2. Preliminaries

In this section we first recall some basic notions and results on optimal stopping problems without aggregation. More details can be found in [Reference Kobylanski and Quenez7] and [Reference Kobylanski, Quenez and Rouy-Mironescu8]. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a Radon measure space, which is equipped with a filtration $\mathbb{F}=\{\mathcal{F}_t\}_{t\in[0,T]}$ satisfying the usual conditions of right continuity and completeness. Throughout this paper, we fix a terminal time T. Let $\mathcal{S}_{0,T}$ be the set of $\mathbb{F}$ -stopping times taking values between 0 and T, and for any $\sigma,\sigma'\in \mathcal{S}_{0,T}$ with $\sigma\leq \sigma'$ , set $\mathcal{S}_{\sigma,\sigma'}=\{\tau\in \mathcal{S}_{0,T}\colon \sigma\leq \tau\leq \sigma'\}$ . For simplicity, we always omit the subscript T.

Remark 2.1. The reason why we consider a Random measure space is that we need the fact that $\mathbb{P}$ is a compact measure and Lusin’s theorem to prove our technical result (see Lemma 3.2). Typically, if $\Omega$ is a Polish space and $\mathcal{F}$ is the $\sigma$ -algebra of Borel sets, then $(\Omega,\mathcal{F},\mathbb{P})$ is a Radon measure space. For example, let $\Omega=C_0^d([0,\infty))$ be the space of all continuous, $\mathbb{R}^d$ -valued functions $\omega$ on $[0,\infty)$ with $\omega_0=0$ , equipped with the distance

\begin{align*} d(\omega^1,\omega^2)=\sum_{n=1}^{\infty}\dfrac{1}{2^n}\max_{0\leq t\leq n}\bigl(|\omega^1_t-\omega^2_t|\wedge 1\bigr).\end{align*}

Let $\mathcal{F}$ be the Borel $\sigma$ -algebra and let $\mathbb{P}$ be the Wiener measure. Then $(\Omega,\mathcal{F},\mathbb{P})$ is a Radon measure space.

Given a set of random variables indexed by stopping times written as $\{X(\tau),\tau\in\mathcal{S}_0\}$ , the basic requirement of this family is to be admissible, defined as follows.

Remark 2.2. (i) Let $\{Y_t\}_{t\in[0,T]}$ be a positive, progressively measurable process. For any $\tau\in \mathcal{S}_0$ , set $X(\tau)\,:\!=\, Y_{\tau}$ . Then the family $\{X(\tau),\tau\in\mathcal{S}_0\}$ is admissible. The assumptions on admissible family can be seen as the requirement for adaptedness and compatibility, respectively. In fact, the definition of admissible family coincides with the notion of $T_0$ -system introduced in [Reference El Karoui5].

(ii) Let $T=1$ and $(\Omega,\mathcal{F},\mathbb{P})=([0,1],\mathcal{L},\lambda)$ , where $\mathcal{L}$ denotes the $\sigma$ -algebra of Lebesgue sets in [0,Reference Ben Latifa, Bonnans and Mnif1] and $\lambda$ denotes the Lebesgue measure. Let $\mathcal{L}_0\subset \mathcal{L}$ be the $\sigma$ -algebra such that $\lambda(B)=0$ or 1 for any $B\in\mathcal{L}_0$ . For each $t\in[0,1]$ , set $\mathcal{F}_t=\mathcal{L}_0$ and $\mathbb{F}=\{\mathcal{F}_t\}_{t\in[0,1]}$ . Define the set A by $A=\{(t,\omega)\in[0,1]\times[0,1]\colon t=\omega\in[0,1/2]\}$ and the function $I^A$ by

\begin{align*}I^A_t(\omega)=\begin{cases}1, & t=\omega\in[0,1/2],\\0, &\text{otherwise}.\end{cases}\end{align*}

We claim that for any $\mathbb{F}$ -stopping time $\tau$ , $I^A_\tau$ is $\mathcal{F}_\tau$ -measurable. It suffices to prove that for any $a\in\mathbb{R}$ and any $t\in[0,1]$ , we have

\begin{align*}\{I^A_\tau\geq a\}\cap \{\tau\leq t\}\in \mathcal{F}_t=\mathcal{L}_0,\end{align*}

which is equivalent to the following equation

(2.1) \begin{equation}\lambda(\{I^A_\tau\geq a\}\cap \{\tau\leq t\})=0\ \text{or 1.}\end{equation}

Clearly (2.1) holds true for any $a\leq 0$ and $a> 1$ . By the definition of $\mathbb{F}$ , for any stopping time $\tau$ , there exists a constant $t_\tau\in[0,1]$ and a set $B_\tau\in \mathcal{L}_0$ with $\lambda(B_\tau)=1$ , such that $\tau(\omega)=t_\tau$ for any $\omega\in B_\tau$ . Then we consider the following two cases.

Case 1. $t\in [0,t_\tau)$ . In this case, we have $\lambda(\{\tau\leq t\})=0$ .

Case 2. $t\in [t_\tau,1]$ . In this case, for any $a\in(0,1]$ , we have

\begin{align*}\{I^A_\tau(\omega)\geq a\}&=\{\tau(\omega)=\omega\in[0,1/2]\}\\&=(\{t_\tau=\omega\in [0,1/2]\}\cap B_\tau)\cup (\{\tau(\omega)=\omega\in [0,1/2]\}\cap B^c_\tau).\end{align*}

It follows that $\lambda(\{I^A_\tau(\omega)\geq a\})=0$ .

Then we conclude that in the above two cases, (2.1) still holds. Hence, for any $\mathbb{F}$ -stopping time $\tau$ , $I^A_\tau\in \mathcal{F}_\tau$ . However, it is worth pointing out that Example 1.38 of [Reference Scheutzow12] indicates that the process $I^A$ is not progressively measurable.

Let $\{X(\tau),\tau\in\mathcal{S}_0\}$ be a given admissible family, which can be regarded as the reward. The value function at time $S\in \mathcal{S}_0$ is defined by

(2.2) \begin{equation}v(S)\,:\!=\, \operatorname{ess\,sup}_{\tau\in \mathcal{S}_S}\mathbb{E}_S[X(\tau)],\end{equation}

where $\mathbb{E}_S[\cdot]$ is the conditional expectation taken under $\mathbb{P}$ with respect to $\mathcal{F}_S$ . The following result indicates that the value function family is admissible.

Definition 2.2. An admissible family $\{X(\tau),\tau\in\mathcal{S}_0\}$ is said to be a supermartingale family (resp. a martingale family) if, for any $\tau,\sigma\in \mathcal{S}_0$ with $\tau\leq \sigma$ a.s., we have

\begin{align*}\mathbb{E}_\tau[X(\sigma)]\leq X(\tau)\ \text{a.s.}\quad (\text{resp.}\ \mathbb{E}_\tau[X(\sigma)]= X(\tau)\ \text{a.s.}).\end{align*}

The following proposition, known as the optimality criterion, provides a characterization of optimal stopping times for the value function v(S).

