# Stopping Times of Boundaries:
Relaxation and Continuity

H. Mete Soner111Department of Operations Research and Financial Engineering,
Princeton University, Princeton, NJ, 08540, USA, email<EMAIL_ADDRESS>Valentin Tissot-Daguette222Department of Operations Research and Financial
Engineering, Princeton University, Princeton, NJ, 08540, USA, email: v.tissot-
<EMAIL_ADDRESS>
###### Abstract

We study boundaries and their hitting times in the context of optimal stopping
and Bermudan-style options. We prove the continuity of the map linking the
boundary to the value of its associated stopping policy. While the supremum
norm is used to compare continuous boundaries, a weaker topology induced by
the so-called relaxed $L^{\infty}$ distance is used in the semicontinuous
case. Relaxed stopping rules and their properties are also discussed.
Incidentally, we provide a convergence analysis for neural stopping boundaries
[Reppen, Soner, and Tissot-Daguette, Neural Optimal Stopping Boundary, 2022]
entailing the universal approximation capability of neural networks and the
notion of inf/sup convolution.

Keywords: Hitting times, relaxed stopping rules, convergence analysis,
Bermudan options, inf/sup convolution.

Mathematics Subject Classification: 35R35, 91G20, 60G57, 68T07.

##  1 Introduction

The optimal stopping problem, whose simplest form is given by

$\sup\left\\{v(\tau)\mathrel{\mathop{\ordinarycolon}}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\varphi(\tau,X_{\tau})]\
\mathrel{\mathop{\ordinarycolon}}\ \tau=\text{ stopping time}\right\\},$ (1)

for a process $X$ and reward function $\varphi$, plays a central role in
stochastic optimal control and has applications in many quantitative
disciplines. In mathematical statistics, sequential tests such as the quickest
detection of a Wiener process can be tackled using the theory of optimal
stopping [36, 30]. In mathematical finance, optimal stopping problems emerge
naturally from the pricing of American or Bermudan contingent claims [11, 22,
30]. In this context, $\varphi$ is the payoff of the claim, $X$ describes the
price evolution of the underlying asset, and the optimal value in (1) gives
the initial price of the contract.

The initial motivation of this paper is to provide a convergence study of the
neural optimal stopping boundary method [33] (see also [32]) to solve optimal
stopping problems arising notably from American options. The boundary
delimiting the stopping (or exercise) region is parameterized by a feedforward
artificial neural network. Further details on the algorithm are given in
Section 5 and at the end of this section. The convergence analysis relies
mainly on relaxation and continuity results for hitting times of boundaries,
presented in Sections 3 and 4, respectively. We then realized that these
findings can be used in more general stochastic control problems, hence
broadening the range of applications. We stress that a discrete time framework
is adopted throughout, in line with the inevitable use of temporal
discretization in numerical methods. Adapting the present results in
continuous time is not straightforward and would certainly involve other
interesting techniques. This is, however, beyond the scope of this work.

In Section 2, we introduce the optimal stopping problem in discrete time and
make natural assumptions on the reward function and stopping region.
Importantly, we suppose that the state space of the underlying process can be
transformed so as to represent the stopping region as the epigraph (or
hypograph) of a particular function$-$the optimal stopping boundary$-$denoted
by $f^{\diamond}$ throughout. Provided that the state process started outside
the stopping region, then the hitting time of $f^{\diamond}$ is an optimal
stopping time for (1). This implies that one can restrict the set of stopping
times in (1) to hitting times of boundaries, namely

$\sup\left\\{v(\tau_{f})\mathrel{\mathop{\ordinarycolon}}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\varphi(\tau_{f},X_{\tau_{f}})]\
\mathrel{\mathop{\ordinarycolon}}\ f=\text{ stopping boundary}\right\\},$ (2)

where $\tau_{f}$ is the hitting time of the epigraph (or hypograph) of $f$.

In Section 3, we discuss a relaxation of (2) which is essential for stochastic
gradient-based algorithms in optimal stopping, such as the deep optimal
stopping method [3, 4] and the neural optimal stopping boundary method [33].
Postponing technical details to Section 3, the relaxed version of problem (1)
takes the form

$\sup\left\\{v(\mu)\mathrel{\mathop{\ordinarycolon}}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\int\varphi(t,X_{t})\mu(dt)\right]\
\mathrel{\mathop{\ordinarycolon}}\ \mu=\text{ random probability
measure}\right\\}.$ (3)

Also called convexification [1] or compactification [10], relaxation
procedures such as (3) are classically used in stochastic control problems to
prove the existence of an optimal solution [13, 17, 28]. Indeed, there may not
exist an optimal control in the strict sense, e.g. when the admissible set is
not convex. Here, convexification has the special advantage that it transforms
optimal stopping problems into a more classical formulation in stochastic
control, opening the door to numerous methods and tools. In addition, we shall
see in Proposition 3 that problems (1) and (3) achieve the same optimal value.
This is a crucial property, valid in many contexts; see, e.g., [16, 18] and
[23, Lemma 1.5.2.] in continuous time optimal stopping and [10, 13, 17] in
general stochastic optimal control. Other relaxation techniques have proved
useful in optimal stopping, such as policies induced by occupation measure of
the stopped process [6] or randomized exercise rates [2]. We can also mention
penalty methods for American options [14, 20] relaxing the otherwise
constrained dominance of the value function (the price) over the payoff
$\varphi$. Unlike convexification, the penalty approach introduces a bias in
the optimal value, which nonetheless vanishes in the limit.

In the neural optimal stopping boundary method, the stopping policy is relaxed
by introducing a fuzzy region around the boundary. The stopping decision
remains the same outside the fuzzy region, while inside, the former is
replaced by stopping intensities proportional to the signed distance to the
boundary. This procedure generates relaxed hitting times of
boundaries$-$namely random probability measures supported on the set of
exercise dates. The relaxed problem (3) thus becomes

$\sup\left\\{v(\mu_{f})\mathrel{\mathop{\ordinarycolon}}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\int\varphi(t,X_{t})\mu_{f}(dt)\right]\
\mathrel{\mathop{\ordinarycolon}}\ \mu_{f}=\text{relaxed hitting time of
$f$}\right\\}.$ (4)

As shifting the boundary ever so slightly changes the stopping intensities
inside the fuzzy region, the above relaxation makes stochastic gradient
methods applicable. The fuzzy region is akin to the mushy zone between two
phases (typically solid and liquid) in phase-field models [5, 40, 38]. The
introduction of a fuzzy region is critical in other deep learning algorithms
as well, such as occupation networks in computer vision [29] or deep level-set
functions for solving Stefan problems [37].

Section 4 is devoted to continuity results. In particular, Theorems 4.1 and
4.2 establish the continuity of the value functional
$\operatorname{\mathfrak{F}}\ni f\mapsto v(\tau_{f})$, where
$\operatorname{\mathfrak{F}}$ is a set of stopping boundaries. Let us give a
condensed statement.

######  Theorem.

(Continuity of $\bm{f\mapsto v(\tau_{f})}$) Let $\mathfrak{m}$ be a metric on
$\operatorname{\mathfrak{F}}$. Then for all $\delta>0$, there exists $\iota>0$
satisfying

$\forall f,f^{\prime}\in\operatorname{\mathfrak{F}}\ \text{ such that }\
\mathfrak{m}(f,f^{\prime})\leq\iota\ \Longrightarrow\
|v(\tau_{f})-v(\tau_{f^{\prime}})|\leq\delta.$ (5)

The first continuity theorem (Theorem 4.1) implies that the value associated
to the relaxed hitting time of boundaries as in (4) converges to the value
using the sharp interface as the width of the fuzzy region tends to zero. This
is the content of Corollary 1.

When the boundaries are continuous in space, the metric in the above theorem
will be the supremum norm in a fixed compact set. Otherwise, a weaker topology
induced by the so-called relaxed $L^{\infty}$ distance is considered; see
Definition 1. Loosely speaking, the relaxed $L^{\infty}$ distance compares
boundaries $f,f^{\prime}\in\operatorname{\mathfrak{F}}$ uniformly while giving
the flexibility to move to nearby points where $f,f^{\prime}$ are closer to
one another. This is done through a transformation of the domain, similar to
reparametrization maps in the Skorokhod $J1$ distance for càdlàg processes.

In Section 5, we combine the above results and provide a convergence analysis
of the neural optimal stopping boundary method [33]. For concreteness, we now
summarize the algorithm while postponing the details to Section 5. The
algorithm looks for a neural stopping boundary, namely a feedforward neural
network $g^{\theta}$ such that

$v(\tau_{g^{\theta}})\approx\sup_{f\in\operatorname{\mathfrak{F}}}v(\tau_{f})\
=v(\tau_{f^{\diamond}}).$

The parameter vector $\theta$ is then trained by maximizing the relaxed reward
$v(\mu_{g^{\theta}})$ using stochastic gradient ascent. When the optimal
stopping boundary $f^{\diamond}$ is continuous, one can apply the celebrated
Universal Approximation Theorem [8, 19] stating that any continuous function
can be approximated arbitrarily well by feedforward neural networks in compact
sets. Combined with the continuity of $f\mapsto v(\tau_{f})$, we can derive a
convergence result (Theorem 5.3) whose statement is now summarized.

######  Theorem.

(Convergence) For all $\delta>0$, there exists $\varepsilon>0$ and a
feedforward neural network $g^{\theta}$ such that $v(\tau_{g^{\theta}})\geq
v(\tau_{f^{\diamond}})-\delta$.

Generally, the optimal stopping boundary $f^{\diamond}$ may fail to be
continuous, preventing the use of the Universal Approximation Theorem.
Nevertheless, it is shown in Proposition 2 that, under mild assumptions, the
stopping boundary is at least semicontinuous. We then adopt the following
strategy: $(i)$ Construct a sequence of continuous functions converging to
$f^{\diamond}$, $(ii)$ Approximate each continuous function in $(i)$ by a
neural network. We therefore expect that the sequence of neural networks
converges to $f^{\diamond}$ in some suitable sense. Regarding step $(i)$, we
employ inf (respectively sup) convolutions when $f^{\diamond}$ is lower (resp.
upper) semicontinuous. Inf/sup convolutions, recalled in Section 5.2, are
effective regularization tools popularized in the theory of viscosity
solutions; see, e.g., [12, Section V.5] and [25]. As explained in Theorem 5.5,
it turns out that lower semicontinuous functions can be approximated
arbitrarily well by neural networks with respect to the relaxed $L^{\infty}$
distance mentioned earlier. This is an interesting result in its own right and
can be applied to other tasks involving the approximation of semicontinuous
functions.

There is a recent body of literature on the convergence of deep learning
methods in stochastic optimal control: Huré et al. [21] study the convergence
of discrete time stochastic control problems involving policy and/or value
iteration; Lapeyre and Lelong [24] prove the convergence of the Longstaff-
Schwartz algorithm [27] using neural networks to approximate the continuation
value; Gonon [15] sheds light on the expressivity and maximal size of neural
networks to produce optimal stopping rules. Let us also mention a result given
in the original deep optimal stopping paper [3, Proposition 4] proving the
near optimality of deep stopping strategies. While all the above algorithms
are based on dynamic programming, the neural optimal stopping boundary method
[33] finds an optimal stopping rule in a forward fashion. Consequently, our
convergence analysis necessitates other techniques and tools, hence differs
greatly from previous works.

Structure of the paper. In Section 2, we introduce the optimal stopping
problem, our main structural assumption on the exercise region, and give
financial examples. In Section 3, we define the fuzzy region and construct
relaxed stopping rules. Section 4 establishes the continuity of the map
$f\mapsto v(\tau_{f})$ as well as important consequences. In Section 5, we
prove the convergence of the neural optimal stopping boundary method in the
continuous and semicontinuous case, respectively. Section 6 provides
concluding remarks and Appendix A contains the proof of Proposition 3.

