[Paper Review] On a Stochastic Representation Theorem for Meyer-measurable Processes and its Applications in Stochastic Optimal Control and Optimal Stopping

Peter Bank, David Beßlich|arXiv (Cornell University)|Oct 19, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance6 references3 citations

TL;DR

This paper establishes a stochastic representation theorem for Meyer-measurable processes, generalizing prior work by allowing random measures with atoms and arbitrary Meyer-σ-fields. It proves existence and maximality of a process $ L $ such that any $ \Lambda $-measurable process $ X $ admits the representation $ X_S = \mathbb{E}\left[\int_{[S,\infty)} g_t(\sup_{v\in[S,t]} L_v) \mu(dt) \,\middle|\, \mathcal{F}^\Lambda_S\right] $ at every $ \Lambda $-stopping time $ S $, with applications to irreversible investment and optimal stopping over divided stopping times.

ABSTRACT

In this paper we study a representation problem first considered in a simpler version by Bank and El Karoui [2004]. A key ingredient to this problem is a random measure $μ$ on the time axis which in the present paper is allowed to have atoms. Such atoms turn out to not only pose serious technical challenges in the proof of the representation theorem, but actually have significant meaning in its applications, for instance, in irreversible investment problems. These applications also suggest to study the problem for processes which are measurable with respect to a Meyer-$σ$-field that lies between the predictable and the optional $σ$-field. Technically, our proof amounts to a delicate analysis of optimal stopping problems and the corresponding optimal divided stopping times and we will show in a second application how an optimal stopping problem over divided stopping times can conversely be obtained from the solution of the representation problem.

Motivation & Objective

• To generalize the stochastic representation theorem of Bank and El Karoui (2004) to allow random measures with atoms and arbitrary Meyer-σ-fields.
• To address the technical challenges arising from atoms in the measure $ \mu $, which disrupt standard optimal stopping arguments and necessitate the use of divided stopping times.
• To establish the maximality of the solution process $ L $, ensuring it dominates all other $ \Lambda $-measurable solutions up to evanescent sets.
• To demonstrate applications in irreversible investment problems where atoms in $ \mu $ represent critical risk assessment times.
• To show that solving the representation problem enables solving generalized optimal stopping problems over divided stopping times, offering an alternative to Snell envelope methods.

Proposed method

• The proof relies on a delicate analysis of optimal stopping problems over $ \Lambda $-stopping times, particularly the auxiliary problem $ Y_S^\ell = \mathrm{ess\,sup}_{T \in \mathcal{S}^\Lambda([S,\infty))} \mathbb{E}\left[ X_T + \int_{[S,T)} g_t(\ell) \mu(dt) \,\middle|\, \mathcal{F}_S^\Lambda \right] $.
• The solution process $ L $ is constructed as $ L_t = \sup\{ \ell \in \mathbb{R} \mid X_t = Y_t^\ell \} $, linking the representation to the value function of the optimal stopping problem.
• Divided stopping times—quadruples of a stopping time and three disjoint sets—are used to handle discontinuities in the running cost when $ \mu $ has atoms.
• The construction leverages properties of $ \Lambda $-measurable processes and the structure of Meyer-σ-fields, which interpolate between predictable and optional $ \sigma $-fields.
• Maximality of $ L $ is proven by showing that any other solution is $ \leq L $ up to an evanescent set, improving on uniqueness under path regularity assumptions in prior work.
• The paper establishes equivalence between optimal stopping over $ \Lambda $-stopping times and over divided stopping times, showing $ \sup_{\tau \in \mathcal{S}^{\Lambda,\mathrm{div}}} \mathbb{E}[X_\tau + \int_{[0,\tau)} g_t(\ell) \mu(dt)] = \sup_{T \in \mathcal{S}^\Lambda} \mathbb{E}[X_T + \int_{[0,T)} g_t(\ell) \mu(dt)] $.

Experimental results

Research questions

• RQ1 Can a stochastic representation theorem be extended to Meyer-measurable processes when the random measure $ \mu $ has atoms, despite the resulting discontinuities in the running cost?
• RQ2 How can optimal stopping problems with atoms in the measure $ \mu $ be handled, especially when standard optimal stopping times fail to exist?
• RQ3 What is the role of divided stopping times in characterizing solutions when $ \mu $ has atoms, and how do they relate to the representation problem?
• RQ4 How does the choice of Meyer-σ-field $ \Lambda $ affect the existence and maximality of the solution process $ L $ in the representation?
• RQ5 Can the solution to the representation problem be used to solve generalized optimal stopping problems over divided stopping times, and vice versa?

Key findings

• The main theorem establishes the existence of a $ \Lambda $-measurable process $ L $ such that $ X_S = \mathbb{E}\left[\int_{[S,\infty)} g_t(\sup_{v\in[S,t]} L_v) \mu(dt) \,\middle|\, \mathcal{F}_S^\Lambda \right] $ holds at every $ \Lambda $-stopping time $ S $, even when $ \mu $ has atoms.
• The solution $ L $ is maximal: any other $ \Lambda $-measurable solution is $ \leq L $ up to an evanescent set, a stronger result than uniqueness under path regularity assumptions in Bank and El Karoui (2004).
• The paper shows that optimal stopping problems over $ \Lambda $-stopping times and over divided stopping times yield the same value, establishing equivalence between the two formulations.
• The process $ L $ is explicitly constructed as $ L_t = \sup\{ \ell \in \mathbb{R} \mid X_t = Y_t^\ell \} $, where $ Y^\ell $ solves the auxiliary optimal stopping problem.
• For the irreversible investment application, $ \sup_{v\in[0,t]} L_v $ yields the optimal investment strategy, with atoms in $ \mu $ corresponding to times of critical risk assessment.
• The proof reveals that $ \tilde{\tau}_\ell = (T_{0,\ell}, \emptyset, \{L_{T_{0,\ell}} \geq \ell\}, \{L_{T_{0,\ell}} < \ell\}) $ is an optimal divided stopping time, and $ T_{0,\ell} = T_\ell $ a.s., linking the representation to optimal stopping time selection.