Note that none of the above results need any regularity assumptions on the reward family $\{X(\tau),\tau\in\mathcal{S}_0\}$ . However, in order to construct the optimal stopping time for the value function defined by (2.2), we need to propose the following condition on the admissible family $\{X(\tau),\tau\in \mathcal{S}_0\}$ .

Definition 2.3. An admissible family $\{X(\tau),\tau\in \mathcal{S}_0\}$ is said to be right (resp. left) upper semicontinuous in expectation along stopping times (RUSCE (resp. LUSCE)) if, for all $\tau\in \mathcal{S}_0$ and for all sequence of stopping times $\{\tau^n\}$ such that $\tau^n\downarrow \tau$ (resp. $\tau^n\uparrow \tau$ ),

\begin{align*}\mathbb{E}[X(\tau)]\geq \limsup_{n\rightarrow \infty} \mathbb{E}[X(\tau^n)].\end{align*}

An admissible family $\{X(\tau),\tau\in \mathcal{S}_0\}$ is said to be upper semicontinuous in expectation along stopping times (USCE) if it is RUSCE and LUSCE.

Now, we give the existence result of an optimal stopping time and the regularity of the value function family.

Theorem 2.1. ([Reference Kobylanski and Quenez7].) Suppose the reward family $\{X(\tau),\tau\in \mathcal{S}_0\}$ is such that $v(0)<\infty$ and is USCE. Let $S\in\mathcal{S}_0$ . The stopping time $\theta^*(S)$ defined by

\begin{align*}\theta^*(S)\,:\!=\, \operatorname{ess\, inf} \{\theta\in\mathcal{S}_S\colon v(\theta)=X(\theta) \ {a.s.}\}\end{align*}

is the minimal optimal stopping time for v(S). Further, the admissible family $\{v(S),S\in \mathcal{S}_0\}$ is USCE.

The optimal stopping time $\theta^*(S)$ can also be constructed by using hitting times of processes if the reward is given by some progressively measurable process. This formulation (see (2.3) below) gives an efficient method to compute an optimal stopping time by numerical methods since it is defined pathwise (see also line 7 of Section 4 in [Reference Kobylanski, Quenez and Rouy-Mironescu8]).

Proposition 2.5 ([Reference Kobylanski and Quenez7].) Let $\{X_t,t\in[0,T]\}$ be a right upper semicontinuous (i.e. for almost every $\omega$ , the function $t\rightarrow X_t(\omega)$ is right upper semicontinuous) progressively measurable process of class $\mathcal{D}$ . Suppose that the associated family $\{X_\tau,\tau\in \mathcal{S}_0\}$ is LUSCE. Then there exists an adapted process $\{v_t,t\in[0,T]\}$ aggregating the value function family $\{v(S),S\in\mathcal{S}_0\}$ , that is, for all $S\in\mathcal{S}_0$ , $v_S=v(S)$ a.s. Moreover, for all $S\in \mathcal{S}_0$ , the stopping time $\tau^*(S)$ defined by

(2.3) \begin{align}\tau^*(S)\,:\!=\, \inf\{t\geq S\colon v_t=X_t\}\end{align}

is the minimal optimal stopping time for v(S).

3. Optimal multiple stopping with reward family

Consider the holder of a swing option who can exercise it $\ell$ times. Let $\delta>0$ be the refracting period which separates two successive exercises. This requirement will exclude the trivial case that the holder exercises her $\ell$ -times rights simultaneously. When she exercises at stopping time $\tau$ , she will get the reward $X(\tau)$ . For $\ell\geq 2$ , let $\mathcal{S}_\tau(\ell,\delta)$ be the collection of all vectors of stopping times $\tau^{(\ell)}=(\tau_1,\ldots,\tau_\ell)$ satisfying

\begin{align*} \tau_1\geq \tau, \tau_i-\tau_{i-1}\geq \delta \text{ on } \{\tau_{i-1}+\delta\leq T\},\quad \tau_i=T_+ \text{ on } \{\tau_{i-1}+\delta >T\}\quad \text{for all $ i=2, \ldots,\ell$.} \end{align*}

For any $\mathbb{F}$ -stopping time $\sigma$ and any family of random variables $\{\xi(\tau),\tau\in \mathcal{S}_0\}$ , we set $\xi(\sigma) I_{\{\sigma>T\}}=0$ . The value of the swing option at time $\tau$ is defined by

(3.1) \begin{equation}Z^{(\ell)}(\tau)\,:\!=\, \operatorname{ess\,sup}_{\tau^{(\ell)}\in \mathcal{S}_\tau(\ell,\delta)} \mathbb{E}_\tau\Biggl[\sum_{i=1}^\ell X(\tau_i)\Biggr].\end{equation}

The objective includes the following two aspects: (i) computing the value function $Z^{(\ell)}(\tau)$ as explicitly as possible, and (ii) finding the optimal multiple stopping time $(\tau^*_1,\ldots,\tau^*_\ell)\in \mathcal{S}_\tau(\ell,\delta)$ at which the supremum in (3.1) is attained. Motivated by [Reference Ben Latifa, Bonnans and Mnif1] and [Reference Carmona and Touzi4], we consider the following auxiliary optimal single stopping problems inductively:

(3.2) \begin{equation}Y^{(0)}\equiv 0\quad \text{and}\quad Y^{(i)}(\tau)=\operatorname{ess\,sup}_{\sigma\in\mathcal{S}_\tau}\mathbb{E}_\tau[X^{(i)}(\sigma)], \quad i=1,\ldots,\ell,\end{equation}

where the ith exercise reward family $X^{(i)}$ is given by

(3.3) \begin{equation}X^{(i)}(\tau)=\begin{cases} X(\tau)+ \mathbb{E}_\tau[Y^{(i-1)}(\tau+\delta)], & 0\leq \tau\leq T-\delta,\\X(\tau), &T-\delta < \tau\leq T.\end{cases}\end{equation}

In the rest of this paper, C will always be a positive constant, which may vary from line to line.

Remark 3.1. (i) Recalling the convention that $\xi(\sigma) I_{\{\sigma>T\}}=0$ for any family $\{\xi(\tau),\tau\in \mathcal{S}_0\}$ , we can write $X^{(i)}(\tau)=X(\tau)+ \mathbb{E}_\tau[Y^{(i-1)}(\tau+\delta)]$ for any $\tau\in \mathcal{S}_0$ .