##  2 Optimal stopping

Let $\operatorname{\mathcal{T}}\subset[0,\infty)$ a finite subset of time
indices containing $0$ and
$(\Omega,\operatorname{\mathbb{F}}=(\operatorname{\mathcal{F}}_{t})_{t\in\operatorname{\mathcal{T}}},\operatorname{\mathbb{Q}})$
a filtered probability space. The state
$X=(X_{t})_{t\in\operatorname{\mathcal{T}}}$ is an
$\operatorname{\mathbb{F}}$-adapted, Markov process which take values in a
Euclidean space $\operatorname{\mathcal{X}}$. We assume for concreteness that
$\operatorname{\mathbb{F}}$ is the filtration generated by $X$. Given a
_reward (or payoff) function_
$\varphi\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{T}}\times\operatorname{\mathcal{X}}\to\operatorname{\mathbb{R}}$,
the associated _optimal stopping problem_ is to

$\text{maximize}\ \
v(\tau)\mathrel{\mathop{\ordinarycolon}}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\varphi(\tau,X_{\tau})],\qquad\text{over}\
\tau\,\in\,\vartheta(\operatorname{\mathcal{T}}),$ (6)

where $\vartheta(\operatorname{\mathcal{T}})$ is the set of
$\operatorname{\mathcal{T}}$-valued $\operatorname{\mathbb{F}}$-stopping
times. Problem $\eqref{eq:OS}$ is well-defined (and non-trivial) if
$\max_{t\in\operatorname{\mathcal{T}}}|\varphi(t,X_{t})|\in
L^{1}(\operatorname{\mathbb{Q}})$. We write $v^{\diamond}$ for the optimal
value of (6). The following assumption on $\varphi$ and the dynamics of $X$
will prove useful.

######  Assumption 1.

There exists $a>0$ such that
$\varphi(t,\cdot)\in\operatorname{\mathcal{L}}_{a}\ \forall\
t\in\operatorname{\mathcal{T}}$, where

$\operatorname{\mathcal{L}}_{a}\mathrel{\mathop{\ordinarycolon}}=\left\\{\phi\in\operatorname{\mathcal{C}}(\operatorname{\mathcal{X}};\operatorname{\mathbb{R}})\
\mathrel{\mathop{\ordinarycolon}}\ |\phi(x)|\leq C[1+|x|^{a}]\ \ \text{for
some}\ C>0\right\\}.$ (7)

Moreover, $\phi\in\operatorname{\mathcal{L}}_{a}$ implies that
$K_{t,s}\phi\in\operatorname{\mathcal{L}}_{a}$ $\forall\
t,s\in\operatorname{\mathcal{T}}$, $t<s$, with the transition semigroup

$K_{t,s}\ \phi(x)=\operatorname{\mathbb{E}}[\phi(X_{s})\mid X_{t}=x],\quad
x\in\operatorname{\mathcal{X}}.$ (8)

Allowing the second component of $K_{\cdot,\cdot}$ in (8) to be a stopping
time, define the value function

$v^{\diamond}(t,x)=\sup_{t\leq\tau\in\vartheta(\operatorname{\mathcal{T}})}K_{t,\tau}\varphi(\cdot,x)\
,$ (9)

and stopping region
$\operatorname{\mathcal{S}}_{t}=\\{x\in\operatorname{\mathcal{X}}\mathrel{\mathop{\ordinarycolon}}\varphi(t,x)=v^{\diamond}(t,x)\\},\
t\in\operatorname{\mathcal{T}}.$ We then have the following result, proved in
[33].

######  Proposition 1.

Under Assumption 1, the value function $v(t,\cdot)$ belongs to
$\operatorname{\mathcal{L}}_{a}$ for each $t\in\operatorname{\mathcal{T}}$. In
particular, $v^{\diamond}(t,\cdot)$ is continuous and the stopping region
$\operatorname{\mathcal{S}}_{t}$ is a relatively closed subset of
$\operatorname{\mathcal{X}}$ .

Similar to [33], we make an important structural assumption which gives the
existence of a stopping boundary (or free boundary) delimiting the stopping
region in some adequate coordinate system.

######  Assumption 2.

There exist measurable functions

$\alpha\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{X}}\to\operatorname{\mathbb{R}}_{+},\quad\Xi\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{X}}\to\operatorname{\mathcal{E}}\mathrel{\mathop{\ordinarycolon}}=\Xi(\operatorname{\mathcal{X}}),\quad
f^{\diamond}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{T}}\times\
\operatorname{\mathcal{E}}\to[0,\infty],$

such that
$A=(\Xi,\alpha)\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{X}}\to\operatorname{\mathcal{E}}\times\operatorname{\mathbb{R}}_{+}$
is a homeomorphism and for every $t\in\operatorname{\mathcal{T}}$,

$\operatorname{\mathcal{S}}_{t}=\\{x\in\operatorname{\mathcal{X}}\
\mathrel{\mathop{\ordinarycolon}}\
\eta\left(f^{\diamond}(t,\Xi(x))-\alpha(x)\right)\leq
0\\},\qquad\eta\in\\{-1,1\\}.$ (10)

Under the above assumption, the optimal stopping boundary is thus given by

$\displaystyle f^{\diamond}(t,\xi)=\begin{cases}\inf\\{a\geq 0\
\mathrel{\mathop{\ordinarycolon}}\
A^{-1}(\xi,a)\in\operatorname{\mathcal{S}}_{t}\\},&\eta=1,\\\\[5.0pt]
\sup\\{a\geq 0\ \mathrel{\mathop{\ordinarycolon}}\
A^{-1}(\xi,a)\in\operatorname{\mathcal{S}}_{t}\\},&\eta=-1.\end{cases}$ (11)

Figure 1: Stopping region of a max-call option on two symmetric assets. The
upper connected component becomes an epigraph through $(\alpha,\Xi)$. Figures
adapted from [33].

(a) Stopping region in $\operatorname{\mathcal{X}}$

(b) Stopping region through $(\alpha,\Xi)$

Consequently, the time $t$ section of the stopping region is the epigraph of
$f^{\diamond}(t,\cdot)$ when $\eta=1$; see Fig. 1(b). Similarly,
$\operatorname{\mathcal{S}}_{t}$ is the hypograph of $f^{\diamond}(t,\cdot)$
when $\eta=-1$. The next proposition sheds light on the regularity of
$f^{\diamond}$. We refer the interested reader to the books [39, 31] for a
thorough treatment of free boundary regularity in one-phase and obstacle
problems, respectively. In the context of American options, see [7, 26].

######  Proposition 2.

For every $t\in\operatorname{\mathcal{T}}$, the map $\xi\mapsto
f^{\diamond}(t,\xi)$ is lower (respectively upper) semicontinuous when
$\eta=1$ (resp. $\eta=-1$). Moreover, if the payoff function $\varphi$ is
convex in $x$, then $f^{\diamond}$ is locally Lipschitz continuous.

###### Proof.

Fix $t\in\operatorname{\mathcal{T}}$ and prove the result for $\eta=1$.
Similar arguments can be applied for the case $\eta=-1$. Let
$\xi\in\operatorname{\mathcal{E}}$ and
$(\xi_{n})\subset\operatorname{\mathcal{E}}$ such that $\xi_{n}\to\xi$ and fix
$\varepsilon>0$. In light of $\eqref{eq:stopBdry}$, there exists
$a_{n}\in\operatorname{\mathbb{R}}_{+}$ such that
$x_{n}\mathrel{\mathop{\ordinarycolon}}=A^{-1}(\xi_{n},a_{n})\in\operatorname{\mathcal{S}}_{t}$
and $f^{\diamond}(t,\xi_{n})\geq a_{n}-\varepsilon$ for every
$n\in\operatorname{\mathbb{N}}$. Defining $a=\liminf_{n\to\infty}a_{n},$ then

$\liminf_{n\to\infty}f^{\diamond}(t,\xi_{n})\geq a-\varepsilon.$

Next, the continuity of $A^{-1}$ yields
$\lim_{k\to\infty}x_{n_{k}}=x\mathrel{\mathop{\ordinarycolon}}=A^{-1}(\xi,a)$
for a subsequence $(n_{k})$ such that $a_{n_{k}}\to a$. As
$(x_{n_{k}})\subset\operatorname{\mathcal{S}}_{t}$ and
$\operatorname{\mathcal{S}}_{t}$ is closed (see Proposition 1), then
$x\in\operatorname{\mathcal{S}}_{t}$. Therefore $a\geq f^{\diamond}(t,\xi)$,
giving $\liminf_{n\to\infty}f^{\diamond}(t,\xi_{n})\geq
f^{\diamond}(t,\xi)-\varepsilon.$ Letting $\varepsilon\to 0$ indeed shows the
lower semicontinuity of $\xi\mapsto f^{\diamond}(t,\xi)$.

For the second assertion, it is shown in [7] that the convexity of
$x\mapsto\varphi(t,x)$ carries over to $\operatorname{\mathcal{S}}_{t}$. In
particular, $f^{\diamond}$ is locally Lipschitz continuous. ∎

Let us assume throughout that $\eta=1$ to simplify the exposition. The case
$\eta=-1$ is analogous. The optimal stopping boundary $f^{\diamond}(t,\xi)$ is
therefore lower semicontinuous (l.s.c.) in $\xi$ for all
$t\in\operatorname{\mathcal{T}}$. Let us introduce
$\operatorname{\mathfrak{F}}=\\{f\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{T}}\times\
\operatorname{\mathcal{E}}\to[0,\infty],\;f(t,\cdot)\text{ l.s.c. }\forall t\
\in\operatorname{\mathcal{T}}\\}$ and associate to each
$f\in\operatorname{\mathfrak{F}}$ the stopping region

$\operatorname{\mathfrak{S}}_{t}(f)=\\{x\in\operatorname{\mathcal{X}}\
\mathrel{\mathop{\ordinarycolon}}\ \alpha(x)\geq f(t,\Xi(x))\\}.$ (12)

Moreover, we define the hitting time
$\tau_{f}=\inf\\{t\in\operatorname{\mathcal{T}}\mathrel{\mathop{\ordinarycolon}}X_{t}\in\operatorname{\mathfrak{S}}_{t}(f)\\}$.
Assumption 2 implies in particular that it is enough to consider hitting times
of boundaries to achieve the optimal value in (6). Precisely, we have that

$v^{\diamond}=\sup_{f\in\operatorname{\mathfrak{F}}}v(\tau_{f})=v(\tau_{f^{\diamond}}).$

See also Lemma 2.3 in [33]. Problem (6) is therefore equivalent to finding
$f\in\operatorname{\mathfrak{F}}$ such that
$\operatorname{\mathfrak{S}}_{t}(f)=\operatorname{\mathcal{S}}_{t}$ for all
$t\in\operatorname{\mathcal{T}}$.

###  2.1 Financial examples

Suppose that $X=(X^{1},\ldots,X^{m})$ describes the price process of $m\geq 2$
financial assets. The state space is therefore
$\operatorname{\mathcal{X}}=\operatorname{\mathbb{R}}^{m}_{+}$. Moreover,
consider payoff functions of the form

$\varphi(t,x)=e^{-rt}(\alpha(x)-\kappa)^{+},$ (13)

where $r\geq 0$ is a constant interest rate, $\kappa>0$ is the strike of the
option, and
$\alpha\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{X}}\to\operatorname{\mathbb{R}}$
is a statistic of the underlying prices. For this class of contracts,
Assumption 2 holds with $\eta=1$, and the homeomorphism $A=(\Xi,\alpha)$,
$\Xi(x)=\frac{x}{\alpha(x)}$. See Theorem 4.1 in [33] for a complete
statement.

######  Example 1.

Consider a max-call option, which is obtained by choosing
$\alpha(x)=\max_{i=1,...,m}x^{i}$, is convex. Hence Proposition 2 tells us
that the stopping boundary is locally Lipschitz continuous. Moreover, the
latent space $\operatorname{\mathcal{E}}=\Xi(\operatorname{\mathcal{X}})$ is
given by
$\\{\xi\in[0,1]^{m}\mathrel{\mathop{\ordinarycolon}}\max_{i=1,...,m}\xi_{i}=1\\}$.
In particular, it is compact. Interestingly, the stopping region consists of
$m$ connected components, as can be seen in Fig. 1(a) in the case of $m=2$
assets. The right panel (Fig. 1(b)) displays the image through $(\alpha,\Xi)$
of the upper connected component of the stopping region in Fig. 1(a).

