**4. The Result and Proof**

Taking into account the facts proved above, we now formulate the main result of the paper, which extends the assertion of (Gapeev and Rodosthenous 2015, Theorem 4.1) to the case of the model with a random independent exponential time horizon and the (*<sup>X</sup>*, *S*, *Q*)-setting.

**Theorem 1.** *Suppose that the coefficients μ*(*<sup>x</sup>*,*s*, *q*) *and <sup>σ</sup>*(*<sup>x</sup>*,*s*, *q*) *of the diffusion-type process X given by (1)–(2) are continuously differentiable functions on* [−∞, ∞]<sup>3</sup> *which are of at most linear growth in x and uniformly bounded in s and q. Let η be a random time with the distribution <sup>P</sup>*(*η* > *t*) = *e*<sup>−</sup>*<sup>α</sup>t, for all t* ≥ 0 *and some α* > 0 *fixed, which is independent of the process X. Then, the joint Laplace transform <sup>V</sup>*∗(*<sup>x</sup>*,*s*, *q*) *from (4) of the associated with X random variables τa* ∧ *η, <sup>S</sup><sup>τ</sup>a*∧*η, and Q<sup>τ</sup>a*∧*<sup>η</sup> such that τa* < *ζb from (3), admits the representation*

$$V\_\*(x,s,q) = \begin{cases} V(x,s,q; \infty), & \text{if } q \le s - a \le x \le q + b \le s \\ V(x,s,q;a), & \text{if } q \le s - a \le x \le s < q + b \\ V(x,s,q;b), & \text{if } s - a < q \le x \le q + b \le s \\ V(x,s,q;0), & \text{if } s - a < q \le x \le s < q + b \end{cases} \tag{64}$$

*for any a*, *b* > 0 *fixed. Here, the function <sup>V</sup>*(*<sup>x</sup>*,*s*, *q*; ∞) *takes the form of (30) with the coefficients Ci*(*<sup>s</sup>*, *q*; <sup>∞</sup>)*, i* = 1, 2*, given by (31)–(32), <sup>V</sup>*(*<sup>x</sup>*,*s*, *q*; *a*) *takes the form of (33) with Ci*(*<sup>s</sup>*, *q*; *<sup>a</sup>*)*, i* = 1, 2*, given by (34) and (40) (or (42) when μ*(*<sup>x</sup>*,*s*, *q*) = *μ*(*<sup>x</sup>*,*<sup>s</sup>*) *and <sup>σ</sup>*(*<sup>x</sup>*,*s*, *q*) = *<sup>σ</sup>*(*<sup>x</sup>*,*<sup>s</sup>*) *as well as κ* = 0 *and b* = ∞*) <sup>V</sup>*(*<sup>x</sup>*,*s*, *q*; *b*) *takes the form of (43) with Ci*(*<sup>s</sup>*, *q*; *b*)*, i* = 1, 2*, given by (44) and (50), and <sup>V</sup>*(*<sup>x</sup>*,*s*, *q*; 0) *takes the form of (52) with Ci*(*<sup>s</sup>*, *q*; <sup>0</sup>)*, i* = 1, 2*, being a unique solution of the two-dimensional system of first-order partial differential equations in (28)–(29) and satisfying the conditions of (53)–(54) together with the property <sup>C</sup>*2(*<sup>r</sup>*,*r*; 0) → 0 *as r* ↓ <sup>−</sup>∞*.*

**Proof.** In order to verify the assertion stated above, it remains to show that the function defined in (64) coincides with the value function in (6). For this purpose, let us denote by *<sup>V</sup>*(*<sup>x</sup>*,*s*, *q*) the right-hand side of the expression in (64). Then, taking into account the fact that the function *<sup>V</sup>*(*<sup>x</sup>*,*s*, *q*) is *C*2,1,1 on *E*3, by applying the change-of-variable formula from (Peskir 2007, Theorem 3.1) to *<sup>e</sup>*<sup>−</sup>*<sup>λ</sup>tV*(*Xt*, *St*, *Qt*), we obtain that the expression