(ii) We provide the motivation for the auxiliary optimal single stopping problems. Consider the multiple stopping problem in the discrete-time case. Suppose that $T,\delta,\ell$ are positive integers such that $(\ell-1)\delta\leq T$ . For any fixed $i=0,\ldots,T$ , let $\mathcal{S}^d_i(\ell,\delta)$ be the collection of all vectors of integer-valued stopping times $\tau^{(\ell)}=(\tau_1,\ldots,\tau_\ell)$ satisfying

\begin{align*} \tau_1\geq i, \tau_j-\tau_{j-1}\geq \delta\quad \text{for all $ j=2, \ldots,\ell$.} \end{align*}

Further, we define $\mathcal{S}^d_i\,:\!=\, \mathcal{S}^d_i(1,\delta)$ . The value function of the optimal multiple stopping problem is given by

\begin{align*}Z^{(\ell)}_d(i)=\operatorname{ess\,sup}_{\tau^{(\ell)}\in \mathcal{S}^d_i(\ell,\delta)} \mathbb{E}_i\Biggl[\sum_{j=1}^\ell X(\tau_j)\Biggr].\end{align*}

By Lemma 3.1 of [Reference Li9], the value function can be obtained recursively. More precisely, for any $j=2,\ldots,\ell$ , we have $Z^{(j)}_d(T)=X(T)$ and

(3.4) \begin{equation}Z^{(j)}_d(i)=\max\bigl\{ X(i)+\mathbb{E}_i\bigl[Z^{(j-1)}_d(i+\delta)\bigr],\mathbb{E}_i\bigl[Z^{(j)}_d(i+1)\bigr]\bigr\}, \quad i=0,\ldots, T-1,\end{equation}

and $Z^{(1)}_d$ is the value function of the optimal single stopping problem with reward $\{X(i)\}_{i=0}^T$ . Motivated by (3.4), $Z^{(j)}_d$ can be seen as the value function of the optimal single stopping problem with reward $\bigl\{X(i)+\mathbb{E}_i\bigl[Z^{(j-1)}_d(i+\delta)\bigr]\bigr\}_{i=0}^T$ .

Therefore, set

\begin{align*}Y^{(0)}_d\equiv 0\quad \text{and}\quad Y^{(j)}_d(i)=\operatorname{ess\,sup}_{\sigma\in\mathcal{S}^d_i}\mathbb{E}_i[X^{(j)}(\sigma)],\quad j=1,\ldots,\ell,\end{align*}

where $X^{(j)}$ is given by

\begin{align*}X^{(j)}(i)=\begin{cases} X(i)+ \mathbb{E}_i[Y^{(j-1)}(i+\delta)], & 0\leq i\leq T-\delta,\\X(i), &T-\delta < i\leq T.\end{cases}\end{align*}

It is obvious that $Z^{(1)}_d\equiv Y^{(1)}_d$ . By induction, we have $Z^{(j)}_d\equiv Y^{(j)}_d$ , $j=2,\ldots,\ell$ . That is, the value function of the optimal multiple stopping problem coincides with that for some optimal single stopping problem with a new reward family.

Proof. By Proposition 2.1, it is sufficient to prove that the family $\{X^{(i)}(\tau),\tau\in \mathcal{S}_0\}$ is admissible, $1\leq i\leq \ell$ . This fact is obvious when $i=1$ . We assume that $\{X^{(k)}(\tau),\tau\in \mathcal{S}_0\}$ are admissible for $1\leq k\leq i-1$ with $i\geq 2$ . It is clear that $X^{(i)}(\tau)$ is non-negative and $\mathcal{F}_\tau$ -measurable. Further, applying Proposition 2.1, the family $\{Y^{(i-1)}(\tau),\tau\in \mathcal{S}_0\}$ is admissible. For any $\tau,\sigma\in \mathcal{S}_0$ , on the set $A=\{\tau=\sigma\}$ , noting that $A\in \mathcal{F}_\tau\cap \mathcal{F}_\sigma$ , we have

\begin{align*} \mathbb{E}_\tau[Y^{(i-1)}(\tau+\delta)]I_A&=\mathbb{E}_\tau[Y^{(i-1)}(\tau+\delta)I_A]\\ &=\mathbb{E}_\tau[Y^{(i-1)}(\sigma+\delta)I_A]\\ &=\mathbb{E}_\tau[Y^{(i-1)}(\sigma+\delta)]I_A\\ &=\mathbb{E}_\sigma[Y^{(i-1)}(\sigma+\delta)]I_A.\end{align*}

Hence the family $\{X^{(i)}(\tau),\tau\in\mathcal{S}_0\}$ is admissible.

In order to construct the optimal stopping time for the auxiliary problem (3.2), we assume that the family $\{X(\tau),\tau\in \mathcal{S}_0\}$ satisfies the following condition:

1. (H1) The admissible family $\{X(\tau),\tau\in\mathcal{S}_0\}$ is USCE and there exists some $p>1$ such that $\sup_{\tau\in\mathcal{S}_0}\mathbb{E}[|X(\tau)|^p]<\infty$ .

As we will see later, the optimal multiple stopping times are constructed by the optimal stopping times for the auxiliary single stopping problems. The existence of optimal stopping times for the single stopping problems needs some regularity property of the admissible family $\{X^{(i)}(\tau),\tau\in \mathcal{S}_0\}$ . More precisely, the admissible family $\{X^{(i)}(\tau),\tau\in \mathcal{S}_0\}$ should be USCE (see Theorem 2.1). The following lemma shows that the iterated reward processes $X^{(i)}$ , $i=2, \ldots,\ell$ inherit conditions proposed on the reward family $\{X(\tau),\tau\in\mathcal{S}_0\}$ .

Proof. Step 1. We first prove that the properties hold when $i=1$ . Since $X^{(1)}=X$ , the properties of $\{X^{(1)}(\tau),\tau\in\mathcal{S}_0\}$ are obvious. By Theorem 2.1, for any stopping time $\tau$ , there exists a stopping time $\tau^*(\tau)$ such that $Y^{(1)}(\tau)=\mathbb{E}_\tau[X(\tau^*(\tau))]$ . It follows that

\begin{align*}\mathbb{E}[|Y^{(1)}(\tau)|^p]&=\mathbb{E}[|\mathbb{E}_\tau[X(\tau^*(\tau))]|^p]\\&\leq \mathbb{E}[\mathbb{E}_\tau[|X(\tau^*(\tau))|^p]]\\&=\mathbb{E}[|X(\tau^*(\tau))|^p]\\&\leq \sup_{\tau\in \mathcal{S}_0}\mathbb{E}[|X(\tau)|^p]\\&<\infty.\end{align*}

Now suppose that the properties hold for the families $\{X^{(k)}(\tau),\tau\in\mathcal{S}_0\}$ , $\{Y^{(k)}(\tau),\tau\in \mathcal{S}_0\}$ for $1\leq k\leq i-1$ with some $i\geq 2$ .