Max-call options are popular benchmark examples for numerical methods
addressing high-dimensional optimal stopping problems; see [33, Section 8] and
the references therein. Indeed, the maximum function in the payoff makes the
problem truly high-dimensional in the sense that the process $\alpha(X)$
cannot be replaced by a one-dimensional diffusion with identical law.

######  Example 2.

If $\alpha(x)=\min_{i=1,...,m}x^{i}$, then the associated contract is a so-
called min-call option. Note that the payoff $\varphi(t,x)$ is not convex in
$x$, leading to pathological properties of the stopping region as studied by
Detemple et al. [9] in the case $m=2$. In particular,
$\partial\operatorname{\mathcal{S}}_{t}$ contains a nonempty singular set,
namely an isolated segment on the diagonal $\\{x_{1}=x_{2}\\}$. Setting
$\Xi(x)=\frac{x_{1}\vee x_{2}}{x_{1}\wedge
x_{2}}\in\operatorname{\mathcal{E}}=[1,\infty)$ for simplicity, this implies
that $\operatorname{\mathcal{E}}\ni\xi\to f^{\diamond}(t,\xi)$ is
discontinuous at $\xi=1$. The lower semicontinuity of the stopping boundary
may therefore be strict, even for seemingly simple examples.

##  3 Relaxed stopping rules

To find an optimal boundary with stochastic gradient techniques as in Section
5, the optimal stopping problem needs to be relaxed. Indeed, the map
$f\mapsto\tau_{f}(\omega)$ is piecewise constant for almost every
$\omega\in\Omega$ as the hitting time of $f$, taking values in the discrete
set $\operatorname{\mathcal{T}}$, is typically unchanged when varying the
stopping boundary ever so slightly. This leads in turn to severe vanishing
gradient problems. One workaround is to introduce a fuzzy region

$\operatorname{\mathfrak{B}}_{t}^{\varepsilon}(f)\mathrel{\mathop{\ordinarycolon}}=\operatorname{\mathfrak{S}}_{t}(f-\varepsilon)\setminus\operatorname{\mathfrak{S}}_{t}(f)=\\{x\in\operatorname{\mathcal{X}}\
\mathrel{\mathop{\ordinarycolon}}\
d(t,x;f)\in(0,\varepsilon]\\},\quad\varepsilon>0,$ (14)

where
$d\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{T}}\times\operatorname{\mathcal{X}}\times\operatorname{\mathfrak{F}}\to[0,\infty]$
gives the distance from $x$ to $\operatorname{\mathfrak{S}}_{t}(f)$ in the
coordinate system induced by $(\alpha,\Xi)$. That is

$d(t,x;f)=(f(t,\Xi(x))-\alpha(x))^{+},\quad(t,x,f)\in\operatorname{\mathcal{T}}\times\operatorname{\mathcal{X}}\times\operatorname{\mathfrak{F}}.$
(15)

See the left panel of Fig. 2 for an illustration. The fuzzy region is
introduced to relax the stopping rule, as we now explain. Let
$\operatorname{\mathcal{P}}(\operatorname{\mathcal{T}})$ be the set of
probability measures on $\operatorname{\mathcal{T}}$ and
$\underline{\vartheta}(\operatorname{\mathcal{T}})$ denotes the set of
$\operatorname{\mathcal{P}}(\operatorname{\mathcal{T}})-$valued random
variables

$\mu(\omega)=\sum_{t\in\operatorname{\mathcal{T}}}P_{t}(\omega)\delta_{t},\quad\omega\in\Omega,\qquad(\delta_{t}\mathrel{\mathop{\ordinarycolon}}\text{Dirac
measure at $t$})$

such that the process $(P_{t})_{t\in\operatorname{\mathcal{T}}}$ is
$\operatorname{\mathbb{F}}$-adapted. In this context,
$\underline{\vartheta}(\operatorname{\mathcal{T}})$ is the set of relaxed (or
randomized) $\operatorname{\mathbb{F}}$-stopping times taking values in
$\operatorname{\mathcal{T}}$. If we identify
$\tau\in\vartheta(\operatorname{\mathcal{T}})$ with the Dirac measure
$\delta_{\tau}$, then $\underline{\vartheta}(\operatorname{\mathcal{T}})$
contains all $\operatorname{\mathbb{F}}$-stopping times. The procedure is
effectively a convexification of the stopping times as every relaxed stopping
rule is a random convex combination of the Dirac measures
$(\delta_{t})_{t\in\operatorname{\mathcal{T}}}$. We can now define the value
functional

$v(\mu)=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\int_{\operatorname{\mathcal{T}}}\varphi(t,X_{t})\mu(dt)\right],\quad\mu\in\underline{\vartheta}(\operatorname{\mathcal{T}}).$
(16)

The _relaxed optimal stopping problem_ amounts to maximizing $v(\mu)$ over all
$\mu\in\underline{\vartheta}(\operatorname{\mathcal{T}})$. Importantly, the
relaxed problem and classical one (6) achieve the same optimal value as in as
we shall see in the next proposition. Similar results in continuous time can
be found in [16], [18] and [23, Lemma 1.5.2.].

######  Proposition 3.

Let $v^{\diamond}$ be the value of the classical optimal stopping problem (6).
Then,

$v^{\diamond}=\sup_{\mu\in\underline{\vartheta}(\operatorname{\mathcal{T}})}v(\mu).$
(17)

Moreover, any maximizer $\mu^{\diamond}$ of $\mu\mapsto v(\mu)$ is a stopping
time, i.e. $\mu^{\diamond}=\delta_{\tau^{\diamond}}$ for some
$\tau^{\diamond}\in\vartheta(\operatorname{\mathcal{T}})$.

###### Proof.

See Appendix A. ∎

We now explain how a relaxed stopping rule can be constructed from a boundary
$f\in\operatorname{\mathfrak{F}}$. Let
$P^{\varepsilon}_{f}=(P_{f,t}^{\varepsilon})_{t\in\operatorname{\mathcal{T}}}$
be an $[0,1]-$valued stochastic process representing stopping probabilities.
The latter are induced by the distance function defined in (15). Specifically,
writing $\operatorname{\mathcal{T}}_{t-}=\operatorname{\mathcal{T}}\cap\
[0,t)\ $ ($\operatorname{\mathcal{T}}_{0-}=\varnothing$), then
$P^{\varepsilon}_{f}$ evolves according to

$\displaystyle
P^{\varepsilon}_{f,t}=\underbrace{p_{f,t}^{\varepsilon}}_{\text{stop at $t$}}\
\
\underbrace{\Big{(}1-\sum_{s\in\operatorname{\mathcal{T}}_{t-}}P_{f,s}^{\varepsilon}\Big{)}}_{\text{continue
until $t$}},\qquad p_{f,t}^{\varepsilon}=(\chi^{\varepsilon}\circ
d)(t,X_{t};f),$ (18)

with the relaxed phase indicator function
$\chi^{\varepsilon}(\delta)=(1-\delta/\varepsilon)^{+}\wedge 1$. In other
words, $\chi^{\varepsilon}$ is equal to one in the stopping region
$\operatorname{\mathfrak{S}}_{t}(f)$, it is linearly decreasing in the fuzzy
region $\operatorname{\mathfrak{B}}_{t}^{\varepsilon}(f)$, and is zero
otherwise. The right panel of Fig. 2 displays the composition
$\chi^{\varepsilon}\circ d$ in the space
$\Xi(\operatorname{\mathcal{X}})\times\alpha(\operatorname{\mathcal{X}})$. We
can finally define the relaxed stopping rule associated to $f$ as

$\mu^{\varepsilon}_{f}(\omega)=\sum_{t\in\operatorname{\mathcal{T}}}P^{\varepsilon}_{f,t}(\omega)\
\delta_{t}.$ (19)

Figure 2: Left panel: fuzzy region (purple) separating the interior of the
continuation region (blue) from the stopping region (red). Right panel:
Illustration of $\chi^{\varepsilon}\circ d(\cdot;f)$ in the coordinates
$(\Xi,\alpha)$, where $d(\cdot;f)$ is the distance function to
$\operatorname{\mathfrak{S}}_{t}(f)$ and
$\chi^{\varepsilon}(\delta)=(1-\delta/\varepsilon)^{+}\wedge 1$.

From (18), it is easily checked that
$\mu_{f}^{\varepsilon}(\operatorname{\mathcal{T}})=1$, thus
$\mu_{f}^{\varepsilon}$ is a random probability measure. Moreover,
$P^{\varepsilon}_{f,t}$ clearly depends only on $\\{X_{s}\
\mathrel{\mathop{\ordinarycolon}}\ s\in\operatorname{\mathcal{T}}\cap\
[0,t]\\}$, from which the $\operatorname{\mathbb{F}}-$adaptedness of
$P^{\varepsilon}_{f}$ follows. Hence is a relaxed
$\operatorname{\mathbb{F}}-$stopping time. In [33, Lemma B.$1$], it is argued
that

$v(\mu_{f}^{\varepsilon})\longrightarrow v(\tau_{f})\quad\text{as }\
\varepsilon\downarrow 0.$ (20)

In words, the value of the relaxed stopping rule $\mu_{f}^{\varepsilon}$
converges to the expected reward from the hitting time of $f$ when the fuzzy
parameter $\varepsilon$ decreases to zero. A refined statement of (20) will be
given in Corollary 1. The latter turns out to be a byproduct of the continuity
of the map $f\mapsto v(\tau_{f})$, which is the focus of the next section.
Towards our main results (Theorems 4.1 and 4.2), we prove a continuity lemma
for relaxed stopping rules with respect to the total variation distance of
probability measures,

$\textnormal{TV}(\mu,\mu^{\prime})=\frac{1}{2}\sum_{t\in\operatorname{\mathcal{T}}}|\mu(\\{t\\})-\mu^{\prime}(\\{t\\})|\in[0,1],\qquad\mu,\mu^{\prime}\in\operatorname{\mathcal{P}}(\operatorname{\mathcal{T}}).$

If $\mu,\mu^{\prime}\in\underline{\vartheta}(\operatorname{\mathcal{T}})$,
then $\textnormal{TV}(\mu,\mu^{\prime})$ is understood as the random variable
$\omega\mapsto\textnormal{TV}(\mu(\omega),\mu^{\prime}(\omega))$.

######  Lemma 1.

Under Assumption 1, there exists $C\in(0,\infty)$ depending on $\varphi$ such
that

$|v(\mu)-v(\mu^{\prime})|\leq C\
\lVert\textnormal{TV}(\mu,\mu^{\prime})\rVert_{L^{2}(\operatorname{\mathbb{Q}})},\quad\forall\
\mu,\mu^{\prime}\in\underline{\vartheta}(\operatorname{\mathcal{T}}).$ (21)

In particular, $|v(\tau)-v(\tau^{\prime})|\leq
C\operatorname{\mathbb{Q}}(\tau\neq\tau^{\prime})^{1/2}$ for any stopping
times $\tau,\tau^{\prime}\in\vartheta(\operatorname{\mathcal{T}})$.

