arXiv subjects

Shuoqing Deng

Publications and source records attributed to Shuoqing Deng.

13 recordsLinked to original sources

arXiv2026

Distribution-constrained optimal multiple stopping: the Root-type solution

We consider the distribution-constrained optimal stopping problem introduced by Bayraktar and Miller (Mathematical Finance, 2019) and Beiglbock et al. (PTRF, 2018). Motivated by the multi-marginal Skorokhod embedding problem (SEP) and applications in financial mathematics, we generalize the Root-type solution to the multi-marginal case. The key difficulty is that the associated stopping barriers need not be ordered, so the problem in general cannot be reduced to a sequence of one-marginal problems. First, we give a probabilistic characterization in terms of sequential optimal stopping for an auxiliary time-reversed process, in the same spirit as Cox et al. (PTRF, 2019). Then, we prove the optimality of this construction by martingale arguments for a general class of reward functions, including multi-marginal versions of all examples in Beiglbock et al. (PTRF, 2018) as special cases.

math.PR↗

arXiv2026

Convex ordering for graphon mean-field systems

We establish marginal and functional convex order comparisons for graphon mean-field systems on $\mathbb{R}^d$, including the infinite-horizon setting. A key difficulty is that convex order cannot in general be transferred through the usual particle approximation, which requires us to work directly with Euler schemes for the graphon mean-field system. Under a suitable dissipativity condition, we establish Euler approximation estimates that are uniform in both time and the agent label, and combine them with forward-backward induction arguments to obtain the convex order comparisons. For the infinite-horizon problem, dissipativity provides the required uniform-in-time stability, while an exponentially weighted $L^2$-space enables trajectory-level convergence and leads to the functional convex order on $[0,\infty)$. As an application, we derive value-function comparisons for a class of one-dimensional linear-quadratic graphon mean-field games.

math.PR↗

arXiv2026

Graphon Mean Field Game of mutual holding

This paper studies the mean field game of mutual holding proposed by Djete and Touzi(AAP, 2024), and consider the case where the interactions among agents are described by a graphon. We adopt the formulation on the enlarged space which is modeled using the joint law of the value process and the graphon label, as in Lacker and Soret(MOR, 2023). Under suitable conditions on the graphon function, we are able to provide the explicit characterization of the optimal strategy, prove the wellposedness of associated Mckean-Vlasov SDE and establish the convergence results of the Nash equilibria. The key technique consists in a detailed analysis of the continuity property under the $\mathcal{WOP}_2$ metric, and tailor-made arguments for different graphon equilibria under different regularities of the model.

math.OC↗

arXiv2026

Stability of supermartingale optimal transport problems

We investigate stability properties of weak supermartingale optimal transport (WSOT) problems on $\mathbb{R}$. For probability measures $\mu,\nu\in\mathcal{P}_r$ satisfying $\mu \leq_{cd} \nu$ (equivalently, $\Pi_S(\mu,\nu)\neq\emptyset$), we consider supermartingale couplings $\pi=\mu(d x)\pi_x(d y)$ and the weak transport functional \[ V_S^C(\mu,\nu) := \inf_{\pi\in\Pi_S(\mu,\nu)} \int_\mathbb{R} C(x,\pi_x)\,\mu(d x), \] for some appropriate cost function $C:\mathbb{R}\times\mathcal{P}_r\to\mathbb{R}$. Our first main contribution is an approximation result in adapted Wasserstein distance: under $W_r$-convergence of marginals $(\mu^k,\nu^k)\to(\mu,\nu)$ with $\mu^k\leq_{cd} \nu^k$, any $\pi\in\Pi_S(\mu,\nu)$ can be approximated by $\pi^k\in\Pi_S(\mu^k,\nu^k)$ such that $A\mathcal{W}_r(\pi^k,\pi)\to0$. As a consequence, we obtain the continuity of the functional $(\mu,\nu) \mapsto V_S^C(\mu,\nu)$, and the monotonicity principle for WSOT.

math.PR↗

arXiv2025

On time-consistent equilibrium stopping under aggregation of diverse discount rates

This paper studies a central planner's decision making on behalf of a group of members with diverse discount rates. In the context of optimal stopping, we work with an aggregation preference to incorporate all discount rates via an attitude function that reflects the aggregation rule chosen by the central planner. The problem formulation is also applicable to single agent's stopping problem with uncertain discount rate, where our aggregation preference coincides with the conventional smooth ambiguity preference. The resulting optimal stopping problem is time inconsistent, for which we develop an iterative approach using consistent planning and characterize all time-consistent mild equilibria as fixed points of an operator in the setting of one-dimensional diffusion processes. We provide some sufficient conditions on the underlying models and the attitude function such that the smallest mild equilibrium attains the optimal equilibrium. In addition, we show that the optimal equilibrium is a weak equilibrium. When the sufficient condition of the attitude function is violated, we illustrate by various examples that the characterization of the optimal equilibrium may differ significantly from some existing results for a single agent, which now sensitively depends on the attitude function and the diversity distribution of discount rates within the group.

q-fin.MF↗

arXiv2022

Supermartingale Brenier's Theorem with full-marginals constraint

We explicitly construct the supermartingale version of the Fr{é}chet-Hoeffding coupling in the setting with infinitely many marginal constraints. This extends the results of Henry-Labordere et al. obtained in the martingale setting. Our construction is based on the Markovian iteration of one-period optimal supermartingale couplings. In the limit, as the number of iterations goes to infinity, we obtain a pure jump process that belongs to a family of local L{é}vy models introduced by Carr et al. We show that the constructed processes solve the continuous-time supermartingale optimal transport problem for a particular family of path-dependent cost functions. The explicit computations are provided in the following three cases: the uniform case, the Bachelier model and the Geometric Brownian Motion case.