Step 2. We then show that $\{X^{(i)}(\tau),\tau\in\mathcal{S}_0\}$ is USCE. Set $Y^{(i-1)}(\sigma)=0$ on the set $\{\sigma> T\}$ . Note that for any $\tau\in \mathcal{S}_0$ we have

\begin{equation*}\mathbb{E}[X^{(i)}(\tau)]=\mathbb{E}[X(\tau)]+\mathbb{E}[Y^{(i-1)}(\tau+\delta)].\end{equation*}

It suffices to show that the family $\{Y^{(i-1)}(\tau+\delta),\tau\in \mathcal{S}_{0}\}$ is USCE. We first prove the left upper semicontinuity. Let $\{\tau^n\}$ be a set of stopping times such that $\tau^n\uparrow \tau$ ; we aim to show that

\begin{align*}\limsup_{n\rightarrow \infty}\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)]\leq \mathbb{E}[Y^{(i-1)}(\tau+\delta)].\end{align*}

For any $\varepsilon \in (0,1)$ , there exists an $\mathcal{F}_\tau$ -measurable compact set B such that $\mathbb{P}(B^c)\leq {{\varepsilon}/{2}}$ . By Lusin’s theorem, there exists some $\mathcal{F}_\tau$ -measurable open sets $\{O_n\}_{n=0}^\infty$ with $\mathbb{P}(O_n)\leq {{\varepsilon}/{2^{n+2}}}$ , such that the restrictions of $\tau_n$ to $O^c_n$ and of $\tau$ to $O^c_0$ are continuous. Set

\begin{align*}O=\biggl(\bigcup_{n=0}^\infty O_n\biggr)\bigcup B^c.\end{align*}

It is easy to check that $\mathbb{P}(O)\leq \varepsilon$ . Since $\{\tau^n\}$ is increasing in n, it converges uniformly on $O^c$ by Dini’s theorem. Hence, on the set $\{\tau+\delta>T\}\cap O^c$ , there exists an integer N independent of $\omega$ , such that for any $n\geq N$ , $\tau^n+\delta>T$ and on the set $\{\tau+\delta\leq T\}$ , we have $\{\tau^n+\delta\leq T\}$ for any n. Now, for $n\geq N$ , recalling that $Y^{(i-1)}(\sigma)=0$ on the set $\{\sigma> T\}$ , it is easy to check that

(3.5) \begin{equation}\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)]=\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)I_{\{\tau+\delta\leq T\}}]+\mathbb{E}[Y^{(i-1)}(\tau^n+\delta) I_{\{\tau+\delta> T\}}I_{O}].\end{equation}

For any n, a simple calculation implies that

(3.6) \begin{align}|\mathbb{E}[Y^{(i-1)}(\tau^n+\delta) I_{\{\tau+\delta> T\}}I_{O}]|&\leq (\mathbb{E}[|Y^{(i-1)}(\tau^n+\delta)|^p])^{1/p} (\mathbb{P}(O))^{1/q} \notag \\&\leq \Bigl(\sup_{\sigma\in \mathcal{S}_0}\mathbb{E}[|Y^{(i-1)}(\sigma)|^p]\Bigr)^{1/p} \varepsilon^{1/q},\end{align}

where $q=p/(p-1)$ . Recalling that $\{X^{(i-1)}(\tau),\tau\in\mathcal{S}_0\}$ is USCE, by Theorem 2.1, the family $\{Y^{(i-1)}(\tau),\tau\in \mathcal{S}_0\}$ is USCE. It is easy to check that $\{\tau+\delta\leq T\}\in \mathcal{F}_\tau$ and $\tau^n+\delta\geq \tau$ for n large enough. By Proposition 2.4 and applying (3.5) and (3.6), we have

\begin{align*}\limsup_{n\rightarrow \infty}\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)]&\leq \limsup_{n\rightarrow \infty}\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)I_{\{\tau+\delta\leq T\}}]+ C\varepsilon^{1/q}\\&\leq \mathbb{E}[Y^{(i-1)}(\tau+\delta)I_{\{\tau+\delta\leq T\}}]+C\varepsilon^{1/q}\\&=\mathbb{E}[Y^{(i-1)}(\tau+\delta)]+C\varepsilon^{1/q},\end{align*}

where $C=(\sup_{\sigma\in \mathcal{S}_0}\mathbb{E}[|Y^{(i-1)}(\sigma)|^p])^{1/p}$ . Letting $\varepsilon$ approach 0, we get the desired result.

Now we prove the right upper semicontinuity of the family $\{Y^{(i-1)}(\tau+\delta),\tau\in \mathcal{S}_{0}\}$ . Let $\{\tau^n\}$ be a set of stopping times such that $\tau^n\downarrow \tau$ . On the set $\{\tau+\delta>T\}$ , for any n, we have $\{\tau^n+\delta> T\}$ . Hence, on the set $\{\tau+\delta> T\}$ , $Y^{(i-1)}(\tau^n+\delta)=Y^{(i-1)}(\tau+\delta)(=0)$ . It follows that

(3.7) \begin{equation}\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)]= \mathbb{E}[Y^{(i-1)}(\tau^n+\delta)I_{\{\tau+\delta\leq T\}}]+\mathbb{E}[Y^{(i-1)}(\tau+\delta) I_{\{\tau+\delta>T\}}].\end{equation}

We define

\begin{align*}\widetilde{\tau}^n=\tau^n I_{\{\tau^n+\delta\leq T\}}+(T-\delta) I_{\{\tau^n+\delta>T\}}=\min\{\tau^n,T-\delta\}.\end{align*}

It is easy to check that $\widetilde{\tau}^n$ is non-increasing in n, $\widetilde{\tau}^n\geq \tau\wedge (T-\delta)$ and $\widetilde{\tau}^n\downarrow \widetilde{\tau}$ , where $\widetilde{\tau}=\tau I_{\{\tau+\delta\leq T\}}+(T-\delta)I_{\{\tau+\delta>T\}}$ . Noting that $Y^{(i-1)}$ is non-negative and $\{\tau^n+\delta\leq T\}\subset \{\tau+\delta\leq T\}$ , we have

\begin{align*}&\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)I_{\{\tau+\delta\leq T\}}]\\&\quad =\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)I_{\{\tau^n+\delta\leq T\}}]+\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)I_{\{\tau+\delta\leq T<\tau^n+\delta\}}]\\&\quad \leq \mathbb{E}[Y^{(i-1)}(\widetilde{\tau}^n+\delta)I_{\{\tau^n+\delta\leq T\}}]+\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)I_{\{\tau^n-\tau>0\}}]\\&\quad \leq \mathbb{E}[Y^{(i-1)}(\widetilde{\tau}^n+\delta)I_{\{\tau+\delta\leq T\}}]+C(\mathbb{P}(\{\tau^n-\tau>0\}))^{1/q},\end{align*}

where $C=(\sup_{\sigma\in \mathcal{S}_0}\mathbb{E}[|Y^{(i-1)}(\sigma)|^p])^{1/p}$ and $1/p+1/q=1$ . It follows that

(3.8) \begin{align}\limsup_{n\rightarrow\infty}\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)I_{\{\tau+\delta\leq T\}}]&\leq \limsup_{n\rightarrow\infty}\mathbb{E}[Y^{(i-1)}(\widetilde{\tau}^n+\delta)I_{\{\tau+\delta\leq T\}}]\notag \\&\leq \mathbb{E}[Y^{(i-1)}(\widetilde{\tau}+\delta)I_{\{\tau+\delta\leq T\}}]\notag \\&= \mathbb{E}[Y^{(i-1)}(\tau+\delta)I_{\{\tau+\delta\leq T\}}],\end{align}

where we have used the dominated convergence theorem in the first inequality and Proposition 2.4 in the second inequality. Combining (3.7) and (3.8) yields

\begin{align*}\limsup_{n\rightarrow\infty}\mathbb{E}[Y^{(i-1)}(\tau^n+\delta)]\leq \mathbb{E}[Y^{(i-1)}(\tau+\delta)],\end{align*}

which is the desired result.