###### Proof.

Define $\Phi=\sum_{t\in\operatorname{\mathcal{T}}}|\varphi(t,X_{t})|$ and
recall from Assumption 1 that
$\varphi(t,\cdot)\in\operatorname{\mathcal{L}}_{a}$ for some $a>0$. Since
$\phi\in\operatorname{\mathcal{L}}_{a}$ implies
$\phi^{2}\in\operatorname{\mathcal{L}}_{2a}$, we obtain for some constant
$C\in(0,\infty)$ that

$\operatorname{\mathbb{E}}[\Phi^{2}]\leq|\operatorname{\mathcal{T}}|\sum_{t\in\operatorname{\mathcal{T}}}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\varphi(t,X_{t})^{2}]=|\operatorname{\mathcal{T}}|\sum_{t\in\operatorname{\mathcal{T}}}\underbrace{K_{0,t}\varphi^{2}(t,x_{0})}_{\in\
\operatorname{\mathcal{L}}_{2a}}\leq
C\sum_{t\in\operatorname{\mathcal{T}}}[1+|x_{0}|^{2a}]<\infty.$

Thus $\Phi\in L^{2}(\operatorname{\mathbb{Q}})$. Next, observe that

$\displaystyle|v(\mu)-v(\mu^{\prime})|\leq\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\int_{0}^{T}|\varphi(t,X_{t})|\
|\mu-\mu^{\prime}|(dt)\right]\leq
2\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\Phi\
\text{TV}(\mu,\mu^{\prime})\right].$

As $\text{TV}(\mu,\mu^{\prime})\leq 1\in L^{2}(\operatorname{\mathbb{Q}})$,
the result follows from the Cauchy-Schwarz inequality and setting
$C=2\lVert\Phi\rVert_{L^{2}(\operatorname{\mathbb{Q}})}$. For the second
claim, it suffices to observe that
$\text{TV}(\delta_{\tau},\delta_{\tau^{\prime}})=\mathds{1}_{\\{\tau\neq\tau^{\prime}\\}}$.
∎

######  Remark 3.1.

If we define
$\overline{\textnormal{TV}}_{2}(\mu,\mu^{\prime})\mathrel{\mathop{\ordinarycolon}}=\lVert\textnormal{TV}(\mu,\mu^{\prime})\lVert_{L^{2}(\operatorname{\mathbb{Q}})}$
for relaxed stopping rules
$\mu,\mu^{\prime}\in\underline{\vartheta}(\operatorname{\mathcal{T}})$, then
Lemma 1 states that $\mu\mapsto v(\mu)$ is Lipschitz continuous with respect
to $\overline{\textnormal{TV}}_{2}$.

##  4 Continuity of the value functional

First, we need a distributional assumption on the process
$\alpha(X)=(\alpha(X_{t}))_{t\in\operatorname{\mathcal{T}}}$.

######  Assumption 3.

For every $t\in\operatorname{\mathcal{T}}$,
$\operatorname{\mathbb{Q}}(\alpha(X_{t})\in\cdot)$ is absolutely continuous
with respect to the Lebesgue measure on $\operatorname{\mathbb{R}}_{+}$. In
particular, the cumulative distribution function
$a\mapsto\operatorname{\mathbb{Q}}(\alpha(X_{t})\in[0,a])$ admits a modulus of
continuity $\rho_{t}\mathrel{\mathop{\ordinarycolon}}[0,\infty]\to[0,1]$.

We are now set to state our first continuity result. Throughout, we adopt the
notation $\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$
if $\operatorname{\mathcal{K}}$ is a compact subset of
$\operatorname{\mathcal{E}}$.

######  Theorem 4.1.

(Continuity of $\bm{f\mapsto v(\tau_{f})}$) Let $C\in(0,\infty)$ given in
Lemma 1 and fix $\delta>0$. Then under Assumptions 3, there exists a compact
set $\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$ such
that for all $\iota>0$ and $f,f^{\prime}\in\operatorname{\mathfrak{F}}$,

$\lVert
f-f^{\prime}\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}})}\leq\iota\
\Longrightarrow\ |v(\tau_{f})-v(\tau_{f^{\prime}})|\leq
C\sqrt{\rho(\iota)+\delta},$ (22)

where $\rho_{t}$ is the modulus of continuity of
$a\mapsto\operatorname{\mathbb{Q}}(\alpha(X_{t})\in[0,a])$ and
$\rho(\iota)=\sum_{t\in\operatorname{\mathcal{T}}}\rho_{t}(\iota)$.

Figure 3: Illustration of the symmetric difference between the stopping
regions $\operatorname{\mathfrak{S}}_{t}(f)$ and
$\operatorname{\mathfrak{S}}_{t}(f^{\prime})$ (purple region) contained in the
interval $[f\wedge f^{\prime},f\wedge f^{\prime}+\iota)$ (orange region); see
proof of Theorem 4.1. The stopping decisions with respect to $f$ and
$f^{\prime}$ coincide in the red (stop) and blue (continue) region.

###### Proof.

Step ${1}$. Since $\operatorname{\mathcal{E}}=\Xi(\operatorname{\mathcal{X}})$
is locally compact (being the continuous image of a Euclidean space), every
finite Borel measure on it is tight. Choosing
$\operatorname{\mathcal{B}}(\operatorname{\mathcal{E}})\ni
B\mapsto\operatorname{\mathbb{Q}}(\
\bigcap_{t\in\operatorname{\mathcal{T}}}\\{\ \Xi(X_{t})\in B\\})$, there
exists $\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$
such that

$\operatorname{\mathbb{Q}}(\Omega_{\operatorname{\mathcal{K}}})\geq
1-\delta,\quad\Omega_{\operatorname{\mathcal{K}}}\mathrel{\mathop{\ordinarycolon}}=\\{\omega\in\Omega\
\mathrel{\mathop{\ordinarycolon}}\
\Xi(X_{t}(\omega))\in\operatorname{\mathcal{K}}\;\ \forall
t\in\operatorname{\mathcal{T}}\\}.$ (23)

Step ${2}$. Fix $\iota>0$ and $f,f^{\prime}\in\operatorname{\mathfrak{F}}$
such that $\lVert
f-f^{\prime}\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}})}\leq\iota$.
Using Lemma 1, then

$|v(\tau_{f})-v(\tau_{f^{\prime}})|\leq
C\operatorname{\mathbb{Q}}(\tau_{f}\neq\tau_{f^{\prime}})^{1/2},$

so it remains to find the appropriate upper bound for
$\operatorname{\mathbb{Q}}(\tau_{f}\neq\tau_{f^{\prime}})$. First, note that
the event $\\{\tau_{f}\neq\tau_{f^{\prime}}\\}$ is contained in

$\tilde{\Omega}=\\{\omega\in\Omega\mathrel{\mathop{\ordinarycolon}}X_{t}(\omega)\in\operatorname{\mathfrak{S}}_{t}(f)\triangle\operatorname{\mathfrak{S}}_{t}(f^{\prime})\
\text{ for some }t\in\operatorname{\mathcal{T}}\\}.$

Indeed, for $\omega\notin\tilde{\Omega}$, the sets
$\\{t\in\operatorname{\mathcal{T}}\mathrel{\mathop{\ordinarycolon}}X_{t}(\omega)\in\operatorname{\mathfrak{S}}_{t}(f)\\}$
and
$\\{t\in\operatorname{\mathcal{T}}\mathrel{\mathop{\ordinarycolon}}X_{t}(\omega)\in\operatorname{\mathfrak{S}}_{t}(f^{\prime})\\}$
are identical. Hence $\tau_{f}(\omega)=\tau_{f^{\prime}}(\omega)$, which shows
that $\Omega\setminus\tilde{\Omega}\subseteq\\{\tau_{f}=\tau_{f^{\prime}}\\}$.
Next, notice that

$\displaystyle x$
$\displaystyle\in\operatorname{\mathfrak{S}}_{t}(f)\triangle\operatorname{\mathfrak{S}}_{t}(f^{\prime})\,\Longleftrightarrow\alpha(x)\in
I_{f,f^{\prime}}(t,\Xi(x)),\quad x\in\operatorname{\mathcal{X}},$

with the interval
$I_{f,f^{\prime}}(t,\xi)\mathrel{\mathop{\ordinarycolon}}=[(f\wedge
f^{\prime})(t,\xi),(f\vee f^{\prime})(t,\xi))$. Since $\lVert
f-f^{\prime}\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}})}\leq\iota$,
then $I_{f,f^{\prime}}(t,\xi)$ has length at most $\iota$ for every
$(t,\xi)\in\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}}$. See
Fig. 3 for an illustration. Using Assumption 3 and recalling that $\rho_{t}$
is the modulus of continuity of
$a\mapsto\operatorname{\mathbb{Q}}(\alpha(X_{t})\in[0,a])$, then

$\operatorname{\mathbb{Q}}(\alpha(X_{t})\in[a,a+\iota))\leq\rho_{t}(\iota),\quad\forall
a\in\operatorname{\mathbb{R}}_{+}.$ (24)

We conclude from (24) and the union bound that

$\displaystyle\operatorname{\mathbb{Q}}(\tilde{\Omega}\cap\Omega_{\operatorname{\mathcal{K}}})\leq\sum_{t\in\operatorname{\mathcal{T}}}\operatorname{\mathbb{Q}}(\\{\alpha(X_{t})\in\operatorname{\mathcal{I}}(t,\Xi(X_{t}))\\}\cap\Omega_{\operatorname{\mathcal{K}}})\leq\sum_{t\in\operatorname{\mathcal{T}}}\rho_{t}(\iota)$
$\displaystyle=\rho(\iota).$

Combining Step 1 with the above observations, we finally obtain

$\operatorname{\mathbb{Q}}(\tau_{f}\neq\tau_{f^{\prime}})\leq\operatorname{\mathbb{Q}}(\\{\tau_{f}\neq\tau_{f^{\prime}}\\}\cap\Omega_{\operatorname{\mathcal{K}}})+\operatorname{\mathbb{Q}}(\Omega\setminus\Omega_{\operatorname{\mathcal{K}}})\leq\operatorname{\mathbb{Q}}(\tilde{\Omega}\cap\Omega_{\operatorname{\mathcal{K}}})+\delta\leq\rho(\iota)+\delta.$

∎

From Theorem 4.1 follows the convergence of the value from relaxed to strict
stopping policies anticipated in (20).

######  Corollary 1.

Under Assumptions 1 and 3, then for all $\varepsilon>0$ and
$f\in\operatorname{\mathfrak{F}}$,

$|v(\mu_{f}^{\varepsilon})-v(\tau_{f})|\leq C\sqrt{\rho(\varepsilon)},$ (25)

with $\rho,C$ given in Theorems 4.1 and 1, respectively.

###### Proof.

First, Lemma 1 gives $|v(\mu_{f}^{\varepsilon})-v(\tau_{f})|\leq
C\lVert\textnormal{TV}(\mu_{f}^{\varepsilon},\delta_{\tau_{f}})\lVert_{L^{2}(\operatorname{\mathbb{Q}})}.$
From the definition of $\mu_{f}^{\varepsilon}$ in (19) and the fact that
$\delta_{\tau_{f}}$ is a (random) Dirac measure, the total variation distance
simplifies to

$\displaystyle\text{TV}(\mu_{f}^{\varepsilon},\delta_{\tau_{f}})$
$\displaystyle=\frac{1}{2}\Big{[}1-P_{f,\tau_{f}}^{\varepsilon}+\sum_{t\in\operatorname{\mathcal{T}}\setminus\\{\tau_{f}\\}}P_{f,t}^{\varepsilon}\Big{]}=1-P_{f,\tau_{f}}^{\varepsilon}\leq\mathds{1}_{\\{\mu_{f}^{\varepsilon}\
\neq\ \delta_{\tau_{f}}\\}}\quad\operatorname{\mathbb{Q}}-\text{a.s.}$ (26)

We now claim that
$\\{\tau_{f-\varepsilon}=\tau_{f}\\}\subseteq\\{\mu_{f}^{\varepsilon}=\delta_{\tau_{f}}\\}$.
Indeed, if $\tau_{f-\varepsilon}(\omega)=\tau_{f}(\omega)$, the discrete path
$(X_{t}(\omega))_{t\in\operatorname{\mathcal{T}}}$ never enters the fuzzy
region
$\operatorname{\mathfrak{B}}_{t}^{\varepsilon}(f)=\operatorname{\mathfrak{S}}_{t}(f-\varepsilon)\setminus\operatorname{\mathfrak{S}}_{t}(f)$
prior to $\tau_{f}(\omega)$. Thus the relaxed stopping rule
$\mu_{f}^{\varepsilon}(\omega)$ coincide with $\tau_{f}(\omega)$.
Consequently,

$\lVert\textnormal{TV}(\mu_{f}^{\varepsilon},\delta_{\tau_{f}})\lVert_{L^{2}(\operatorname{\mathbb{Q}})}^{2}\leq\operatorname{\mathbb{Q}}(\mu_{f}^{\varepsilon}\
\neq\
\delta_{\tau_{f}})\leq\operatorname{\mathbb{Q}}(\tau_{f-\varepsilon}\neq\tau_{f}).$

We can therefore follow Step 2 in the proof of Theorem 4.1 with
$f^{\prime}=f-\varepsilon$, $\iota=\varepsilon$ to conclude that
$\operatorname{\mathbb{Q}}(\tau_{f-\varepsilon}\neq\tau_{f})\leq\rho(\varepsilon)$.
As $\lVert
f-f^{\prime}\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{E}})}=\varepsilon$,
no localization is needed here. ∎

###  4.1 Relaxed $L^{\infty}$ distance and second continuity result

The continuity result in Theorem 4.1 only applies when boundaries are close in
the $L^{\infty}$ sense. Although reasonable for continuous boundaries, this
condition is typically too restrictive for semicontinuous boundaries. We
therefore introduce a weaker topology.

