsymbol{q}) \Big|\_{\mathbf{x} = \boldsymbol{q}} \right) = \frac{\boldsymbol{a} \boldsymbol{\chi}}{\boldsymbol{a} + \boldsymbol{\lambda}} \, e^{-\theta \boldsymbol{\mathfrak{s}} - \boldsymbol{\chi} \boldsymbol{q}} \tag{28}$$

for 0 < *s* − *q* < *a*,

$$\sum\_{i=1}^{2} \left( \partial\_{s} \mathbb{C}\_{i}(s, q) \, \middle| \, \mathbb{Y}\_{i}(s, s, q) + \mathbb{C}\_{i}(s, q) \, \partial\_{s} \partial\_{s} \mathbb{Y}\_{i}(x, s, q) \big|\_{x=s} \right) = \frac{a\theta}{a + \lambda} e^{-\theta s - \kappa q} \tag{29}$$

for 0 < *s* − *q* < *b*.

### *3.2. The Solution to the Boundary-Value Problem*

We now derive the solution of the boundary-value problem in (18)–(22). For this purpose, we recall that the second and third components of the process (*<sup>X</sup>*, *S*, *Q*) can increase and decrease only at the planes *d*31 and *d*32, that is, when *Xt* = *St* and *Xt* = *Qt* for *t* ≥ 0, respectively.

**(i)** Let us first consider the domain *a* ∨ *b* ≤ *s* − *q* ≤ *a* + *b*. In this case, solving the system of equations in (26) and (27), we conclude that the candidate value function admits the representation

$$\mathcal{V}(\mathbf{x}, \mathbf{s}, \boldsymbol{\eta}; \infty) = \mathbb{C}\_1(\mathbf{s}, \boldsymbol{\eta}; \infty) \,\,\Psi\_1(\mathbf{x}, \mathbf{s}, \boldsymbol{\eta}) + \mathbb{C}\_2(\mathbf{s}, \boldsymbol{\eta}; \infty) \,\,\Psi\_2(\mathbf{x}, \mathbf{s}, \boldsymbol{\eta}) + \frac{\boldsymbol{\alpha}}{\boldsymbol{\alpha} + \boldsymbol{\lambda}} \,\varepsilon^{-\theta \mathbf{s} - \mathbf{x}\boldsymbol{\eta}} \tag{30}$$

in the region *<sup>R</sup>*<sup>3</sup>(∞) = {(*<sup>x</sup>*,*s*, *q*) ∈ *E*<sup>3</sup> | *q* ≤ *s* − *a* ≤ *x* ≤ *q* + *b* ≤ *<sup>s</sup>*}, with

$$\mathbb{C}\_{1}(s,q;\infty) = \frac{e^{-\theta s - \kappa q} (\lambda \Psi\_{2}(q+b,s,q) + a\Psi\_{2}(s-a,s,q))/(a+\lambda)}{\Psi\_{1}(s-a,s,q)\Psi\_{2}(q+b,s,q) - \Psi\_{1}(q+b,s,q)\Psi\_{2}(s-a,s,q)}\tag{31}$$

and

$$\mathbb{C}\_{2}(s,q;\infty) = \frac{e^{-\theta s - \kappa q} (\lambda \Psi\_{1}(q+b,s,q) + a\Psi\_{1}(s-a,s,q))/(a+\lambda)}{\Psi\_{1}(q+b,s,q)\Psi\_{2}(s-a,s,q) - \Psi\_{1}(s-a,s,q)\Psi\_{2}(q+b,s,q)}\tag{32}$$

for all *q* + *a* ∨ *b* ≤ *s* ≤ *q* + *a* + *b* (see Figures 1 and 2 below).

**Figure 1.** A computer drawing of the state space of the process (*<sup>X</sup>*, *S*, *Q*), for some *q* ∈ R fixed and *a* < *b*.