Step 3. It remains to prove that $\sup_{\tau\in\mathcal{S}_0}\mathbb{E}[|X^{(i)}(\tau)|^p]<\infty$ and $\sup_{\tau\in\mathcal{S}_0}\mathbb{E}[|Y^{(i)}(\tau)|^p]<\infty$ . A simple calculation implies that

\begin{align*}\mathbb{E}[|X^{(i)}(\tau)|^p]&\leq C\{\mathbb{E}[|X(\tau)|^p]+\mathbb{E}[|\mathbb{E}_\tau[Y^{(i-1)}(\tau+\delta)]|^p]\}\\&\leq C\{\mathbb{E}[|X(\tau)|^p]+\mathbb{E}[|Y^{(i-1)}(\tau+\delta)|^p]\}.\end{align*}

It follows that $\sup_{\tau\in\mathcal{S}_0}\mathbb{E}[|X^{(i)}(\tau)|^p]<\infty$ . Since the admissible family $\{X^{(i)}(\tau),\tau\in\mathcal{S}_0\}$ is USCE with $\sup_{\tau\in\mathcal{S}_0}\mathbb{E}[|X^{(i)}(\tau)|^p]<\infty$ , by Theorem 2.1, for each $\tau\in \mathcal{S}_0$ , there exists an optimal stopping time $\tau^*$ , i.e. $Y^{(i)}(\tau)=\mathbb{E}_\tau[X^{(i)}(\tau^*)]$ . By an analysis similar to that of the proof of Step 1, we have

\begin{align*}\sup_{\tau\in\mathcal{S}_0}\mathbb{E}[|Y^{(i)}(\tau)|^p]\leq \sup_{\tau\in\mathcal{S}_0}\mathbb{E}[|X^{(i)}(\tau)|^p]<\infty.\end{align*}

The proof is complete.

In the following we aim to construct the optimal multiple stopping times for the problem (3.1). Recall the double stopping problem in Remark 3.1. It is easy to check that the optimal double stopping times should be $(\tau^*_1(S),\tau^*_2(S))$ , where

\begin{align*}\tau_1^*(S)&=\operatorname{ess\,inf}\{\tau\in \mathcal{S}_S\colon Z^{(2)}(\tau)=X^{(2)}(\tau)\},\\\tau_2^*(S)&=\operatorname{ess\,inf}\{\tau\in \mathcal{S}_{\tau^*_1(S)+\delta}\colon Z^{(1)}(\tau)=X(\tau)\}.\end{align*}

Motivated by this construction (or the construction in [Reference Carmona and Touzi4] and [Reference Kobylanski, Quenez and Rouy-Mironescu8]), for the general case, given $\tau\in \mathcal{S}_0$ , we set

\begin{align*}\tau^*_1(\tau)=\operatorname{ess\,inf}\{\sigma\in \mathcal{S}_\tau\colon X^{(\ell)}(\sigma)=Y^{(\ell)}(\sigma)\}.\end{align*}

For $2\leq i\leq \ell$ , we define

\begin{align*}\tau^*_i(\tau)&=\operatorname{ess\,inf}\{\sigma\in \mathcal{S}_{\tau^*_{i-1}(\tau)+\delta}\colon X^{(\ell-i+1)}(\sigma)=Y^{(\ell-i+1)}(\sigma)\}I_{\{\delta+\tau^*_{i-1}(\tau)\leq T\}} + (T_+) I_{\{\delta+\tau^*_{i-1}(\tau)>T\}}.\end{align*}

It is easy to check that $\vec{\tau}^*(\tau)=(\tau^*_1(\tau),\ldots,\tau^*_\ell(\tau))\in \mathcal{S}_\tau(\ell,\delta)$ . Actually, we claim that $\vec{\tau}^*(\tau)$ is the optimal multiple stopping time for the problem (3.1). In order to prove this assertion, we first show that $\tau^*_i$ is optimal for the auxiliary single stopping problem (3.2).

Theorem 3.1. Under assumption (H1), for each $\tau\in\mathcal{S}_0$ , there exists an optimal stopping time for $Y^{(\ell-i+1)}(\tau)$ , $1\leq i\leq \ell$ . Moreover, $\tau^*_i(\tau)$ is the minimal optimal stopping time for $Y^{(\ell-i+1)}(\tau^*_{i-1}(\tau)+\delta)$ (by convention $\tau^*_0(\tau)+\delta=\tau$ ), and the family

\begin{align*}\{Y^{(\ell-i+1)}(\sigma\wedge \tau^*_i(\tau))\colon \tau^*_{i-1}(\tau)+\delta\leq \sigma\leq T\}\end{align*}

is a martingale family.

Theorem 3.2. Let assumption (H1) hold. Then, for each $\tau\in\mathcal{S}_0$ , we have

\begin{align*}Z^{(\ell)}(\tau)=Y^{(\ell)}(\tau)=\mathbb{E}_\tau\Biggl[\sum_{i=1}^\ell X(\tau^*_i(\tau))\Biggr],\end{align*}

which implies that $\vec{\tau}^*(\tau)$ is the optimal multiple stopping time.

Proof. The proof is similar to those for Theorem 2.1 of [Reference Carmona and Touzi4] and Theorem 4.10 of [Reference Ben Latifa, Bonnans and Mnif1]. For the readers’ convenience, we give a short proof here. For any $\vec{\tau}=(\tau_1,\ldots,\tau_\ell)\in \mathcal{S}_\tau(\ell,\delta)$ , we first note that for any $1\leq i\leq \ell$ ,

(3.9) \begin{equation}\mathbb{E}_\tau\Biggl[\sum_{j=1}^\ell X(\tau_j)\Biggr]\leq \mathbb{E}_\tau\Biggl[X^{(i)}(\bar{\tau}_i)+\sum_{j=i+1}^\ell X(\bar{\tau}_j)\Biggr],\end{equation}

where $\bar{\tau}_i\,:\!=\, \tau_{\ell-i+1}$ and $\sum_{j=\ell+1}^\ell X(\bar{\tau}_j)\,:\!=\, 0$ . The proof of (3.9) is postponed (to the end of this proof). Now, choosing $i=\ell$ , by Theorem 3.1, we have