######  Definition 1.

Given a compact set
$\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$, the
relaxed $L^{\infty}$ distance (in $\operatorname{\mathcal{K}}$) is given by

${d}_{\operatorname{\mathcal{K}}}(h,h^{\prime})=\sup_{r>0}\
\inf_{\psi\in\Psi_{r}^{\operatorname{\mathcal{K}}}}\ \lVert
h\circ\psi-h^{\prime}\circ\psi\rVert_{L^{\infty}(\operatorname{\mathcal{K}})},\quad
h,h^{\prime}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{E}}\to\operatorname{\mathbb{R}},$
(27)

with the ball
$\Psi_{r}^{\operatorname{\mathcal{K}}}=\\{\psi\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{K}}\to\operatorname{\mathcal{K}}\
\mathrel{\mathop{\ordinarycolon}}\
\lVert\psi-\text{id}\rVert_{L^{\infty}(\operatorname{\mathcal{K}})}\leq r\\}$
and identity map $\text{id}(\xi)=\xi,\ \xi\in\operatorname{\mathcal{K}}$. For
$f,f^{\prime}\in\operatorname{\mathfrak{F}}$, we define similarly

$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f,f^{\prime})=\max_{t\in\operatorname{\mathcal{T}}}\
{d}_{\operatorname{\mathcal{K}}}(f(t,\cdot),f^{\prime}(t,\cdot)).$ (28)

It is immediate that $d_{\operatorname{\mathcal{K}}}$ is dominated by the
$L^{\infty}$ metric. Strictly speaking, ${d}_{\operatorname{\mathcal{K}}}$ is
not a distance, as neither the triangular inequality nor the positivity axiom
is satisfied. This is nevertheless a natural notion for semicontinuous
functions and closely related to the definition of inf convolutions discussed
in Section 5.2. Importantly, we will see in Theorem 4.2 that the space-time
relaxed $L^{\infty}$ distance
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}$ yields a
continuity result, namely the analogue of Theorem 4.1 for l.s.c. boundaries.

The transformations
$\psi\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{K}}\to\operatorname{\mathcal{K}}$
may echo the reparametrization maps appearing in the Skorokhod $J1$ distance.
One important difference is that unlike the Skorokhod metric, $\psi$ may not
be invertible as we shall see in the proof of Lemma 2. We finally stress that
relaxed $L^{\infty}$ distance may be defined for general subsets, i.e. non
compact. For our purposes, however, $d_{\operatorname{\mathcal{K}}}$ and
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}$ will be
exclusively used for compact sets, whence the assumption.

We can now adapt Theorem 4.1 to obtain a continuity result of $f\mapsto
v(\tau_{f})$ with respect to the relaxed $L^{\infty}$ distance.

######  Theorem 4.2.

(Continuity of $\bm{f\mapsto v(\tau_{f})}$) Let $C\in(0,\infty)$ given in
Lemma 1 and fix $\delta>0$. Then under Assumptions 3, there exists
$\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$ such that
for all $\iota>0$ and $f,f^{\prime}\in\operatorname{\mathfrak{F}}$,

$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f,f^{\prime})\leq\iota\
\Longrightarrow\ |v(\tau_{f})-v(\tau_{f^{\prime}})|\leq
C\sqrt{\rho(\iota)+\delta},$

where $(\rho_{t})_{t\in\operatorname{\mathcal{T}}}$ are the moduli of
continuity of
$(a\mapsto\operatorname{\mathbb{Q}}(\alpha(X_{t})\in[0,a]))_{t\in\operatorname{\mathcal{T}}}$
and $\rho(\iota)=\sum_{t\in\operatorname{\mathcal{T}}}\rho_{t}(\iota)$.

###### Proof.

Let $\delta^{\prime}$ be arbitrary. As in Step 1 in the proof of Theorem 4.1,
there exists
$\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$ such that

$\operatorname{\mathbb{Q}}(\Omega_{\operatorname{\mathcal{K}}})\geq
1-\delta^{\prime},\quad\Omega_{\operatorname{\mathcal{K}}}=\\{\omega\in\Omega\
\mathrel{\mathop{\ordinarycolon}}\
\Xi(X_{t}(\omega))\in\operatorname{\mathcal{K}}\;\ \forall
t\in\operatorname{\mathcal{T}}\\}.$

Step ${1}$. Let $f,f^{\prime}\in\operatorname{\mathfrak{F}}$ such that
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f,f^{\prime})\leq\iota$.
From the definition of
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}$, we can construct
a sequence
$\\{\psi_{k}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}}\to\operatorname{\mathcal{K}}\\}_{k\geq
1}$ such that $|f(t,\psi_{k}(t,\xi))-f^{\prime}(t,\psi_{k}(t,\xi))|\leq\iota$
for all $\xi\in\operatorname{\mathcal{K}}$ and
$\psi_{k}(t,\cdot)\in\Psi_{1/k}^{\operatorname{\mathcal{K}}}$ $\forall\
t\in\operatorname{\mathcal{T}}$. In other words,

$\displaystyle\lVert
f\circ\psi_{k}-f^{\prime}\circ\psi_{k}\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}})}\leq\iota,\quad\
k\geq 1,$ (29)
$\displaystyle\lVert\psi_{k}(t,\cdot)-\text{id}\rVert_{L^{\infty}(\operatorname{\mathcal{K}})}\leq
1/k,\qquad\forall\
t\in\operatorname{\mathcal{T}}.\qquad(\text{$\text{id}(\xi)=\xi$})$ (30)

Next, let
$\operatorname{\mathcal{X}}_{\operatorname{\mathcal{K}}}=\\{x\in\operatorname{\mathcal{X}}\
\mathrel{\mathop{\ordinarycolon}}\ \Xi(x)\in\operatorname{\mathcal{K}}\\}$ and
consider the transformations

$\phi_{k}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{T}}\times\operatorname{\mathcal{X}}_{\operatorname{\mathcal{K}}}\to\operatorname{\mathcal{X}}_{\operatorname{\mathcal{K}}},\quad\phi_{k}(t,x)=A^{-1}(\alpha(x),\psi_{k}(t,\Xi(x))),\quad
k\geq 1.$

In light of (30), observe that $\phi_{k}(t,x)\to A^{-1}(\alpha(x),\Xi(x))=x$
pointwise in $\operatorname{\mathcal{X}}_{\operatorname{\mathcal{K}}}$ since
$A^{-1}$ is continuous. Noting that
$\omega\in\Omega_{\operatorname{\mathcal{K}}}$ if and only if
$X_{t}(\omega)\in\operatorname{\mathcal{X}}_{\operatorname{\mathcal{K}}}$, the
bounded convergence theorem yields for every
$B\in\operatorname{\mathcal{B}}(\operatorname{\mathcal{X}})$,

$\lim_{k\to\infty}\operatorname{\mathbb{Q}}(\\{\phi_{k}(t,X_{t})\in
B\\}\cap\Omega_{\operatorname{\mathcal{K}}})=\operatorname{\mathbb{Q}}(\\{X_{t}\in
B\\}\cap\Omega_{\operatorname{\mathcal{K}}}),\quad\forall\
t\in\operatorname{\mathcal{T}}.$ (31)

Thus for fixed $\gamma>0$ and $t\in\operatorname{\mathcal{T}}$, there exists
$k\geq 1$ such that

$\operatorname{\mathbb{Q}}(\\{\phi_{k}(t,X_{t})\in
B\\}\cap\Omega_{\operatorname{\mathcal{K}}})\geq\operatorname{\mathbb{Q}}(\\{X_{t}\in
B\\}\cap\Omega_{\operatorname{\mathcal{K}}})-\gamma.$ (32)

Step ${2}$. Recall that

$\\{\tau_{f}\neq\tau_{f^{\prime}}\\}\subseteq\tilde{\Omega}=\\{\omega\in\Omega\mathrel{\mathop{\ordinarycolon}}X_{t}(\omega)\in\operatorname{\mathfrak{S}}_{t}(f)\triangle\operatorname{\mathfrak{S}}_{t}(f^{\prime})\
\text{ for some }t\in\operatorname{\mathcal{T}}\\}.$

Choosing
$B=\operatorname{\mathfrak{S}}_{t}(f)\triangle\operatorname{\mathfrak{S}}_{t}(f^{\prime})$
for each $t\in\operatorname{\mathcal{T}}$ and
$\gamma=\frac{\delta^{\prime}}{|\operatorname{\mathcal{T}}|}$ in (32), we have
for $k$ large enough that

$\displaystyle\operatorname{\mathbb{Q}}(\tilde{\Omega}\cap\Omega_{\operatorname{\mathcal{K}}})$
$\displaystyle\leq\delta^{\prime}+\sum_{t\in\operatorname{\mathcal{T}}}\operatorname{\mathbb{Q}}(\\{\phi_{k}(t,X_{t})\in\operatorname{\mathfrak{S}}_{t}(f)\triangle\operatorname{\mathfrak{S}}_{t}(f^{\prime})\\}\cap\Omega_{\operatorname{\mathcal{K}}}).$
(33)

Defining $I_{f,f^{\prime}}(t,\xi)=[(f\wedge f^{\prime})(t,\xi),(f\vee
f^{\prime})(t,\xi))$ as in the proof of Theorem 4.1, observe that

$\displaystyle\phi_{k}(t,x)$
$\displaystyle\in\operatorname{\mathfrak{S}}_{t}(f)\triangle\operatorname{\mathfrak{S}}_{t}(f^{\prime})\,\Longleftrightarrow\alpha(x)\in
I_{f,f^{\prime}}(t,\psi_{k}(t,\Xi(x))),\quad x\in\operatorname{\mathcal{X}}.$

Because of (29), $I_{f,f^{\prime}}(t,\psi_{k}(t,\xi))$ has length at most
$\iota$ for all
$(t,\xi)\in\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}}$.
Together with the absolute continuity of
$\operatorname{\mathbb{Q}}(\alpha(X_{t})\in\cdot)$, this yields

$\sum_{t\in\operatorname{\mathcal{T}}}\operatorname{\mathbb{Q}}(\\{\phi_{k}(t,X_{t})\in\operatorname{\mathfrak{S}}_{t}(f)\triangle\operatorname{\mathfrak{S}}_{t}(f^{\prime})\\}\cap\Omega_{\operatorname{\mathcal{K}}})\leq\sum_{t\in\operatorname{\mathcal{T}}}\rho_{t}(\iota)=\rho(\iota).$
(34)

From (33),(34), and the rationale in the proof of Theorem 4.1, we conclude
with $\delta^{\prime}=\frac{\delta}{2}$ that

$|v(\tau_{f})-v(\tau_{f^{\prime}})|\leq
C\operatorname{\mathbb{Q}}(\tau_{f}\neq\tau_{f^{\prime}})^{1/2}\leq
C\left[\operatorname{\mathbb{Q}}(\tilde{\Omega}\cap\Omega_{\operatorname{\mathcal{K}}})+\delta^{\prime}\right]^{1/2}\leq
C\sqrt{\rho(\iota)+\delta}.$

∎

##  5 Application to neural stopping boundaries

We apply the above results to study the convergence of the Neural Optimal
Stopping Boundary method [33] which is now outlined. Under a suitable
coordinate system $(\alpha,\Xi)$ as in Assumption 2, the algorithm looks for a
neural stopping boundary, namely a feedforward neural network
$g^{\theta}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{T}}\times\operatorname{\mathcal{E}}\to\operatorname{\mathbb{R}}_{+}$
in a (sufficiently large) hypothesis space $\operatorname{\mathfrak{N}}$ such
that