**(ii)** Let us now consider the domain *a* ≤ *s* − *q* < *b*. In this case, it follows from the equations in (26) and (29) that the candidate value function admits the representation

$$\mathcal{V}(\mathbf{x}, \mathbf{s}, q; a) = \mathbb{C}\_1(\mathbf{s}, q; a) \,\,\Psi\_1(\mathbf{x}, \mathbf{s}, q) + \mathbb{C}\_2(\mathbf{s}, q; a) \,\,\Psi\_2(\mathbf{x}, \mathbf{s}, q) + \frac{a}{a + \lambda} \,\, e^{-\theta \mathbf{s} - \mathbf{x}q} \tag{33}$$

in the region *<sup>R</sup>*<sup>3</sup>(*a*) = {(*<sup>x</sup>*,*s*, *q*) ∈ *E*<sup>3</sup> | *q* ≤ *s* − *a* ≤ *x* ≤ *s* < *q* + *b*}, with

$$\mathbb{C}\_{2}(s,q;a) = \frac{\lambda}{a+\lambda} \frac{e^{-\theta s - \kappa q}}{\mathbb{Y}\_{2}(s-a,s,q)} - \mathbb{C}\_{1}(s,q;a) \frac{\mathbb{Y}\_{1}(s-a,s,q)}{\mathbb{Y}\_{2}(s-a,s,q)}\tag{34}$$

for *q* + *a* ≤ *s* < *q* + *b*, where *<sup>C</sup>*1(*<sup>s</sup>*, *q*; *a*) solves the first-order linear ordinary differential equation

$$\partial\_s \mathbb{C}\_1(s, q; a) \, H\_{1,2}(s, q; a) + \mathbb{C}\_1(s, q; a) \, H\_{1,1}(s, q; a) = H\_{1,0}(s, q; a) \tag{35}$$

with

$$H\_{1,2}(s,q;a) = \Psi\_1(s,s,q) - \Psi\_2(s,s,q)\frac{\Psi\_1(s-a,s,q)}{\Psi\_2(s-a,s,q)}\tag{36}$$

$$\left.H\_{1,1}(\mathbf{s}, q; \mathbf{a}) = \left. \partial\_{\mathbf{s}} \Psi\_1(\mathbf{x}, \mathbf{s}, q) \right|\_{\mathbf{x} = \mathbf{s}} \tag{37}$$

$$\begin{split} -\partial\_{\mathbf{s}} \left( \frac{\Psi\_{1}(\mathbf{s} - a, \mathbf{s}, q)}{\Psi\_{2}(\mathbf{s} - a, \mathbf{s}, q)} \right) \Psi\_{2}(\mathbf{s}, \mathbf{s}, q) - \frac{\Psi\_{1}(\mathbf{s} - a, \mathbf{s}, q)}{\Psi\_{2}(\mathbf{s} - a, \mathbf{s}, q)} \partial\_{\mathbf{s}} \Psi\_{1}(\mathbf{x}, \mathbf{s}, q) \big|\_{\mathbf{x} = \mathbf{s}} \\ \mathcal{H}\_{1,0}(\mathbf{s}, q; a) = \frac{\lambda}{\mathfrak{a} + \lambda} \left( \theta \, e^{-\theta \mathfrak{s} - \mathbf{x} \underline{q}} \\ -\partial\_{\mathbf{s}} \left( \frac{e^{-\theta \mathfrak{s} - \mathbf{x} \underline{q}}}{\Psi\_{2}(\mathbf{s} - a, \mathbf{s}, q)} \right) \Psi\_{2}(\mathbf{s}, \mathbf{s}, q) - \frac{e^{-\theta \mathfrak{s} - \mathbf{x} \underline{q}}}{\Psi\_{2}(\mathbf{s} - a, \mathbf{s}, q)} \partial\_{\mathbf{s}} \Psi\_{2}(\mathbf{x}, \mathbf{s}, q) \big|\_{\mathbf{x} = \mathbf{s}} \right) \end{split} \tag{38}$$

for all *q* + *a* ≤ *s* < *q* + *b*. Observe that the process (*<sup>X</sup>*, *S*, *Q*) can exit the region *<sup>R</sup>*<sup>3</sup>(*a*) by passing to the region *<sup>R</sup>*<sup>3</sup>(∞) in part (i) of this subsection only through the point *x* = *s* = *q* + *b*, by hitting the plane *d*31 so that increasing its second component *S*. Thus, the candidate function *<sup>V</sup>*(*<sup>x</sup>*,*s*, *q*) should be continuous at the point (*q* + *b*, *q* + *b*, *q*), that is expressed by the equality