*e* <sup>−</sup>*<sup>λ</sup>*(*<sup>τ</sup>a*<sup>∧</sup>*ζb*∧*<sup>t</sup>*) *<sup>V</sup>*(*<sup>X</sup><sup>τ</sup>a*<sup>∧</sup>*ζb*∧*t*, *<sup>S</sup><sup>τ</sup>a*<sup>∧</sup>*ζb*∧*t*, *Q<sup>τ</sup>a*<sup>∧</sup>*ζb*∧*<sup>t</sup>*) = *<sup>V</sup>*(*<sup>x</sup>*,*s*, *q*) + *<sup>M</sup><sup>τ</sup>a*<sup>∧</sup>*ζb*∧*<sup>t</sup>* + *<sup>τ</sup>a*<sup>∧</sup>*ζb*∧*<sup>t</sup>* 0 *e*<sup>−</sup>*λ<sup>u</sup>* (L*V* − (*α* + *λ*)*V* + *α e*<sup>−</sup>*θSu*−*<sup>κ</sup>Qu* )(*Xu*, *Su*, *Qu*) *I*(*Xu* = *Su*, *Xu* = *Qu*) *du* + *<sup>τ</sup>a*<sup>∧</sup>*ζb*∧*<sup>t</sup>* 0 *e*<sup>−</sup>*λ<sup>u</sup> <sup>∂</sup>q<sup>V</sup>*(*Xu*, *Su*, *Qu*) *I*(*Xu* = *Qu*) *dQu* + *<sup>τ</sup>a*<sup>∧</sup>*ζb*∧*<sup>t</sup>* 0 *e*<sup>−</sup>*λ<sup>u</sup> ∂s<sup>V</sup>*(*Xu*, *Su*, *Qu*) *I*(*Xu* = *Su*) *dSu* (65)

holds, for the stopping times *τa* and *ζb* given by (3), and all *t* ≥ 0. Here, the process *M* = (*Mt*)*t*≥0 defined by

$$M\_t = \int\_0^t e^{-\lambda u} \partial\_x V(X\_{\mathfrak{u}\prime} S\_{\mathfrak{u}\prime} Q\_{\mathfrak{u}}) \, I(X\_{\mathfrak{u}} \neq S\_{\mathfrak{u}\prime} X\_{\mathfrak{u}} \neq Q\_{\mathfrak{u}}) \, \sigma(S\_{\mathfrak{u}\prime} Q\_{\mathfrak{u}}) \, dB\_{\mathfrak{u}} \tag{66}$$

is a continuous local martingale under *Px*,*s*,*q*. Note that, since the time spent by the process *X* at the hyperplanes *d*3*k*, *k* = 1, 2, is of Lebesgue measure zero, the indicators in the second line of the expression in (65) and in (66) can be ignored. Moreover, since the processes *S* and *Q* change their values only on the hyperplanes *d*31 and *d*32, respectively, the indicators appearing in the third and fourth lines of (65) can be set equal to one.

By virtue of straightforward calculations and the arguments of the previous section, it is verified that the function *<sup>V</sup>*(*<sup>x</sup>*,*s*, *q*) solves the ordinary differential equation in (18) and satisfies the normal-reflection conditions in (21)–(22). Observe that the process (*<sup>M</sup><sup>τ</sup>a*<sup>∧</sup>*ζb*∧*<sup>t</sup>*)*t*≥<sup>0</sup> is a uniformly integrable martingale, since the derivative and the coefficient in (66) are bounded functions on the compact set {(*<sup>x</sup>*,*s*, *q*) ∈ R<sup>3</sup> | *a* ∨ *q* ≤ *x* ≤ *s* ∧ *b*}. Then, using the properties of the indicators mentioned above and taking the expectation with respect to *Px*,*s*,*<sup>q</sup>* in (65), by means of the optional sampling theorem (see, e.g., Liptser and Shiryaev [1977] 2001, chp. III, Theorem 3.6 or Karatzas and Shreve 1991, chp. I, Theorem 3.22), we ge<sup>t</sup>