$v(\tau_{g^{\theta}})\approx\sup_{f\in\operatorname{\mathfrak{F}}}v(\tau_{f})\
=v^{\diamond}.$

Additional details on the architecture and size of the neural networks for
this problem are given in [33]. The parameter vector $\theta$ is thereafter
trained using stochastic gradient ascent. As explained at the beginning of
Section 3, the stopping rule needs to be relaxed to avoid vanishing gradient
issues. For some fixed fuzzy region width $\varepsilon>0$, we therefore aim to

$\text{maximize}\ \
v(\mu_{g}^{\varepsilon})=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\int_{\operatorname{\mathcal{T}}}\varphi(t,X_{t})\mu_{g}^{\varepsilon}(dt)\right]\qquad\text{over}\
g\in\operatorname{\mathfrak{N}},$ (35)

with the relaxed stopping rule $\mu_{g}^{\varepsilon}$ given in (19). The
vanishing gradient problem is henceforth resolved and owing to well-known
convergence results in stochastic optimization [34, 35], we can find for every
$\gamma>0$ an $\gamma-$maximizer
$g^{\varepsilon,\gamma}\in\operatorname{\mathfrak{N}}$, that is

$v(\mu^{\varepsilon}_{g^{\varepsilon,\gamma}})\geq\sup_{g\in\operatorname{\mathfrak{N}}}\
v(\mu^{\varepsilon}_{g})-\gamma.$ (36)

Finally, the value of the optimal stopping problem is computed using the
strict stopping rule $\tau_{g^{\varepsilon,\gamma}}$ instead of
$\mu^{\varepsilon}_{g^{\varepsilon,\gamma}}$. The method is summarized in
Algorithm 1. For further details, see Appendix C in [33]. The goal of this
section is to show that $v(\tau_{g^{\varepsilon,\gamma}})$ can be made
arbitrarily close to the optimal value $v^{\diamond}$ when
$\varepsilon,\gamma$ are small enough.

######  Remark 5.1.

When computing the value associated to the trained stopping rule, say
$\tau_{g}$, an additional inaccuracy comes naturally from the Monte Carlo
simulations; see step III in Algorithm 1. We nevertheless have a clear control
of this error. Indeed, it is classical that

$|v(\tau_{g})-\hat{v}_{J}(\tau_{g})|=\operatorname{\mathcal{O}}(J^{-1/2}),$

when the random reward $\varphi(\tau_{g},X_{\tau_{g}})$ is square integrable.
Under Assumption 3, this is indeed the case (see also the proof of Lemma 1).

######  Remark 5.2.

In [33], a two-sided fuzzy region is considered, namely
$\\{x\in\operatorname{\mathcal{X}}\ \mathrel{\mathop{\ordinarycolon}}\
|d(t,x;f)|<\varepsilon\\},$ where $d(t,\cdot;f)$ is the signed distance
function to $\partial\operatorname{\mathfrak{S}}_{t}(f)$. Although the two
representations are equivalent, one-sided fuzzy regions are more practical
towards a convergence analysis.

Algorithm 1 Neural Optimal Stopping Boundary Algorithm (summary)

Given: $\varphi=$ payoff, $\operatorname{\mathcal{T}}=$ exercise dates,
$\Theta=$ parameter set, $I=\\#$ training iterations, $\zeta=$ learning rate
process, $\varepsilon=$ fuzzy region width, $B=$ batch size, $J=\\#$
simulations for the value

  * I.

Initialize $\theta_{0}\in\Theta$

  * II.

For $i=0,\ldots,I-1$:

    1. 1.

Simulate
$X(\omega_{b})=\\{X_{t}(\omega_{b})\\}_{t\in\operatorname{\mathcal{T}}}$,
$\omega_{b}\in\Omega$, $b=1,\ldots,B$

    2. 2.

Compute
$\hat{v}_{B}(\mu^{\varepsilon}_{g^{\theta_{i}}})\mathrel{\mathop{\ordinarycolon}}=\frac{1}{B}\sum\limits_{b=1}^{B}\int_{\operatorname{\mathcal{T}}}\varphi(t,X_{t}(\omega_{b}))\
\mu^{\varepsilon}_{g^{\theta_{i}}}(\omega_{b})(dt)$

    3. 3.

Gradient step:
$\theta_{i+1}=\theta_{i}+\zeta_{i}\nabla_{\theta}\hat{v}_{B}(\mu^{\varepsilon}_{g^{\theta_{i}}})$

  * III.

Return $g=g^{\theta_{I}}$ and
$\hat{v}_{J}(\tau_{g})\mathrel{\mathop{\ordinarycolon}}=\frac{1}{J}\sum\limits_{j=1}^{J}\varphi(\tau_{g}(\omega_{j}),X_{\tau_{g}}(\omega_{j})),\
\omega_{j}\in\Omega$, $\ j=1,\ldots,J$

###  5.1 Convergence analysis: continuous boundaries

We first set out our strategy to prove convergence of the Neural Optimal
Stopping Boundary method when the optimal stopping boundary $f^{\diamond}$ is
continuous. First, the hypothesis space $\operatorname{\mathfrak{N}}$ (neural
networks in this case) must satisfy a universal approximation property; see
Assumption 4 below. Thus, there exists
$g^{\diamond}\in\operatorname{\mathfrak{N}}$ arbitrarily close to
$f^{\diamond}$ in compact sets. We can therefore invoke our first continuity
result (Theorem 4.1) to ensure that the values of the stopping strategies
$\tau_{f^{\diamond}}$, $\tau_{g^{\diamond}}$ are close as well. Next, we relax
the problem and use Corollary 1 to guarantee that
$v(\mu^{\varepsilon}_{g^{\diamond}})\approx v(\tau_{g^{\diamond}})$ for small
fuzzy parameter $\varepsilon$. We then train the neural boundary to obtain
$g^{\varepsilon,\gamma}\in\operatorname{\mathfrak{N}}$ such that
$v(\mu^{\varepsilon}_{g^{\diamond}})\approx
v(\mu^{\varepsilon}_{g^{\varepsilon,\gamma}})$ and finally conclude from
Corollary 1 that $v(\mu_{g^{\varepsilon,\gamma}}^{\varepsilon})\approx
v(\tau_{g^{\varepsilon,\gamma}})$. The above steps can be summarized as
follows:

$\displaystyle v^{\diamond}$ $\displaystyle=v(\tau_{f^{\diamond}})$
$\displaystyle\approx v(\tau_{g^{\diamond}})$ $\displaystyle\longleftarrow\
\textnormal{approximation and continuity (Assumption \ref{asm:UAP} and
\lx@cref{creftype~refnum}{thm:unifcontV})}$ $\displaystyle\approx
v(\mu_{g^{\diamond}}^{\varepsilon})$ $\displaystyle\longleftarrow\
\textnormal{relaxation (\lx@cref{creftype~refnum}{cor:relaxUnif})}$
$\displaystyle\approx v(\mu_{g^{\varepsilon,\gamma}}^{\varepsilon})$
$\displaystyle\longleftarrow\ \textnormal{stochastic optimization
\cite[cite]{[\@@bibref{Number}{RobbinsMonro,RobbinsSiegmund}{}{}]}}$
$\displaystyle\approx v(\tau_{g^{\varepsilon,\gamma}}).$
$\displaystyle\longleftarrow\ \textnormal{relaxation
(\lx@cref{creftype~refnum}{cor:relaxUnif})}$

We start off with the approximation of $f^{\diamond}$. Let
$\operatorname{\mathfrak{F}}_{c}=\\{f\in\operatorname{\mathfrak{F}}\mathrel{\mathop{\ordinarycolon}}f(t,\cdot)\in\operatorname{\mathcal{C}}(\operatorname{\mathcal{E}})\;\forall
t\in\operatorname{\mathcal{T}}\\}$ denotes the set of continuous boundaries
and fix throughout a hypothesis space
$\operatorname{\mathfrak{N}}\subset\operatorname{\mathfrak{F}}_{c}$ satisfying
the following property.

######  Assumption 4.

For all $f\in\operatorname{\mathfrak{F}}_{c}$, compact set
$\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$ and
$\iota>0$, there exists $g\in\operatorname{\mathfrak{N}}$ such that

$\lVert
f-g\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}})}\leq\iota.$
(37)

We refer to elements in a family
$\operatorname{\mathfrak{N}}\subset\operatorname{\mathfrak{F}}_{c}$ such that
Assumption 4 holds as universal approximators. The Stone-Weierstrass theorem
gives an example of universal approximators, namely the polynomials. This
class is however unsuitable numerically when the Hausdorff dimension of
$\operatorname{\mathcal{E}}$ becomes large. In our case,
$\operatorname{\mathfrak{N}}$ is evidently the family of all feedforward
neural networks
$g^{\theta}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{T}}\times\
\operatorname{\mathcal{E}}\to\operatorname{\mathbb{R}}_{+}$ with finite-
dimensional parameter vector $\theta$. Assumption 4 follows classically from
the Universal Approximation Theorem (UAT) for neural networks [8, 19].
Importantly, neural networks do not suffer much from the curse of
dimensionality; see, e.g., [15].

In this section, we assume that $f^{\diamond}(t,\xi)$ is continuous in $\xi$,
i.e. $f^{\diamond}\in\operatorname{\mathfrak{F}}_{c}$. We can therefore
directly use $f^{\diamond}$ in Assumption 4. Together with the results from
Section 4, the next convergence theorem follows naturally.

######  Theorem 5.3.

Let $\delta>0$,
$\operatorname{\mathfrak{N}}\subset\operatorname{\mathfrak{F}}_{c}$ satisfying
Assumption 4, and suppose that
$f^{\diamond}\in\operatorname{\mathfrak{F}}_{c}$. Then, under Assumption 3,
there exists $\varepsilon,\gamma>0$ such that for any $\gamma-$maximizer
$g^{\varepsilon,\gamma}\in\operatorname{\mathfrak{N}}$ of
$\underset{g\in\operatorname{\mathfrak{N}}}{\sup}\ v(\mu^{\varepsilon}_{g})$,

$v(\tau_{g^{\varepsilon,\gamma}})\geq v^{\diamond}-\delta.$ (38)

###### Proof.

Step ${1}$. Let $\delta^{\prime}>0$ be arbitrary. From Theorem 4.1, there
exist $\delta^{\prime\prime}>0$,
$\operatorname{\mathcal{K}}\subseteq\operatorname{\mathcal{E}}$ compact, and
$\iota>0$ such that

$\lVert
f-f^{\prime}\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}})}\leq\iota\
\Longrightarrow\ |v(\tau_{f})-v(\tau_{f^{\prime}})|\leq
C\sqrt{\rho(\iota)+\delta^{\prime\prime}}\leq\delta^{\prime}.$ (39)

Invoking Assumption 4, there exists
$g^{\diamond}\in\operatorname{\mathfrak{N}}$ such that $\lVert
f^{\diamond}-g^{\diamond}\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}})}\leq\iota.$
We conclude from Theorem 4.1 that $v(\tau_{g^{\diamond}})\geq
v^{\diamond}-\delta^{\prime}$.

Step ${2}$. Define $v^{\varepsilon}=\underset{g\ \in\
\operatorname{\mathfrak{N}}}{\sup}\ v(\mu^{\varepsilon}_{g}),$
$\varepsilon>0$. For any $f\in\operatorname{\mathfrak{F}}$, Corollary 1
implies that $v(\mu^{\varepsilon}_{f})\to v(\tau_{f})$ as $\varepsilon\to 0$.
In particular, taking $f=g^{\diamond}$ from Step 1 yields

$\liminf_{\varepsilon\to 0}v^{\varepsilon}\geq\liminf_{\varepsilon\to
0}v(\mu_{g^{\diamond}}^{\varepsilon})=v(\tau_{g^{\diamond}}).$

Thus, one can pick $\bar{\varepsilon}>0$ such that $v^{\varepsilon}\geq
v(\tau_{g^{\diamond}})-\delta^{\prime}$ for all
$\varepsilon\in(0,\bar{\varepsilon})$.