$$\mathbb{C}\_{1}(q+b,q;a)\,\Psi\_{1}(q+b,q+b,q) + \mathbb{C}\_{2}(q+b,q;a)\,\Psi\_{2}(q+b,q+b,q) = -\frac{a}{a+\lambda}\,e^{-\theta(q+b)-\kappa q} \tag{39}$$

for all *q* ∈ R (see Figure 1 above). Hence, solving the differential equation in (35) together with the system of equations in (34) with *s* = *q* + *b* and (39), we obtain

$$\begin{split} \mathcal{C}\_{1}(s,q;a) &= \mathcal{C}\_{1}(q+b,q;a) \exp\left(\int\_{s}^{q+b} \frac{H\_{1,1}(u,q;a)}{H\_{1,2}(u,q;a)} du\right) \\ &- \int\_{s}^{q+b} \frac{H\_{1,0}(u,q;a)}{H\_{1,2}(u,q;a)} \exp\left(\int\_{s}^{u} \frac{H\_{1,1}(v,q;a)}{H\_{1,2}(v,q;a)} dv\right) du \end{split} \tag{40}$$

for all *q* + *a* ≤ *s* < *q* + *b*, where *<sup>C</sup>*1(*q* + *b*, *q*; *a*) is given by

$$\begin{aligned} \mathbb{C}\_{1}(q+b,q;a) & \mathbb{V}\_{1}(q,a) \\ \mathbb{V}\_{2} &= \frac{e^{-\theta(q+b)-\chi q} \left(\lambda \mathbb{V}\_{2}(q+b,q+b,q) + a\mathbb{V}\_{2}(q+b-a,q+b,q)\right)/(a+\lambda)}{\mathbb{V}\_{1}(q+b-a,q+b,q)\mathbb{V}\_{2}(q+b,q+b,q) - \mathbb{V}\_{1}(q+b,q+b,q)\mathbb{V}\_{2}(q+b-a,q+b,q)} \end{aligned} \tag{41}$$

for all *q* ∈ R.

Note that in the case in which *μ*(*<sup>s</sup>*, *q*) = *μ*(*s*) and *<sup>σ</sup>*(*<sup>s</sup>*, *q*) = *σ*(*s*) in (1) as well as *κ* = 0 and *b* = ∞ in (6), the candidate value function admits the representation of (33) with *<sup>V</sup>*(*<sup>x</sup>*,*s*, *q*; *a*) = *<sup>U</sup>*(*<sup>x</sup>*,*s*; *a*) and *Ci*(*<sup>s</sup>*, *q*; *a*) = *Di*(*s*; *a*) as well as <sup>Ψ</sup>*i*(*<sup>x</sup>*,*s*, *q*) = <sup>Φ</sup>*i*(*<sup>x</sup>*,*<sup>s</sup>*), *i* = 1, 2. Moreover, we observe that *<sup>D</sup>*1(<sup>∞</sup>; *a*) = 0 should hold in (33), since otherwise *<sup>U</sup>*(*<sup>x</sup>*,*s*; *a*) → ±∞ as *x* = *s* ↑ <sup>∞</sup>, which must be excluded, by virtue of the obvious fact that the value function *<sup>V</sup>*∗(*<sup>x</sup>*,*s*, *q*) = *<sup>U</sup>*∗(*<sup>x</sup>*,*<sup>s</sup>*) in (6) is bounded. Therefore, using arguments similar to the ones above, we conclude that the function *<sup>C</sup>*2(*<sup>s</sup>*, *q*; *a*) = *<sup>D</sup>*2(*<sup>s</sup>*; *a*) has the form of (34) with *<sup>C</sup>*1(*<sup>s</sup>*, *q*; *a*) = *<sup>D</sup>*1(*<sup>s</sup>*; *a*) given by

$$D\_1(s;a) = -\int\_s^\infty \frac{G\_{1,0}(u;\infty)}{G\_{1,2}(u;\infty)} \exp\left(\int\_s^u \frac{G\_{1,1}(v;\infty)}{G\_{1,2}(v;\infty)} dv\right) du\tag{42}$$

and *<sup>H</sup>*1,*j*(*<sup>s</sup>*, *q*; *a*) = *<sup>G</sup>*1,*j*(*<sup>s</sup>*; *<sup>a</sup>*), *j* = 0, 1, 2, from (36)–(38), for all *s* ∈ R.

