*2.1. Formulation of the Problem*

Let us consider a probability space (<sup>Ω</sup>, F, *P*) with a standard Brownian motion *B* = (*Bt*)*t*≥0 and a positive random time *η* such that *<sup>P</sup>*(*η* > *t*) = *<sup>e</sup>*<sup>−</sup>*αt*, for all *t* ≥ 0 and some *α* > 0 fixed (*B* and *η* are supposed to be independent). Assume that there exists a process *X* = ( *Xt*)*t*≥0 solving the stochastic differential equation

$$dX\_t = \mu(X\_t, \mathcal{S}\_t, \mathcal{Q}\_t) \, dt + \sigma(X\_t, \mathcal{S}\_t, \mathcal{Q}\_t) \, dB\_t \quad (X\_0 = x) \tag{1}$$

where *x* ∈ R is fixed, and *μ*(*<sup>x</sup>*,*s*, *q*) and *<sup>σ</sup>*(*<sup>x</sup>*,*s*, *q*) > 0 are continuously differentiable functions on [− <sup>∞</sup>, ∞] 3 which are of at most linear growth in *x* and uniformly bounded in *s* and *q*. Here, the associated with *X running maximum* process *S* = (*St*)*t*≥0 and the *running minimum* process *Q* = ( *Qt*)*t*≥0 are defined by

$$S\_t = s \lor \max\_{0 \le u \le t} X\_u \quad \text{and} \quad Q\_t = q \land \min\_{0 \le u \le t} X\_u \tag{2}$$

for arbitrary *q* ≤ *x* ≤ *s*. It follows from the result of (Liptser and Shiryaev [1977] 2001, chp. IV, Theorem 4.8) that the equation in (1) admits a pathwise unique (strong) solution. We also define the associated first hitting (stopping) times

$$
\pi\_d = \inf\{t \ge 0 \mid S\_t - X\_t \ge a\} \quad \text{and} \quad \mathbb{Z}\_b = \inf\{t \ge 0 \mid X\_t - Q\_t \ge b\} \tag{3}
$$

for some *a*, *b* > 0 fixed.

The purpose of the present paper is to derive closed-form expressions for the joint Laplace transform of the random time *τa* ∧ *ζb* ∧ *η* and the random variables *<sup>S</sup><sup>τ</sup>a*<sup>∧</sup>*ζb*∧*<sup>η</sup>* and *Q<sup>τ</sup>a*<sup>∧</sup>*ζb*∧*η*. We therefore need to compute the value function of the following stopping problem for the (time-homogeneous strong) Markov process (*<sup>X</sup>*, *S*, *Q*)=( *Xt*, *St*, *Qt*)*t*≥0 given by

$$V\_\*(\mathbf{x}, \mathbf{s}, q) = E\_{\mathbf{x}, \mathbf{s}, \eta} \left[ e^{-\lambda \left( \mathbf{r}\_a \wedge \eta \right) - \theta S\_{\mathbf{m} \wedge \eta} - \kappa Q\_{\mathbf{m} \wedge \eta}} I \left( \mathbf{r}\_a < \mathbb{Z}\_b \right) \right] \tag{4}$$

for any (*<sup>x</sup>*,*s*, *q*) ∈ *E*<sup>3</sup> and some *λ*, *θ*, *κ* > 0 fixed, where *<sup>I</sup>*(·) denotes the indicator function. Here, *Ex*,*s*,*<sup>q</sup>* denotes the expectation under the assumption that the (three-dimensional) Markov process (*<sup>X</sup>*, *S*, *Q*) defined in (1)–(2) starts at (*<sup>x</sup>*,*s*, *q*) ∈ *E*3, where we assume that the state space of (*<sup>X</sup>*, *S*, *Q*) is essentially *E*<sup>3</sup> = {(*<sup>x</sup>*,*s*, *q*) ∈ R<sup>3</sup> | *q* ≤ *x* ≤ *s*} with its border planes *d*3 1 = {(*<sup>x</sup>*,*s*, *q*) ∈ R<sup>3</sup> | *x* = *s*} and *d*3 2 = {(*<sup>x</sup>*,*s*, *q*) ∈ R<sup>3</sup> | *x* = *q*}.

It follows from the independence of the process *X* and the random time *η* that the value function in (4) admits the representation

$$W\_\*(\mathbf{x}, \mathbf{s}, q) = \int\_0^\infty \mathcal{W}\_\*(T; \mathbf{x}, \mathbf{s}, q) \, \mathbf{a} \, e^{-\mathbf{a}T} \, dT \tag{5}$$

where we set

$$\mathcal{W}\_{\*}(T; \mathbf{x}, \mathbf{s}, q) = E\_{\mathbf{x}, \mathbf{s}, \mathbf{q}} \left[ e^{-\lambda \left( \mathbf{r}\_{\mathbf{d}} \wedge T \right) - \theta \mathbf{S}\_{\mathbf{r}\_{\mathbf{d}} \wedge T} - \mathbf{x} \mathbf{Q}\_{\mathbf{r}\_{\mathbf{d}} \wedge T}} I \left( \mathbf{r}\_{\mathbf{d}} \prec\_{\sim} \mathbf{f}\_{b} \right) \right] \tag{6}$$

for any (*<sup>x</sup>*,*s*, *q*) ∈ *E*3, and each *T* > 0 fixed.