Step ${3}$. Using again Corollary 1, there exists $\
\varepsilon\in(0,\bar{\varepsilon})$ such that for all
$f\in\operatorname{\mathfrak{F}}$,

$|v(\tau_{f})-v(\mu^{\varepsilon}_{f})|\leq
C\sqrt{\rho(\varepsilon)}\leq\delta^{\prime}.$ (40)

Consequently, if $g^{\varepsilon,\gamma}\in\operatorname{\mathfrak{N}}$ is a
$\gamma-$maximizer of $v^{\varepsilon}$, then

$v(\tau_{g^{\varepsilon,\gamma}})\geq
v(\mu^{\varepsilon}_{g^{\varepsilon,\gamma}})-\delta^{\prime}\geq
v^{\varepsilon}-\gamma-\delta^{\prime}.$

Step ${4}$. Collecting the above inequalities, we obtain

$\displaystyle v(\tau_{g^{\varepsilon,\gamma}})\geq
v^{\varepsilon}-\gamma-\delta^{\prime}\geq
v(\tau_{g^{\diamond}})-\gamma-2\delta^{\prime}\geq
v^{\diamond}-\gamma-3\delta^{\prime}.$

We can therefore set $\delta^{\prime}=\gamma=\frac{\delta}{4}$. ∎

###  5.2 Convergence analysis: semicontinuous boundaries

This section treats the case where the optimal boundary
$f^{\diamond}=f^{\diamond}(t,\xi)$ is strictly lower continuous in $\xi$. To
project $f^{\diamond}$ onto $\operatorname{\mathfrak{N}}$, we first consider a
sequence of continuous boundaries converging to $f^{\diamond}$. These
continuous boundaries can thereafter be approximated by elements in
$\operatorname{\mathfrak{N}}$. To construct such sequence, we borrow the
notion of inf/sup convolution, which is central in the theory of viscosity
solutions; see [12, Section V.5] and [25].

Let
$h\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{E}}\to\operatorname{\mathbb{R}}$
be a lower semicontinuous function and define its inf convolution as

$h_{\delta}(\xi)\mathrel{\mathop{\ordinarycolon}}=\inf_{\xi^{\prime}\in\operatorname{\mathcal{E}}}\left[h(\xi^{\prime})+\frac{1}{\delta}|\xi^{\prime}-\xi|^{2}\right]\leq
h(\xi),\quad\delta>0.$ (41)

It is easily seen that $h_{\delta}$ is semiconcave, hence continuous. For
$f\in\operatorname{\mathfrak{F}}$, the inf convolution of $f$ is understood as
$f_{\delta}(t,\cdot)=f(t,\cdot)_{\delta}$ for each
$t\in\operatorname{\mathcal{T}}$. The next proposition shows that
$f_{\delta}\uparrow f$ as $\delta\to 0$. See Fig. 4 for an illustration. In
particular, the hitting time $\tau_{f_{\delta}}$ of
$\operatorname{\mathfrak{S}}(f_{\delta})$ increases to $\tau_{f}$.

######  Proposition 4.

Let
$h\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{E}}\to\operatorname{\mathbb{R}}$
be lower semicontinuous and $h_{\delta}$ its inf convolution with parameter
$\delta$ as in (41). Then $h_{\delta}\uparrow h$ pointwise.

###### Proof.

It is clear that $h_{\delta}$ increases with $\delta$, hence any sequence
$(h_{\delta_{l}})$ with $\delta_{l}\downarrow 0$ is non-decreasing. Let
$\xi_{l}$ be a $\delta_{l}-$minimizer of $h_{\delta_{l}}(\xi)$, i.e.
$h(\xi_{l})+\frac{1}{\delta_{l}}|\xi-\xi_{l}|^{2}\leq
h_{\delta_{l}}(\xi)+\delta_{l}$. Then,

$h(\xi)\geq h_{\delta_{l}}(\xi)\geq
h(\xi_{l})+\frac{1}{\delta_{l}}|\xi-\xi_{l}|^{2}-\delta_{l}\geq\frac{1}{\delta_{l}}|\xi-\xi_{l}|^{2}-\delta_{l},$

which implies that $|\xi-\xi_{l}|^{2}\leq\delta_{l}(\delta_{l}+h(\xi))$.
Therefore $\xi_{l}\to\xi$, and the lower semicontinuity of $h$ gives

$h(\xi)\leq\liminf_{l\to\infty}h(\xi_{l})\leq\liminf_{l\to\infty}\left[h(\xi_{l})+\frac{1}{\delta_{l}}|\xi-\xi_{l}|^{2}\right]\leq\liminf_{l\to\infty}\left[h_{\delta_{l}}(\xi)+\delta_{l}\right]\leq
h(\xi).$

∎

More can be said when restricting l.s.c. boundaries to a compact set.
Accordingly, we define

$h_{\delta}^{\operatorname{\mathcal{K}}}(\xi)\mathrel{\mathop{\ordinarycolon}}=\inf_{\xi^{\prime}\in\
\operatorname{\mathcal{K}}}\left[h(\xi^{\prime})+\frac{1}{\delta}|\xi^{\prime}-\xi|^{2}\right],\quad\delta>0,\quad\xi\in\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}.$
(42)

The inf convolution of $f\in\operatorname{\mathfrak{F}}$ in
$\operatorname{\mathcal{K}}$ is understood as
$f_{\delta}^{\operatorname{\mathcal{K}}}(t,\cdot)=f(t,\cdot)_{\delta}^{\operatorname{\mathcal{K}}}$
for each $t\in\operatorname{\mathcal{T}}$. Since $\xi^{\prime}\mapsto
h(\xi^{\prime})+\frac{1}{\delta}|\xi^{\prime}-\xi|^{2}$ is also l.s.c., it
attains its minimum on $\operatorname{\mathcal{K}}$. This leads us to the
following corollary. Throughout, we write $B_{r}(\xi)$ for the closed ball of
radius $r\geq 0$ centered at $\xi\in\operatorname{\mathcal{E}}$.

######  Corollary 2.

Let $\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$ and
$h\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{E}}\to\operatorname{\mathbb{R}}$
a lower semicontinuous function such that
$M\mathrel{\mathop{\ordinarycolon}}=\lVert
h\rVert_{L^{\infty}(\operatorname{\mathcal{K}})}<\infty$. Then for all
$\delta>0$, $\xi\in\operatorname{\mathcal{K}}$, any minimizer achieving
$h_{\delta}^{\operatorname{\mathcal{K}}}(\xi)$ in (42) is contained in
$B_{\sqrt{\delta M}}(\xi)$.

###### Proof.

Let $\xi_{\delta}\in\operatorname{\mathcal{K}}$ be any minimizer in (42), i.e.
$h_{\delta}^{\operatorname{\mathcal{K}}}(\xi)=h(\xi_{\delta})+\frac{1}{\delta}|\xi_{\delta}-\xi|^{2}$.
Then

$M\geq h(\xi)\geq
h_{\delta}^{\operatorname{\mathcal{K}}}(\xi)=h(\xi_{\delta})+\frac{1}{\delta}|\xi_{\delta}-\xi|^{2}\geq\frac{1}{\delta}|\xi_{\delta}-\xi|^{2}.$

∎

Figure 4: Inf convolution of $f(t,\cdot)$ for some
$t\in\operatorname{\mathcal{T}}$. The monotonicity of $\delta\to f_{\delta}$
implies that the hitting time $\tau_{f_{\delta}}$ increases to $\tau_{f}$
(pathwise) as $\delta\downarrow 0$.

######  Remark 5.4.

When the stopping region is the hypograph of the optimal boundary ($\eta=-1$
in 2) one can employ sup convolutions instead, that is

$h^{\delta}(\xi)\mathrel{\mathop{\ordinarycolon}}=\sup_{\xi^{\prime}\in\
\operatorname{\mathcal{E}}}\left[h(\xi^{\prime})-\frac{1}{\delta}|\xi^{\prime}-\xi|^{2}\right]\geq
h(\xi),\quad\delta>0.$

We may also define the restriction $h^{\delta,\operatorname{\mathcal{K}}}$ to
some compact set $\operatorname{\mathcal{K}}$. This case is simpler since
u.s.c functions are automatically bounded above in compact sets.

The next lemma states that inf convolutions approximate lower semicontinuous
functions arbitrarily well with respect to the relaxed $L^{\infty}$ distance
seen in Definition 1.

######  Lemma 2.

Let $\iota>0$ and
$\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$. If
$f\in\operatorname{\mathfrak{F}}$ bounded in
$\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}}$, then there
exists $\delta>0$ such that
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f,f_{\delta}^{\operatorname{\mathcal{K}}})\leq\iota.$

###### Proof.

Let $\iota>0$ and $r>0$ be arbitrary. Since $\operatorname{\mathcal{K}}$ is
compact, one can choose
$(\xi^{i})_{i=1}^{I}\subseteq\operatorname{\mathcal{K}}$, $I<\infty$ such that
$\operatorname{\mathcal{K}}\subseteq\bigcup_{i=1}^{I}B_{r}(\xi^{i})$ where we
recall that $B_{r}(\xi^{i})=\\{\xi\in\operatorname{\mathcal{E}}\
\mathrel{\mathop{\ordinarycolon}}\ |\xi-\xi_{i}|\leq r\\}$. Now fix
$t\in\operatorname{\mathcal{T}}$ and write $h(\xi)=f(t,\xi)$ for brevity. As
$h$ is l.s.c., there exist $\\{r_{i}\\}_{i=1}^{I}\subset(0,\infty)$ satisfying

$h(\xi^{i})\leq h(\xi^{\prime})+\iota,\quad\forall\ \xi^{\prime}\in
B_{r_{i}}(\xi^{i}),\quad i=1,\ldots,I.$ (43)

If we set $\delta_{t}=\frac{1}{M}\min_{i}r_{i}^{2}>0$, $M=\lVert
h\rVert_{L^{\infty}(\operatorname{\mathcal{K}})}$, then
$\sqrt{\delta_{t}M}\leq r_{i}$ for all $i=1,\ldots,I$. For each
$i\in\\{1,\ldots,I\\}$, take a minimizer $\xi^{i}_{\delta_{t}}$ achieving
$h_{\delta_{t}}^{\operatorname{\mathcal{K}}}(\xi^{i})$ in (42). In view of
Corollary 2, $\xi^{i}_{\delta_{t}}$ belongs to
$B_{\sqrt{\delta_{t}M}}(\xi^{i})$. But
$B_{\sqrt{\delta_{t}M}}(\xi^{i})\subseteq B_{r_{i}}(\xi^{i})$, hence

$h_{\delta_{t}}^{\operatorname{\mathcal{K}}}(\xi^{i})=h(\xi^{i}_{\delta_{t}})+\frac{1}{\delta_{t}}|\xi^{i}-\xi^{i}_{\delta_{t}}|^{2}\geq
h(\xi^{i}_{\delta_{t}})\overset{\eqref{eq:lscIota}}{\geq}h(\xi^{i})-\iota.$

Consequently, $0\leq
h(\xi^{i})-h_{\delta_{t}}^{\operatorname{\mathcal{K}}}(\xi^{i})\leq\iota$ for
all $i=1,\ldots,I$. To finish off, define the projection map

$\underline{\psi}\mathrel{\mathop{\ordinarycolon}}\operatorname{\mathcal{K}}\to\operatorname{\mathcal{K}},\quad\underline{\psi}(\xi)=\underset{(\xi^{i})_{i=1}^{I}}{\text{argmin}}\
|\xi-\xi^{i}|.$

It is clear that $|\underline{\psi}(\xi)-\xi|\leq r$ for all
$\xi\in\operatorname{\mathcal{K}}$, hence
$\underline{\psi}\in\Psi_{r}^{\operatorname{\mathcal{K}}}$. As
$|h(\xi^{i})-h_{\delta_{t}}^{\operatorname{\mathcal{K}}}(\xi^{i})|\leq\iota$
for all $i=1,\ldots,I$, we conclude that

$\inf_{\psi\in\Psi_{r}^{\operatorname{\mathcal{K}}}}\lVert h\circ\psi-
h_{\delta_{t}}^{\operatorname{\mathcal{K}}}\circ\psi\rVert_{L^{\infty}(\operatorname{\mathcal{K}})}\leq\lVert
h\circ\underline{\psi}-h_{\delta_{t}}^{\operatorname{\mathcal{K}}}\circ\underline{\psi}\rVert_{L^{\infty}(\operatorname{\mathcal{K}})}\leq\iota.$

The proof is complete after setting
$\delta=\min_{t\in\operatorname{\mathcal{T}}}\delta_{t}>0$. ∎

The next approximation result follows immediately.