**(iii)** Let us now consider the domain *b* ≤ *s* − *q* < *a*. In this case, it follows from the equations in (27) and (28) that the candidate value function admits the representation

$$\mathcal{V}(\mathbf{x}, \mathbf{s}, q; b) = \mathbb{C}\_1(\mathbf{s}, q; b) \,\,\Psi\_1(\mathbf{x}, \mathbf{s}, q) + \mathbb{C}\_2(\mathbf{s}, q; b) \,\,\Psi\_2(\mathbf{x}, \mathbf{s}, q) + \frac{\mathfrak{a}}{\mathfrak{a} + \lambda} e^{-\theta \mathfrak{s} - \mathbf{x}q} \tag{43}$$

in the region *R*<sup>3</sup>(*b*) = {(*<sup>x</sup>*,*s*, *q*) ∈ *E*<sup>3</sup> |*s* − *a* < *q* ≤ *x* ≤ *q* + *b* ≤ *<sup>s</sup>*}, with

$$\mathcal{C}\_{2}(s,q;b) = -\frac{a}{a+\lambda} \frac{e^{-\theta s - \kappa q}}{\Psi\_{2}(q+b,s,q)} - \mathcal{C}\_{1}(s,q;b) \frac{\Psi\_{1}(q+b,s,q)}{\Psi\_{2}(q+b,s,q)}\tag{44}$$

for *q* + *b* ≤ *s* < *q* + *a*, where *<sup>C</sup>*1(*<sup>s</sup>*, *q*; *b*) solves the first-order linear ordinary differential equation

$$\partial\_{\eta} \mathbb{C}\_{1}(s, q; b) \, H\_{2,2}(s, q; b) + \mathbb{C}\_{1}(s, q; b) \, H\_{2,1}(s, q; b) = H\_{2,0}(s, q; b) \tag{45}$$

with

$$H\_{2,2}(s, q; b) = \Psi\_1(q, s, q) - \Psi\_2(q, s, q) \frac{\Psi\_1(q + b, s, q)}{\Psi\_2(q + b, s, q)}\tag{46}$$

$$\begin{split} \left. \Pi\_{2,1}(s, q; b) = \left. \partial\_{q} \Psi\_{1}(\mathbf{x}, s, q) \right|\_{\mathbf{x} = q} \\ -\left. \partial\_{q} \left( \frac{\Psi\_{1}(q + b, s, q)}{\Psi\_{2}(q + b, s, q)} \right) \right|\_{\mathbf{Y}} \Psi\_{2}(q, s, q) - \frac{\Psi\_{1}(q + b, s, q)}{\Psi\_{2}(q + b, s, q)} \left. \partial\_{q} \Psi\_{2}(\mathbf{x}, s, q) \right|\_{\mathbf{x} = q} \end{split} \tag{47}$$

$$\begin{aligned} \,^{\alpha\_{\overline{q}}} \Big( \,^{\Psi\_{2}} \Psi\_{2}(q+b,s,q) \Big) \,^{\Psi\_{2}(q,s,q)} \, \quad \Psi\_{2}(q+b,s,q) \, ^{\alpha\_{\overline{q}}} \, ^{\Psi\_{2}(\infty,s,q)} \Big) \, \_{x=q} \\ \,^{\alpha\_{\overline{q}}} H\_{2,0}(s,q;b) &= \frac{a}{a+\lambda} \left( \,^{\varphi} \, \epsilon^{-\theta s-\overline{\kappa}q} \\ &+ \partial\_{q} \left( \frac{\epsilon^{-\theta s-\overline{\kappa}q}}{\Psi\_{2}(q+b,s,q)} \right) \, \Psi\_{2}(q,s,q) + \frac{\epsilon^{-\theta s-\overline{\kappa}q}}{\Psi\_{2}(q+b,s,q)} \, \partial\_{q} \Psi\_{2}(\mathbf{x},s,q) \Big) \, \right. \end{aligned} \tag{48}$$