(3.10) \begin{align}\mathbb{E}_\tau\Biggl[\sum_{j=1}^\ell X(\tau_j)\Biggr]&\leq \mathbb{E}_\tau[X^{(\ell)}(\bar{\tau}_\ell)]\notag \\&\leq Y^{(\ell)}(\tau)=\mathbb{E}_\tau[X^{(\ell)}(\tau^*_1(\tau))]\notag \\&=\mathbb{E}_\tau[X(\tau^*_1(\tau))+\mathbb{E}_{\tau^*_1(\tau)}[Y^{(\ell-1)}(\tau^*_1(\tau)+\delta)]]\notag \\&=\mathbb{E}_\tau[X(\tau^*_1(\tau))]+\mathbb{E}_{\tau}[Y^{(\ell-1)}(\tau^*_1(\tau)+\delta)].\end{align}

By Theorem 3.1 and (3.3), it is easy to check that for any $1\leq i\leq \ell-1$ ,

(3.11) \begin{equation}\mathbb{E}_\tau[Y^{(\ell-i)}(\tau^*_{i}(\tau)+\delta)]=\mathbb{E}_\tau[X^{(\ell-i)}(\tau^*_{i+1}(\tau))]=\mathbb{E}_\tau[X(\tau^*_{i+1}(\tau))+Y^{(\ell-i-1)}(\tau^*_{i+1}(\tau)+\delta)].\end{equation}

Using (3.10), (3.11), and by a recursive method, we have

\begin{align*}\mathbb{E}_\tau\Biggl[\sum_{j=1}^\ell X(\tau_j)\Biggr]\leq Y^{(\ell)}(\tau)\leq \mathbb{E}_\tau\Biggl[\sum_{j=1}^\ell X(\tau^*_j(\tau))\Biggr]\leq Z^{(\ell)}(\tau).\end{align*}

Since $\vec{\tau}=(\tau_1,\ldots,\tau_\ell)\in \mathcal{S}_\tau(\ell,\delta)$ is chosen arbitrarily, taking the essential supremum over all $\vec{\tau}$ in $\mathcal{S}_\tau(\ell,\delta)$ , we get the desired result.

It remains to prove that (3.9) holds by an induction method. Clearly this inequality holds when $i=1$ . Suppose that (3.9) is true for $1\leq k\leq i-1$ , with $2\leq i\leq \ell$ . Since we have

(3.12) \begin{equation}\mathbb{E}_\tau\Biggl[\sum_{j=1}^\ell X(\tau_j)\Biggr]=\mathbb{E}_\tau\Biggl[\sum_{j=1}^{i-1} X(\bar{\tau}_j)+\sum_{j=i}^\ell X(\bar{\tau}_j)\Biggr],\end{equation}

it suffices to prove that

\begin{align*}\mathbb{E}_\tau\Biggl[\sum_{j=1}^{i-1} X(\bar{\tau}_j)\Biggr]\leq \mathbb{E}_\tau[X^{(i)}(\bar{\tau}_{i})-X(\bar{\tau}_{i})].\end{align*}

Recalling that (3.9) holds for $1\leq k\leq i-1$ and noting that (3.12), we have

\begin{align*}\mathbb{E}_\tau\Biggl[\sum_{j=1}^{i-1} X(\bar{\tau}_j)\Biggr]\leq \mathbb{E}_\tau[X^{(i-1)}(\bar{\tau}_{i-1})].\end{align*}

Therefore, we only need to show that

(3.13) \begin{equation}\mathbb{E}_\tau[X^{(i-1)}(\bar{\tau}_{i-1})]\leq \mathbb{E}_\tau[X^{(i)}(\bar{\tau}_{i})-X(\bar{\tau}_{i})].\end{equation}

For the case $T-\delta< \bar{\tau}_{i}\leq T$ , we have $X^{(i)}(\bar{\tau}_{i})=X(\bar{\tau}_{i})$ and $\bar{\tau}_{i-1}=\tau_{\ell-i+2}\geq \tau_{\ell-i+1}+\delta=\bar{\tau}_{i}+\delta>T$ . It follows that

\begin{align*}\mathbb{E}_\tau[X^{(i-1)}(\bar{\tau}_{i-1})]=0= \mathbb{E}_\tau[X^{(i)}(\bar{\tau}_{i})-X(\bar{\tau}_{i})].\end{align*}

For the case $0\leq \bar{\tau}_{i}\leq T-\delta$ , noting that $\bar{\tau}_{i-1}=\tau_{\ell-i+2}\geq \tau_{\ell-i+1}+\delta=\bar{\tau}_{i}+\delta\geq \tau$ , we have

\begin{align*}\mathbb{E}_\tau[X^{(i-1)}(\bar{\tau}_{i-1})]=\mathbb{E}_\tau[\mathbb{E}_{\bar{\tau}_{i}+\delta}[X^{(i-1)}(\bar{\tau}_{i-1})]]\leq \mathbb{E}_\tau[Y^{(i-1)}(\bar{\tau}_{i}+\delta)]= \mathbb{E}_\tau[X^{(i)}(\bar{\tau}_{i})-X(\bar{\tau}_{i})].\end{align*}

Hence (3.13) holds true. The proof is complete.

4. Optimal multiple stopping with reward process

In this section we consider optimal multiple stopping problems for the particular case where the reward is given by a progressively measurable process $\{X_t,t\in[0,T]\}$ . Applying the results obtained in the previous section, we construct the optimal multiple stopping times in terms of first hitting times under the following assumption:

1. (H2) X is a non-negative, progressively measurable process with right upper semicontinuous path (RUSC) a.s. For some $p>1$ , $\mathbb{E}[\bar{X}^p]<\infty$ , where $\bar{X}=\sup_{t\in[0,T]}X_t$ . Furthermore, X is left upper semicontinuous in expectation along stopping times (LUSCE), that is, for any $\{\tau_n\}_{n\in \mathbb{N}}\subset \mathcal{S}_0$ with $\tau_n\uparrow \tau$ , we have

   \begin{align*}\limsup_{n\rightarrow \infty}\mathbb{E}[X_{\tau_n}]\leq \mathbb{E}[X_\tau].\end{align*}

Set $X_t=0$ , $t>T$ , and consider the following value function of the optimal multiple stopping problem associated with the reward process X,

\begin{equation*} Z^{(\ell)}_0\,:\!=\, \sup_{\tau^{(\ell)}\in \mathcal{S}_0(\ell,\delta)} \mathbb{E}\Biggl[\sum_{i=1}^\ell X_{\tau_i}\Biggr],\end{equation*}

and the auxiliary optimal single stopping problems,

(4.1) \begin{equation}Y^{(0)}\equiv 0 \quad \text{and}\quad Y^{(i)}_t=\operatorname{ess\,sup}_{\sigma\in\mathcal{S}_t}\mathbb{E}_t[X^{(i)}_\sigma], \quad i=1,\ldots,\ell,\end{equation}

where the ith exercise reward process $X^{(i)}$ is given by

(4.2) \begin{equation}X_t^{(i)}=\begin{cases} X_t+ \mathbb{E}_t[Y^{(i-1)}_{t+\delta}], & t\in[0,T-\delta],\\X_t, &t\in(T-\delta,T].\end{cases}\end{equation}

We will show that the value $Z^{(\ell)}_0$ coincides with $Y^{(\ell)}_0$ and the optimal multiple stopping time exists under assumption (H2). In fact these results are obtained in [Reference Carmona and Touzi4] when the reward process X is continuous and in [Reference Ben Latifa, Bonnans and Mnif1] when X is right continuous and left continuous in expectation along stopping times (LCE for short), that is, for any $\{\tau_n\}_{n\in \mathbb{N}}\subset \mathcal{S}$ with $\tau_n\uparrow \tau$ , we have

\begin{align*}\lim_{n\rightarrow \infty}\mathbb{E}[X_{\tau_n}]= \mathbb{E}[X_\tau].\end{align*}

We first list some properties which do not essentially depend on the regularity of the reward process.