$$E\_{\mathbf{x},\mathbf{s},\mathbf{q}}\left[e^{-\lambda\left(\tau\_{\mathbf{z}}\wedge\tilde{\zeta}\_{\mathbf{b}}\wedge t\right)}V(X\_{\tau\_{\mathbf{z}}\wedge\tilde{\zeta}\_{\mathbf{b}}\wedge t\prime}S\_{\tau\_{\mathbf{z}}\wedge\tilde{\zeta}\_{\mathbf{b}}\wedge t\prime}Q\_{\tau\_{\mathbf{z}}\wedge\tilde{\zeta}\_{\mathbf{b}}\wedge t}\right)\right]\tag{67}$$

$$=V(\mathbf{x},\mathbf{s},\mathbf{q})+E\_{\mathbf{x},\mathbf{s},\mathbf{q}}\left[M\_{\tau\_{\mathbf{z}}\wedge\tilde{\zeta}\_{\mathbf{b}}\wedge t}\right]=V(\mathbf{x},\mathbf{s},\mathbf{q})$$

for all (*<sup>x</sup>*,*s*, *q*) ∈ *E*3. Therefore, letting *t* go to infinity and using the instantaneous-stopping conditions in (19)–(20) as well as the fact that *<sup>e</sup>*<sup>−</sup>*<sup>λ</sup>*(*<sup>τ</sup>a*<sup>∧</sup>*ζb*)*V*(*<sup>X</sup><sup>τ</sup>a*<sup>∧</sup>*ζ<sup>b</sup>* , *<sup>S</sup><sup>τ</sup>a*<sup>∧</sup>*ζ<sup>b</sup>* , *Q<sup>τ</sup>a*<sup>∧</sup>*ζ<sup>b</sup>* ) = 0 on {*<sup>τ</sup>a* ∧ *ζb* = ∞} (*Px*,*s*,*<sup>q</sup>*-a.s.), we can apply the Lebesgue dominated convergence theorem for (67) to obtain the equalities

$$E\_{\mathbf{x},\mathbf{s},\mathbf{q}}\left[e^{-\lambda\left(\mathbf{r}\_{\mathbf{d}}\wedge\boldsymbol{\zeta}\_{\mathbf{b}}\right)-\theta S\_{\mathbf{t}\_{\mathbf{d}}\wedge\boldsymbol{\zeta}\_{\mathbf{b}}}-\kappa Q\_{\mathbf{t}\_{\mathbf{d}}\wedge\boldsymbol{\zeta}\_{\mathbf{b}}}}\,I\left(\mathbf{r}\_{\mathbf{d}}<\boldsymbol{\zeta}\_{\mathbf{b}}\right)\right]\tag{68}$$

$$I = E\_{\mathbf{x},\mathbf{s},\mathbf{q}}\left[e^{-\lambda\left(\mathbf{r}\_{\mathbf{d}}\wedge\boldsymbol{\zeta}\_{\mathbf{b}}\right)}\,V\left(X\_{\mathbf{t}\_{\mathbf{d}}\wedge\boldsymbol{\zeta}\_{\mathbf{b}},\boldsymbol{\zeta}\_{\mathbf{t}\_{\mathbf{d}}\wedge\boldsymbol{\zeta}\_{\mathbf{b}}}},Q\_{\mathbf{t}\_{\mathbf{d}}\wedge\boldsymbol{\zeta}\_{\mathbf{b}}}\right)\right] = V(\mathbf{x},\mathbf{s},\boldsymbol{q})$$

for all (*<sup>x</sup>*,*s*, *q*) ∈ *E*3, which directly implies the desired assertion.

**Author Contributions:** P.V.G.: writing—original draft; N.R.: writing—review and editing; V.L.R.C.: writing —conceptualization.

**Funding:** This research was supported by a Small Grant from the Suntory and Toyota International Centres for Economics and Related Disciplines (STICERD) at the London School of Economics and Political Science.

**Acknowledgments:** The authors are grateful to Florin Avram and two anonymous referees for their valuable suggestions which helped to essentially improve the presentation of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