for all *q* + *b* ≤ *s* < *q* + *a*. Observe that the process (*<sup>X</sup>*, *S*, *Q*) can exit *R*<sup>3</sup>(*b*) by passing to the region *<sup>R</sup>*<sup>3</sup>(∞) in part (i) of this subsection only through the point *x* = *q* = *s* − *a*, by hitting the plane *d*32 so that decreasing its third component *Q*. Then, the candidate value function should be continuous at the point (*s* − *a*,*s*,*s* − *<sup>a</sup>*), that is expressed by the equality

$$\mathbb{C}\_{1}(\mathbf{s}, \mathbf{s} - a; b) \, \mathbb{V}\_{1}(\mathbf{s} - a, \mathbf{s}, \mathbf{s} - a) + \mathbb{C}\_{2}(\mathbf{s}, \mathbf{s} - a; b) \, \mathbb{V}\_{2}(\mathbf{s} - a, \mathbf{s}, \mathbf{s} - a) = -\frac{\mathfrak{A}}{a + \lambda} e^{-\mathfrak{A}\mathbf{s} - \mathbf{x}(\mathbf{s} - a)} \tag{49}$$

for all *s* ∈ R (see Figure 2 below). Hence, solving the differential equation in (45) together with the system of equations in (44) with *q* = *s* − *a* and (49), we obtain

$$\begin{split} \mathbb{C}\_{1}(\mathbf{s}, q; b) &= \mathbb{C}\_{1}(\mathbf{s}, \mathbf{s} - a; b) \, \exp\left(-\int\_{s-a}^{q} \frac{H\_{2,1}(\mathbf{s}, u; b)}{H\_{2,2}(\mathbf{s}, u; b)} du\right) \\ &+ \int\_{s-a}^{q} \frac{H\_{2,0}(\mathbf{s}, u; b)}{H\_{2,2}(\mathbf{s}, u; b)} \exp\left(-\int\_{u}^{q} \frac{H\_{2,1}(\mathbf{s}, v; b)}{H\_{2,2}(\mathbf{s}, v; b)} dv\right) du \end{split} \tag{50}$$

for all *q* + *b* ≤ *s* < *q* + *a*, where *<sup>C</sup>*1(*<sup>s</sup>*,*<sup>s</sup>* − *a*; *b*) is given by

$$\begin{aligned} \mathbb{C}\_{1}(\mathbf{s}, \mathbf{s} - a; b) \\ \mathbb{V}\_{1} &= \frac{e^{-\theta \mathbf{s} - \mathbf{x}(\mathbf{s} - a)} \left(\lambda \Psi\_{2}(\mathbf{s} - a + b, \mathbf{s}, \mathbf{s} - a) + a \Psi\_{2}(\mathbf{s} - a, \mathbf{s}, \mathbf{s} - a)\right) / (a + \lambda)}{\Psi\_{1}(\mathbf{s} - a, \mathbf{s}, \mathbf{s} - a) \Psi\_{2}(\mathbf{s} - a + b, \mathbf{s}, \mathbf{s} - a) - \Psi\_{1}(\mathbf{s} - a + b, \mathbf{s}, \mathbf{s} - a) \Psi\_{2}(\mathbf{s} - a, \mathbf{s}, \mathbf{s} - a)} \end{aligned} \tag{51}$$

for *s* ∈ R.

**Figure 2.** A computer drawing of the state space of the process (*<sup>X</sup>*, *S*, *Q*), for some *q* ∈ R fixed and *b* ≤ *a*.

**(iv)** Let us now consider the domain 0 ≤ *s* − *q* < *a* ∧ *b*. In this case, it follows that the candidate value function admits the representation

$$V(\mathbf{x}, \mathbf{s}, q; 0) = \mathbb{C}\_1(\mathbf{s}, q; 0) \,\,\Psi\_1(\mathbf{x}, \mathbf{s}, q) + \mathbb{C}\_2(\mathbf{s}, q; 0) \,\,\Psi\_2(\mathbf{x}, \mathbf{s}, q) + \frac{\mathfrak{a}}{\mathfrak{a} + \lambda} \,\,\varepsilon^{-\theta \mathfrak{s} - \mathbf{x}q} \tag{52}$$