Lemma 4.1. Under assumption (H2), we have for all $i=1,\ldots,\ell$ ,

\begin{align*}\mathbb{E}[(\bar{X}^{(i)})^p]<\infty,\quad {where}\ \bar{X}^{(i)}=\sup_{t\in[0,T]}X^{(i)}_t\end{align*}

and p is the same as in (H2).

Proof. The proof is similar to those for Lemma 2.1 of [Reference Carmona and Touzi4] and Lemma 4.2 of [Reference Ben Latifa, Bonnans and Mnif1]. For the readers’ convenience, we give a short proof here. We proceed by induction. For $i=1$ , the result is obvious since $X^{(1)}=X$ . Let us assume that $\mathbb{E}[(\bar{X}^{(i-1)})^p]<\infty$ for some $2\leq i\leq \ell$ . We shall prove that $\mathbb{E}[|\bar{X}^{(i)}|^p]<\infty$ . For any $\tau\in \mathcal{S}_0$ and $t\in[0,T]$ , it is easy to check that

\begin{align*}\mathbb{E}_t[X^{(i-1)}_\tau]\leq \mathbb{E}_t[\bar{X}^{(i-1)}].\end{align*}

Since $\tau$ is chosen arbitrarily, we have $Y^{(i-1)}_t\leq \mathbb{E}_t[\bar{X}^{(i-1)}]$ . The martingale inequality yields

\begin{align*}\mathbb{E}\Bigl[\sup_{t\in[0,T]}|Y^{(i-1)}_t|^p\Bigr]&\leq \mathbb{E}\Bigl[\sup_{t\in[0,T]}|\mathbb{E}_t[\bar{X}^{(i-1)}] |^p\Bigr]\\&\leq \mathbb{E}\Bigl[\sup_{t\in[0,T]}\mathbb{E}_t[|\bar{X}^{(i-1)}|^p] \Bigr]\\&\leq \biggl(\dfrac{p}{p-1}\biggr)^p\mathbb{E}[|\bar{X}^{(i-1)}|^p]\\&<\infty.\end{align*}

A simple calculation indicates that

\begin{align*}\mathbb{E}[|\bar{X}^{(i)}|^p]& \leq C\Bigl\{\mathbb{E}[|\bar{X}|^p]+\mathbb{E}\Bigl[\sup_{t\in[0,T]}|\mathbb{E}_t[Y^{(i-1)}_{t+\delta}]|^p\Bigr]\Bigr\}\\&\leq C\Bigl\{\mathbb{E}[|\bar{X}|^p]+\mathbb{E}\Bigl[\sup_{t\in[0,T]}\mathbb{E}_t\Bigl[\sup_{s\in[0,T]}|Y^{(i-1)}_{s}|^p\Bigr]\Bigr]\Bigr\}\\&\leq C\Bigl\{\mathbb{E}[|\bar{X}|^p]+\mathbb{E}\Bigl[\sup_{t\in[0,T]}|Y^{(i-1)}_{t}|^p\Bigr]\Bigr\}\\&<\infty,\end{align*}

where C is a constant depending only on p. The proof is complete.

In the following, we aim to show that $X^{(i)}$ possesses all the properties of X, especially the regularity property, which makes it possible to represent the optimal stopping time for $Y^{(i)}$ as the first hitting time using Proposition 2.5. Before establishing this result, we first list some results about the single stopping problem when the reward is given by a progressively measurable process (see [Reference Kobylanski and Quenez7]).

Let $\{X_t,t\in[0,T]\}$ be a non-negative progressively measurable process. Set $\phi(\tau)\,:\!=\, X_\tau$ , for any $\tau\in \mathcal{S}_0$ . Then the family $\{\phi(\tau),\tau\in\mathcal{S}_0\}$ is admissible. Let $v_\phi$ be the value function associated with the family $\{\phi(\tau),\tau\in\mathcal{S}_0\}$ . Suppose that $v_\phi(0)<\infty$ . Then, by Proposition B.1 of [Reference Kobylanski and Quenez7], there exists a progressively measurable process $\{v_t,t\in[0,T]\}$ aggregating the value function family $\{v_\phi(\tau),\tau\in\mathcal{S}_0\}$ , i.e. $v_\phi(S)=v_S$ for all $S\in\mathcal{S}_0$ . Further, the process $\{v_t,t\in[0,T]\}$ is a strong supermartingale and there exists a right limited and left limited version of $\{v_t,t\in[0,T]\}$ , still denoted by $\{v_t,t\in[0,T]\}$ (see P23 in [Reference Kobylanski and Quenez7]).

Further, we assume additionally that $\{X_t,t\in[0,T]\}$ is of class $\mathcal{D}$ and the family $\{X_\tau,\tau\in\mathcal{S}_0\}$ is RUSCE. Then $\{v_t,t\in[0,T]\}$ is a strong supermartingale of class $\mathcal{D}$ . Furthermore, by Proposition 2.26 of [Reference El Karoui5] (see also P25 in [Reference Kobylanski and Quenez7]), there exists a uniformly integrable RCLL martingale M, a predictable right continuous non-decreasing process A with $A_0=0$ and $\mathbb{E}[A_T]<\infty$ and an adapted non-decreasing process C, which is purely discontinuous with $C_0=0$ and $\mathbb{E}[C_T]<\infty$ , such that

\begin{align*}v_t=M_t-A_t-C_{t-}, \quad t\in[0,T].\end{align*}

Moreover, the above decomposition is unique. With this decomposition, it is easy to check that $\{v_t,t\in[0,T]\}$ is RUSC.

To summarize, we have the following lemma.

Lemma 4.2. Suppose that the non-negative, progressively measurable process $\{X_t,t\in[0,T]\}$ is of class $\mathcal{D}$ and RUSCE. Then $\{v_t,t\in[0,T]\}$ is a strong supermartingale of class $\mathcal{D}$ and is RUSC, where, for any $t\in[0,T]$ ,

\begin{align*}v_t=\operatorname{ess\,sup}_{\tau\in \mathcal{S}_t}\mathbb{E}_t[X_\tau].\end{align*}

Now, we may prove that $X^{(i)}$ inherits all the properties of X.

