*2.2. The Boundary-Value Problems*

By means of standard arguments based on the application of Itô's formula (see, e.g., Karatzas and Shreve 1991, chp. V, sct. 5.1), it is shown that the infinitesimal operator L of the process (*<sup>X</sup>*, *S*, *Q*) acts on a function *<sup>F</sup>*(*<sup>x</sup>*,*s*, *q*) from the class *C*2,1,1 on the interior of *E*<sup>3</sup> according to the rule

$$
\mu(\mathbb{L}F)(\mathbf{x}, \mathbf{s}, q) = \mu(\mathbf{x}, \mathbf{s}, q) \, \partial\_{\mathbf{x}} F(\mathbf{x}, \mathbf{s}, q) + \frac{\sigma^2(\mathbf{x}, \mathbf{s}, q)}{2} \, \partial\_{\mathbf{x}\mathbf{x}} F(\mathbf{x}, \mathbf{s}, q) \tag{7}
$$

for all *q* < *x* < *s*. It follows from the results of general theory of Markov processes (see, e.g., Dynkin 1965, chp. V) that the value function *<sup>W</sup>*∗(*<sup>T</sup>*; *x*,*s*, *q*) in (6) solves the equivalent parabolic-type boundary-value problem

$$(\mathbb{L}\mathcal{W} - \lambda\mathcal{W} - \partial\_T \mathcal{W})(T; \mathbf{x}, \mathbf{s}, q) = 0 \quad \text{for} \quad (\mathbf{s} - a) \lor q < \mathbf{x} < \mathbf{s} \land (q + b) \tag{8}$$

$$\left. \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \right|\_{\mathbf{x} = (s - a) +} = e^{-\theta s - \mathbf{x}q} \quad \text{for} \quad s - q \ge a \tag{9}$$

$$\left. \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \right|\_{\mathbf{x} = (q+b)-} = 0 \quad \text{for} \quad \mathbf{s} - q \ge b \tag{10}$$

$$\left. \partial\_{\eta} \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \right|\_{\mathbf{x} = q +} = 0 \quad \text{for} \quad 0 < \mathbf{s} - q < a \tag{11}$$

$$\left. \partial\_s \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \right|\_{\mathbf{x} = \mathbf{s} - \mathbf{s}} = 0 \quad \text{for} \quad 0 < \mathbf{s} - q < b \tag{12}$$

for all *T* > 0. In this case, using the integration-by-parts formula, and taking into account the assumption that the value function in (6) is bounded, we have

$$\int\_{0}^{\infty} \partial\_{T} \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \, \mathbf{a} \, e^{-\mathbf{a}T} \, dT \tag{13}$$

$$\begin{split} \mathcal{V} &= \left[ \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \, \mathbf{a} \, e^{-\mathbf{a}T} \right]\_{0}^{\infty} + \int\_{0}^{\infty} \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \, \mathbf{a}^{2} \, e^{-\mathbf{a}T} \, dT \\ &= -\mathbf{a} \, e^{-\theta \mathbf{s} - \kappa q} + \int\_{0}^{\infty} \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \, \mathbf{a}^{2} \, e^{-\mathbf{a}T} \, dT = -\mathbf{a} \, e^{-\theta \mathbf{s} - \kappa q} + \mathbf{a} \, V(\mathbf{x}, \mathbf{s}, q) \end{split} \tag{14}$$

$$\begin{split} \int\_{0}^{\infty} \partial\_{\mathbf{x}} \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \, \mathbf{a} \, e^{-\mathbf{a}T} \, dT = \partial\_{\mathbf{x}} V(\mathbf{x}, \mathbf{s}, q) \end{split} \tag{14}$$

$$\int\_{0}^{\infty} \partial\_{\text{xx}} \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \, q \, e^{-aT} \, dT = \partial\_{\text{xx}} V(\mathbf{x}, \mathbf{s}, q) \tag{15}$$

$$\int\_0^\infty \partial\_\eta \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \, d\mathbf{x} \, e^{-\mathbf{a}T} \, dT = \partial\_\emptyset V(\mathbf{x}, \mathbf{s}, q) \tag{16}$$

and 
$$\int\_{0}^{\infty} \partial\_{\sf s} \mathcal{W}(T; \mathbf{x}, \mathbf{s}, q) \, \mathbf{a} \, e^{-\mathbf{a}T} \, dT = \partial\_{\sf s} V(\mathbf{x}, \mathbf{s}, q) \tag{17}$$

for all (*<sup>x</sup>*,*s*, *q*) ∈ *E*3. Hence, it follows from the boundary-value problem in (8)–(12), that the value function *<sup>V</sup>*∗(*<sup>x</sup>*,*s*, *q*) in (6) solves the equivalent inhomogeneous ordinary boundary-value problem

$$(\mathbb{L}V - (\mathfrak{a} + \lambda)\,V)(\mathbf{x}, \mathbf{s}, \boldsymbol{\eta}) = -a\,\mathsf{e}\,e^{-\theta\mathbf{s} - \mathbf{x}\boldsymbol{\eta}}\quad\text{for}\quad(\mathbf{s} - \mathbf{a}) \vee \boldsymbol{\eta} < \mathbf{x} < \mathbf{s} \wedge (\boldsymbol{\eta} + \boldsymbol{b})\tag{18}$$

$$\left.V(\mathbf{x},\mathbf{s},q)\right|\_{\mathbf{x}=(s-a)+} = e^{-\theta\mathbf{s}-\mathbf{x}q} \quad \text{for} \quad s-q \ge a \tag{19}$$

$$\left.V(\mathbf{x}, \mathbf{s}, q)\right|\_{\mathbf{x}=(q+b)-} = 0 \quad \text{for} \quad \mathbf{s} - q \ge b \tag{20}$$

$$\left. \partial\_q V(\mathbf{x}, \mathbf{s}, q) \right|\_{\mathbf{x} = q + \tau} = 0 \quad \text{for} \quad 0 < \mathbf{s} - q < a \tag{21}$$

$$\left. \partial\_s V(\mathbf{x}, s, q) \right|\_{\mathbf{x} = \mathbf{s} - \mathbf{u}} = 0 \quad \text{for} \quad 0 < \mathbf{s} - q < b \tag{22}$$

for *a*, *b* > 0 fixed. Note that the homogeneous version of the ordinary differential boundary-value problem in (18)–(22) in a model with more general diffusion-type processes *X* was explicitly solved in (Gapeev and Rodosthenous 2015, sct. 3).