math.PR↗

arXiv2022

Supermartingale shadow couplings: the decreasing case

For two measures $μ$ and $ν$ that are in convex-decreasing order, Nutz and Stebegg (Canonical supermartingale couplings, Ann. Probab., 46(6):3351--3398, 2018) studied the optimal transport problem with supermartingale constraints and introduced two canonical couplings, namely the increasing and decreasing transport plans, that are optimal for a large class of cost functions. In the present paper we provide an explicit construction of the decreasing coupling $π^D$ by establishing a Brenier-type result: (a generalised version of) $π^D$ concentrates on the graphs of two functions. Our construction is based on the concept of the supermartingale \textit{shadow} measure and requires a suitable extension of the results by Juillet (Stability of the shadow projection and the left-curtain coupling, Ann. Inst. H. Poincaré Probab. Statist., 52(4):1823--1843, November 2016) and Beiglböck and Juillet (Shadow couplings, Trans. Amer. Math. Soc., 374:4973--5002, 2021) established in the martingale setting. In particular, we prove the stability of the supermartingale shadow measure with respect to initial and target measures $μ,ν$, introduce an infinite family of lifted supermartingale couplings that arise via shadow measure, and show how to explicitly determine the `martingale points' of each such coupling.

math.PR↗

arXiv2022

Optimal Consumption with Reference to Past Spending Maximum

This paper studies the infinite-horizon optimal consumption with a path-dependent reference under exponential utility. The performance is measured by the difference between the nonnegative consumption rate and a fraction of the historical consumption maximum. The consumption running maximum process is chosen as an auxiliary state process, and hence the value function depends on two state variables. The Hamilton-Jacobi-Bellman (HJB) equation can be heuristically expressed in a piecewise manner across different regions to take into account all constraints. By employing the dual transform and smooth-fit principle, some thresholds of the wealth variable are derived such that a classical solution to the HJB equation and the feedback optimal investment and consumption strategies can be obtained in closed form in each region. A complete proof of the verification theorem is provided, and numerical examples are presented to illustrate some financial implications.

q-fin.MF↗

arXiv2022

A potential-based construction of the increasing supermartingale coupling

The increasing supermartingale coupling, introduced by Nutz and Stebegg (Canonical supermartingale couplings, Annals of Probability, 46(6):3351--3398, 2018) is an extreme point of the set of `supermartingale' couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively. Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglböck et al. (The potential of the shadow measure, Electron. Commun. Probab., 27, paper no. 16, 1--12, 2022) obtained in the martingale setting.

math.PR↗

arXiv2019

Utility maximization with proportional transaction costs under model uncertainty

We consider a discrete time financial market with proportional transaction costs under model uncertainty, and study a numéraire-based semi-static utility maximization problem with an exponential utility preference. The randomization techniques recently developed in \cite{BDT17} allow us to transform the original problem into a frictionless counterpart on an enlarged space. By suggesting a different dynamic programming argument than in \cite{bartl2016exponential}, we are able to prove the existence of the optimal strategy and the convex duality theorem in our context with transaction costs. In the frictionless framework, this alternative dynamic programming argument also allows us to generalize the main results in \cite{bartl2016exponential} to a weaker market condition. Moreover, as an application of the duality representation, some basic features of utility indifference prices are investigated in our robust setting with transaction costs.

q-fin.MF↗

arXiv2018

A sparse grid approach to balance sheet risk measurement

In this work, we present a numerical method based on a sparse grid approximation to compute the loss distribution of the balance sheet of a financial or an insurance company. We first describe, in a stylised way, the assets and liabilities dynamics that are used for the numerical estimation of the balance sheet distribution. For the pricing and hedging model, we chose a classical Black & Scholes model with a stochastic interest rate following a Hull & White model. The risk management model describing the evolution of the parameters of the pricing and hedging model is a Gaussian model. The new numerical method is compared with the traditional nested simulation approach. We review the convergence of both methods to estimate the risk indicators under consideration. Finally, we provide numerical results showing that the sparse grid approach is extremely competitive for models with moderate dimension.

q-fin.RM↗

arXiv2017

Super-replication with proportional transaction cost under model uncertainty

We consider a discrete time financial market with proportional transaction cost under model uncertainty, and study a super-replication problem. We recover the duality results that are well known in the classical dominated context. Our key argument consists in using a randomization technique together with the minimax theorem to convert the initial problem to a frictionless problem set on an enlarged space. This allows us to appeal to the techniques and results of Bouchard and Nutz (2015) to obtain the duality result.

math.PR↗

arXiv2017

Robust pricing--hedging duality for American options in discrete time financial markets

We investigate pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, e.g. a family of European options, only statically. In the first part of the paper we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a non-dominated family of probability measures. Our first insight is that by considering a (universal) enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of the original problem. Our second insight is that lack of duality is caused by the lack of dynamic consistency and hence a different enlargement with dynamic consistency is sufficient to recover duality: it is enough to consider (fictitious) extensions of the market in which all the assets are traded dynamically. In the second part of the paper we study two important examples of robust framework: the setup of Bouchard and Nutz (2015) and the martingale optimal transport setup of Beiglböck et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.

math.OC↗