Proof. It is clear that the process $X^{(i)}$ is non-negative and progressively measurable. We prove by induction that $X^{(i)}$ is RUSC and LUSCE. For $i=1$ , we are done since $X^{(1)}=X$ . Now assume that $X^{(i-1)}$ is RUSC and LUSCE for $i=2,\ldots, \ell$ . Let us first prove that $X^{(i)}$ is RUSC. By the definition of $X^{(i)}$ (see (4.2) and note that it is clear that $X^{(i-1)}_{T-\delta}=X_{T-\delta}+\mathbb{E}_{T-\delta}[X_T]\geq \limsup_{t\downarrow T-\delta} X^{(i-1)}_t$ ), it suffices to prove that $\{\mathbb{E}_t[Y^{(i-1)}_{t+\delta}]\}$ is RUSC on $[0,T-\delta)$ . First, by Lemma 4.2, $Y^{(i-1)}$ is an RUSC strong supermartingale. For any $t_n\downarrow t$ as n goes to infinity, there exists some N such that for $n\geq N$ , $t_n\leq t+\delta$ . Then we have

\begin{align*}&\limsup_{n\rightarrow \infty}\bigl(\mathbb{E}_{t_n}\bigl[Y^{(i-1)}_{t_n+\delta}\bigr]-\mathbb{E}_t\bigl[Y^{(i-1)}_{t+\delta}\bigr]\bigr)\\&\quad \leq \limsup_{n\rightarrow \infty}\bigl(\mathbb{E}_{t_n}\bigl[Y^{(i-1)}_{t_n+\delta}\bigr]-\mathbb{E}_{t_n}\bigl[Y^{(i-1)}_{t+\delta}\bigr]\bigr)+\limsup_{n\rightarrow \infty}\bigl(\mathbb{E}_{t_n}\bigl[Y^{(i-1)}_{t+\delta}\bigr]-\mathbb{E}_{t}\bigl[Y^{(i-1)}_{t+\delta}\bigr]\bigr)\\&\quad =\limsup_{n\rightarrow \infty}\bigl(\mathbb{E}_{t_n}\bigl[\mathbb{E}_{t+\delta}\bigl[Y^{(i-1)}_{t_n+\delta}\bigr]\bigr]-\mathbb{E}_{t_n}\bigl[Y^{(i-1)}_{t+\delta}\bigr]\bigr)\leq 0,\end{align*}

where we have used the fact that $Y^{(i-1)}$ is a supermartingale in the last inequality. Therefore $\bigl\{\mathbb{E}_t\bigl[Y^{(i-1)}_{t+\delta}\bigr]\bigr\}$ is RUSC on $[0,T-\delta)$ . Recalling that we assume $X^{(i-1)}$ is RUSC and LUSCE, it follows that the family $\bigl\{X^{(i-1)}_\tau,\tau\in\mathcal{S}_0\bigr\}$ is USCE. Then the LUSCE property of $X^{(i)}$ is obtained from Lemma 3.2. The proof is complete.

Remark 4.2. In order to show that $\{X^{(i)}_t,t \in [0,T]\}$ is progressively measurable, it suffices to prove that $\{\mathbb{E}_t[\eta_{t+\delta}]\}_{t\in[0,T-\delta]}$ is progressively measurable for any $\eta \in M_p$ , where $p\geq 1$ and

\begin{align*}M_p=\biggl\{\eta\colon \eta \ \text{is progressively measurable and}\ \mathbb{E}\biggl[\int_0^T |\eta_t|^p\,{\mathrm{d}} t\biggr]<\infty\biggr\}.\end{align*}

In fact, if $\eta$ is a simple process, it is clear that $\{\mathbb{E}_t[\eta_{t+\delta}]\}_{t\in[0,T-\delta]}$ is progressively measurable. For the general case, let $\{\eta^n\}_{n\in \mathbb{N}}$ be a sequence of simple process such that

\begin{align*}\lim_{n\rightarrow \infty}\mathbb{E}\biggl[\int_0^T|\eta^n_t-\eta_t|^p \,{\mathrm{d}} t\biggr]=0.\end{align*}

It is easy to check that for any $n\in\mathbb{N}$ , $\{\mathbb{E}_t[\eta^n_{t+\delta}]\}_{t\in[0,T-\delta]}$ is progressively measurable and

\begin{align*}\lim_{n\rightarrow \infty}\mathbb{E}\biggl[\int_0^{T-\delta}|\mathbb{E}_t[\eta^n_{t+\delta}]-\mathbb{E}_t[\eta_{t+\delta}]|^p \,{\mathrm{d}} t\biggr]\leq \lim_{n\rightarrow \infty}\mathbb{E}\biggl[\int_0^{T-\delta}|\eta^n_t-\eta_t|^p \,{\mathrm{d}} t\biggr]=0.\end{align*}

It follows that $\{\mathbb{E}_t[\eta_{t+\delta}]\}_{t\in[0,T-\delta]}$ is progressively measurable.

Building upon Proposition 2.5, Lemma 4.1, and Lemma 4.3, we may construct the optimal stopping time for the auxiliary optimal stopping problems (4.1).

Theorem 4.1. Under assumption (H2), for each $\tau\in \mathcal{S}_0$ , there exists an optimal stopping time for $Y^{(\ell-i+1)}_\tau$ , $i=1,\ldots,\ell$ . Moreover, $\tau^*_i$ is the minimal optimal stopping time for $Y^{(\ell-i+1)}_{\tau^*_{i-1}+\delta}$ (with convention $\tau^*_0+\delta=0$ ), where

\begin{align*}\tau^*_1=\inf\bigl\{t\geq 0\colon Y^{(\ell)}_t=X^{(\ell)}_t\bigr\}\end{align*}

and for $2\leq i\leq \ell$ ,

\begin{align*}\tau^*_i=\inf\bigl\{t\geq \tau^*_{i-1}+\delta\colon Y^{(\ell-i+1)}_t=X^{(\ell-i+1)}_t\bigr\} I_{\{\tau^*_{i-1}+\delta\leq T\}}+T_+ I_{\{\tau^*_{i-1}+\delta>T\}}.\end{align*}

Further, $(\tau^*_1,\ldots,\tau^*_\ell)\in \mathcal{S}_0(\ell,\delta)$ and the process $\bigl\{Y^{(\ell-i+1)}_{t\wedge \tau^*_i}, \tau^*_{i-1}+\delta\leq t\leq T\bigr\}$ is a martingale.

Now we can state the final result in this section, which indicates that when the reward is given by a progressively measurable process satisfying some weak regularity conditions, the optimal multiple stopping times can be interpreted by the first hitting times.

Theorem 4.2. Under assumption (H2), we have

\begin{align*}Z^{(\ell)}_0=Y^{(\ell)}_0=\mathbb{E}\Biggl[\sum_{i=1}^\ell X_{\tau^*_i}\Biggr],\end{align*}

where $(\tau^*_1,\ldots,\tau^*_\ell)$ is defined in Theorem 4.1. That is, $(\tau^*_1,\ldots,\tau^*_\ell)$ is the optimal exercise strategy for the optimal multiple stopping problem.