######  Theorem 5.5.

(Universal Approximation of l.s.c. Functions) Let
$\operatorname{\mathfrak{N}}\subseteq\operatorname{\mathfrak{F}}_{c}$
satisfying Assumption 4. Then for all $f\in\operatorname{\mathfrak{F}}$,
$\iota>0$, and compact set
$\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$ such that
$f$ is bounded in $\operatorname{\mathcal{K}}$, there exists
$g\in\operatorname{\mathfrak{N}}$ such that
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f,g)\leq\iota$.

###### Proof.

Fix $r,\iota^{\prime}>0$,
$\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$, and
$\xi\in\operatorname{\mathcal{K}}$. Using Lemma 2, then
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f,f_{\delta}^{\operatorname{\mathcal{K}}})\leq\iota^{\prime}$
for some $\delta>0$. For every $\gamma>0$, one can find
$\psi_{\gamma}\in\Psi_{r}^{\operatorname{\mathcal{K}}}$ such that

$\lVert
f(t,\cdot)\circ\psi_{\gamma}-f_{\delta}^{\operatorname{\mathcal{K}}}(t,\cdot)\circ\psi_{\gamma}\rVert_{L^{\infty}(\operatorname{\mathcal{K}})}\leq
d_{\operatorname{\mathcal{K}}}(f(t,\cdot),f_{\delta}^{\operatorname{\mathcal{K}}}(t,\cdot))+\gamma\leq\iota^{\prime}+\gamma,\qquad\forall\
t\in\operatorname{\mathcal{T}}.$

Moreover, as $f_{\delta}^{\operatorname{\mathcal{K}}}(t,\xi)$ is continuous in
$\xi$, we can invoke Assumption 4 giving the existence of
$g\in\operatorname{\mathfrak{N}}\subseteq\operatorname{\mathfrak{F}}_{c}$ such
that $\lVert
f_{\delta}^{\operatorname{\mathcal{K}}}-g\rVert_{L^{\infty}(\operatorname{\mathcal{T}}\times\operatorname{\mathcal{K}})}\leq\iota^{\prime}$.
The triangular inequality therefore yields

$\displaystyle\lVert
f(t,\cdot)\circ\psi_{\gamma}-g(t,\cdot)\circ\psi_{\gamma}\rVert_{L^{\infty}(\operatorname{\mathcal{K}})}\leq
2\iota^{\prime}+\gamma,\qquad\forall\ t\in\operatorname{\mathcal{T}}.$

As $r$ is arbitrary, we conclude that
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f,g)\leq
2\iota^{\prime}+\gamma$. Choosing $\iota^{\prime}=\frac{\iota}{2}$ and letting
$\gamma\downarrow 0$ proves the result. ∎

At last, this leads us to the following convergence result for l.s.c.
boundaries.

######  Theorem 5.6.

Let $\delta>0$ and a family
$\operatorname{\mathfrak{N}}\subset\operatorname{\mathfrak{F}}_{c}$ of
universal approximators. Under Assumption 3, there exists
$\varepsilon,\gamma>0$, and a $\gamma-$maximizer $g^{\varepsilon,\gamma}$ of
$\underset{g\in\operatorname{\mathfrak{N}}}{\sup}\ v(\mu^{\varepsilon}_{g})$
such that $v(\tau_{g^{\varepsilon,\gamma}})\geq v^{\diamond}-\delta$.

###### Proof.

Let $\delta^{\prime}>0$ be arbitrary. Using Theorem 4.2, there exist
$\delta^{\prime\prime}>0$,
$\operatorname{\mathcal{K}}\subset\subset\operatorname{\mathcal{E}}$, and
$\iota>0$ such that

$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f,f^{\prime})\leq\iota\
\Longrightarrow\ |v(\tau_{f})-v(\tau_{f^{\prime}})|\leq
C\sqrt{\rho(\iota)+\delta^{\prime\prime}}\leq\delta^{\prime},\quad\forall
f,f^{\prime}\in\operatorname{\mathfrak{F}}.$ (44)

As $f^{\diamond}$ is locally bounded, we can apply Theorem 5.5 giving the
existence of $g^{\diamond}\in\operatorname{\mathfrak{N}}$ such that
$d_{\operatorname{\mathcal{K}},\operatorname{\mathcal{T}}}(f^{\diamond},g^{\diamond})\leq\iota.$
We conclude from (44) that $v(\tau_{g^{\diamond}})\geq
v^{\diamond}-\delta^{\prime}$. As $g^{\diamond}$ is continuous, we can follow
Step 2, 3, and 4 in the proof of Theorem 5.3. ∎

##  6 Conclusion

We analyze hitting times of boundaries arising in optimal stopping. After
introducing relaxed stopping rules and establishing preliminary results, the
continuity of the value functional $f\mapsto v(\tau_{f})$ is shown with
respect to the topology of uniform convergence and the novel relaxed
$L^{\infty}$ distance. The main results are later applied to study the
convergence of the neural optimal stopping boundary method [33]. When the
optimal stopping boundary is continuous, we can directly invoke the universal
approximation theorem of neural networks and obtain a convergence theorem.
When the optimal stopping boundary is strictly semicontinuous, inf/sup
convolutions are combined with the relaxed $L^{\infty}$ distance, leading to a
more general result.

A natural extension of the present work would be to show continuity and
relaxation results for optimal stopping problems in continuous time. For
instance, it would be relevant to investigate the convergence of the neural
optimal stopping method for American options (that is,
$\operatorname{\mathcal{T}}=[0,T]$) as the number of time steps to discretize
$\operatorname{\mathcal{T}}$ goes to infinity.

##  Appendix A Proof of Proposition 3

###### Proof.

We write
$\underline{v}^{\diamond}\mathrel{\mathop{\ordinarycolon}}=\sup_{\mu\in\underline{\vartheta}(\operatorname{\mathcal{T}})}v(\mu)$
and show that $\underline{v}^{\diamond}=v^{\diamond}$ by induction on
$N=|\operatorname{\mathcal{T}}|$. If $N=1$, then $\operatorname{\mathcal{T}}$
is a singleton, hence
$\underline{\vartheta}(\operatorname{\mathcal{T}})=\vartheta(\operatorname{\mathcal{T}})$.
We now suppose the result true for $N\geq 1$ and prove the claim for $N+1$.
Let $\operatorname{\mathcal{T}}=\\{t_{1}<\ldots<t_{N+1}\\}\subset[0,T]$ and
fix $\mu=\sum_{t\in\operatorname{\mathcal{T}}}P_{t}\
\delta_{t}\in\underline{\vartheta}(\operatorname{\mathcal{T}})$ where each
$P_{t}\mathrel{\mathop{\ordinarycolon}}\Omega\to[0,1]$ is
$\operatorname{\mathcal{F}}_{t}-$measurable. Next, write
$\operatorname{\mathcal{T}}_{\\!+}=\operatorname{\mathcal{T}}\setminus\\{t_{1}\\}$
and consider the continuation value functional

$v_{1}(\mu_{+},x)=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\int_{\operatorname{\mathcal{T}}_{\\!+}}\varphi(t,X_{t})\mu^{\prime}(dt)\
\big{|}\
X_{t_{1}}=x\right],\quad\mu_{+}\in\underline{\vartheta}(\operatorname{\mathcal{T}}_{\\!+}).$

Since $|\operatorname{\mathcal{T}}_{\\!+}|=N$, we have by induction that

$v_{1}(\mu_{+},x)\leq
v_{1}^{\diamond}(x)\mathrel{\mathop{\ordinarycolon}}=\sup_{\tau\in\vartheta(\operatorname{\mathcal{T}}_{\\!+})}\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}[\varphi(\tau,X_{\tau})\
|\ X_{t_{1}}=x],\quad\forall\
\mu_{+}\in\underline{\vartheta}(\operatorname{\mathcal{T}}_{\\!+}).$ (45)

Then, the tower property of conditional expectations implies that

$\displaystyle v(\mu)$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[P_{t_{1}}\varphi(t_{1},X_{t_{1}})+\int_{\operatorname{\mathcal{T}}_{\\!+}}\varphi(t,X_{t})\mu(dt)\right]$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[P_{t_{1}}\varphi(t_{1},X_{t_{1}})+(1-P_{t_{1}})\int_{\operatorname{\mathcal{T}}_{\\!+}}\varphi(t,X_{t})\mu_{+}(dt)\right]$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[P_{t_{1}}\varphi(t_{1},X_{t_{1}})+(1-P_{t_{1}})v_{1}(\mu_{+},X_{t_{1}})\right],$
(46)

with the relaxed stopping rule

$\underline{\vartheta}(\operatorname{\mathcal{T}}_{\\!+})\ni\mu_{+}(\omega)=\sum_{t\in\operatorname{\mathcal{T}}_{\\!+}}\frac{P_{t}(\omega)}{1-P_{t_{1}}(\omega)}\
\delta_{t},\quad P_{t_{1}}<1.$

Note that if $P_{t_{1}}(\omega)=1$, then $\mu_{+}(\omega)$ would not appear in
(46). Together with (45), we obtain

$\displaystyle v(\mu)$
$\displaystyle\leq\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[P_{t_{1}}\varphi(t_{1},X_{t_{1}})+(1-P_{t_{1}})v_{1}^{\diamond}(X_{t_{1}})\right]$
$\displaystyle=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[P_{t_{1}}(\varphi(t_{1},X_{t_{1}})-v_{1}^{\diamond}(X_{t_{1}}))\right]+\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[v_{1}^{\diamond}(X_{t_{1}})\right].$
(47)

We can now maximize the first term in (47) over all
$\operatorname{\mathcal{F}}_{t_{1}}-$measurable functions
$P_{t_{1}}\mathrel{\mathop{\ordinarycolon}}\Omega\to[0,1]$. It is
straightforward to see and also classical in optimal stopping [30] that the
unique maximizer is given by333In financial terms, this means that one should
exercise a Bermudan contract whenever the intrinsic value
$\varphi(t_{1},X_{t_{1}})$ exceeds the continuation value
$v_{1}^{\diamond}(X_{t_{1}})$.

$P_{t_{1}}^{\diamond}(\omega)=\begin{cases}1,\quad&\text{if
}\varphi(t_{1},X_{t_{1}}(\omega))\ \geq\
v_{1}^{\diamond}(X_{t_{1}}(\omega)),\\\ 0,\ &\text{otherwise}.\end{cases}$

In particular, it is $\\{0,1\\}-$valued. This leads to the dynamic programming
equation

$v^{\diamond}=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[P_{t_{1}}^{\diamond}(\varphi(t_{1},X_{t_{1}})-v_{1}^{\diamond}(X_{t_{1}}))\right]+\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[v_{1}^{\diamond}(X_{t_{1}})\right]=\operatorname{\mathbb{E}}^{\operatorname{\mathbb{Q}}}\left[\varphi(t_{1},X_{t_{1}})\vee
v_{1}^{\diamond}(X_{t_{1}})\right]\geq v(\mu).$

As $v^{\diamond}\leq\underline{v}^{\diamond}$ is evident, (17) follows. Moving
to the second assertion, the induction hypothesis implies that every optimal
relaxed stopping rule of
$\sup_{\mu_{+}\in\underline{\vartheta}(\operatorname{\mathcal{T}}_{\\!+})}v_{1}(\mu_{+},x)$
is in fact a genuine stopping time, say $\tau_{+}$. Together with the above
arguments, we gather that the same holds true when adding back the first
exercise date $t_{1}$. Precisely, we have
$v(\tau^{\diamond})=\underline{v}^{\diamond}$ with the stopping time
$\tau^{\diamond}=P_{t_{1}}^{\diamond}t_{1}+(1-P_{t_{1}}^{\diamond})\tau_{+}$.
∎

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