in the region *R*<sup>3</sup>(0) = {(*<sup>x</sup>*,*s*, *q*) ∈ *E*<sup>3</sup> |*s* − *a* < *q* ≤ *x* ≤ *s* < *q* + *b*}, where *Ci*(*<sup>s</sup>*, *q*; <sup>0</sup>), *i* = 1, 2, solve the first-order linear partial differential equations in (28) and (29), for all 0 < *s* − *q* < *a* ∧ *b*. Observe that, the process (*<sup>X</sup>*, *S*, *Q*) can exit *R*<sup>3</sup>(0) by passing to the region *<sup>R</sup>*<sup>3</sup>(*a* ∧ *b*) in part (ii) or (iii) of this subsection only through the points *x* = *s* = *q* + *a* ∧ *b* and *x* = *q* = *s* − *a* ∧ *b*, by hitting the plane *d*31 or *d*32, so that increasing its second or third components, *S* or *Q*, respectively. Then, the candidate value function should be continuous at the points (*q* + *a* ∧ *b*, *q* + *a* ∧ *b*, *q*) and (*s* − *a* ∧ *b*,*s*,*<sup>s</sup>* − *a* ∧ *b*), that is expressed by the equalities

$$\begin{aligned} &\mathbb{C}\_1(q+a\wedge b,q;0)\,\Psi\_1(q+a\wedge b,q+a\wedge b,q) \\ &+\mathbb{C}\_2(q+a\wedge b,q;0)\,\Psi\_2(q+a\wedge b,q+a\wedge b,q) \\ &=\mathbb{C}\_1(q+a\wedge b,q;a\wedge b)\,\Psi\_1(q+a\wedge b,q+a\wedge b,q) \\ &+\mathbb{C}\_2(q+a\wedge b,q;a\wedge b)\,\Psi\_2(q+a\wedge b,q+a\wedge b,q) \end{aligned}$$

for all *q* ∈ R, and

$$\begin{aligned} &\mathbb{C}\_{1}(\mathbf{s},\mathbf{s}-a\wedge b;0)\,\Psi\_{1}(\mathbf{s}-a\wedge b,\mathbf{s},\mathbf{s}-a\wedge b) \\ &+\mathbb{C}\_{2}(\mathbf{s},\mathbf{s}-a\wedge b;0)\,\Psi\_{2}(\mathbf{s}-a\wedge b,\mathbf{s},\mathbf{s}-a\wedge b) \\ &=\mathbb{C}\_{1}(\mathbf{s},\mathbf{s}-a\wedge b;a\wedge b)\,\Psi\_{1}(\mathbf{s}-a\wedge b,\mathbf{s},\mathbf{s}-a\wedge b) \\ &+\mathbb{C}\_{2}(\mathbf{s},\mathbf{s}-a\wedge b;a\wedge b)\,\Psi\_{2}(\mathbf{s}-a\wedge b,\mathbf{s},\mathbf{s}-a\wedge b) \end{aligned}$$

for all *s* ∈ R, where *Ci*(*q* + *a* ∧ *b*, *q*; *a* ∧ *b*) and *Ci*(*<sup>s</sup>*,*<sup>s</sup>* − *a* ∧ *b*; *a* ∧ *b*), *i* = 1, 2, are found in (34)+ (40) or (44)+(50). Moreover, we have the property *<sup>C</sup>*2(*<sup>r</sup>*,*r*; 0) → 0 as *r* ↓ <sup>−</sup>∞, since otherwise *<sup>V</sup>*(*<sup>r</sup>*,*r*,*r*; 0) → ±<sup>∞</sup>, that must be excluded by virtue of the obvious fact that the value function in (6) is bounded (see Figures 1 and 2 above). We may therefore conclude that the candidate value function admits the representation of (52) in the region *R*<sup>3</sup>(0) above, where *Ci*(*<sup>s</sup>*, *q*; <sup>0</sup>), *i* = 1, 2, provide a unique solution of the two-dimensional system of first-order linear partial differential equations in (21) and (22) with the boundary conditions of (53)–(54) and *<sup>C</sup>*2(*<sup>r</sup>*,*r*; 0) → 0 as *r* ↓ <sup>−</sup>∞. Here, the existence and uniqueness

of solutions to such special kinds of systems of equations follow from the classical existence and uniqueness results of solutions to appropriate boundary-value problems for first-order linear partial differential equations.
