ABSTRACT Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value‐at‐Risk, Expected Shortfall (ES) and Range Value‐at‐Risk being prime examples. They are induced by law‐based risk measures, called their generators, evaluated on the tail distribution. This paper establishes joint identifiability and elicitability results of tail risk measures together with the corresponding quantile, provided that their generators are identifiable and elicitable, respectively. As an example, we establish the joint identifiability and elicitability of the tail expectile together with the quantile. The corresponding consistent scores constitute a novel class of weighted scores, nesting the known class of scores of Fissler and Ziegel for the ES together with the quantile. For statistical purposes, our results pave the way to easier model fitting for tail risk measures via regression and the generalized method of moments, but also model comparison and model validation in terms of established backtesting procedures.

ABSTRACT This paper studies Merton's problem in an extended formulation by incorporating a benchmark tracking on the wealth process. We consider a tracking formulation where the fund manager aims to maximize the trade‐off between the expected utility of consumption and the expected largest shortfall in wealth relative to the benchmark level. Equivalently, the problem can be interpreted as a mixed stochastic control problem if a fictitious capital injection singular control is allowed, subject to the dynamic constraint that the wealth process compensated by the costly capital injection outperforms the benchmark at all times. By considering an auxiliary state process, we formulate an equivalent stochastic control problem with state reflections at zero. For general utility functions and Itô's diffusion benchmark process, we develop a convex duality theorem, new to the literature, for the auxiliary stochastic control problem with state reflections in which the dual process also exhibits reflections from above. For CRRA utility and geometric Brownian motion benchmark process, we further derive the optimal portfolio and consumption in feedback form using the new duality theorem, allowing us to discuss some interesting financial implications induced by the additional risk‐taking from the capital injection and the goal of tracking.

ABSTRACTDoes retail order internalization benefit (via price improvement) or harm (via reduced liquidity) retail traders? To answer this question, we compare two market designs that differ in their mode of liquidity provision: In the setting capturing retail order internalization, liquidity is provided by market makers (wholesalers) competing for the retail order flow in a Bertrand fashion. Instead, in the open exchange setting, price‐taking competitive agents act as liquidity providers. We discover that, when liquidity providers are risk averse, routing of marketable orders to wholesalers is preferred by all retail traders: informed, uninformed, and noise. Furthermore, most measures of liquidity are unaffected by the market design. We also identify a universal parameter that allows comparison of market liquidity, profit and value of information across different markets.

ABSTRACTBuilding on the macroscopic market making framework as a control problem, this paper investigates its extension to stochastic games. In the context of price competition, each agent is benchmarked against the best quote offered by the others. We begin with the linear case. While constructing the solution directly, the ordering property and the dimension reduction in the equilibrium are revealed. For the nonlinear case, we extend the decoupling approach by introducing a multidimensional characteristic equation to analyze the well‐posedness of the forward–backward stochastic differential equations. Properties of the coefficients in this characteristic equation are derived using tools from nonsmooth analysis. Several new well‐posedness results are presented. Finally, we discuss applications to price impacts and the optimal execution problem.

ABSTRACTThis paper considers finitely many investors who perform mean‐variance portfolio selection under relative performance criteria. That is, each investor is concerned about not only her terminal wealth, but how it compares to the average terminal wealth of all investors. At the inter‐personal level, each investor selects a trading strategy in response to others' strategies. This selected strategy additionally needs to yield an equilibrium intra‐personally, so as to resolve time inconsistency among the investor's current and future selves (triggered by the mean‐variance objective). A Nash equilibrium we look for is thus a tuple of trading strategies under which every investor achieves her intra‐personal equilibrium simultaneously. We derive such a Nash equilibrium explicitly in the idealized case of full information (i.e., the dynamics of the underlying stock is perfectly known) and semi‐explicitly in the realistic case of partial information (i.e., the stock evolution is observed, but the expected return of the stock is not precisely known). The formula under partial information consists of the myopic trading and intertemporal hedging terms, both of which depend on an additional state process that serves to filter the true expected return and whose influence on trading is captured by a degenerate Cauchy problem. Our results identify that relative performance criteria can induce downward self‐reinforcement of investors' wealth—if every investor suffers a wealth decline simultaneously, then everyone's wealth tends to decline further. This phenomenon, as numerical examples show, is negligible under full information but pronounced under partial information.

ABSTRACTWe develop a cross‐border market model for two countries based on a continuous trading mechanism, in which the transmission capacities that enable transactions between market participants from different countries are limited. Our market model can be described by a regime‐switching process alternating between active and inactive regimes, in which cross‐border trading is possible, respectively prohibited. Starting from a reduced‐form representation of the two national limit order books, we derive a high‐frequency approximation of the microscopic model, assuming that the size of an individual order converges to zero while the order arrival rate tends to infinity. If transmission capacities are available, the limiting dynamics are as follows: the queue size processes at the top of the two limit order books follow a four‐dimensional linear Brownian motion in the positive orthant with oblique reflection at the axes. Each time the two best ask queues or the two best bid queues simultaneously hit zero, the queue size process is reinitialized. The capacity process can be described as a linear combination of local times and ishence of finite variation. The analytic tractability of the limiting dynamics allows us to compute key quantities of interest.

ABSTRACTThis paper investigates the optimal investment problem in a market with two types of illiquidity: transaction costs and search frictions. We analyze a power‐utility maximization problem where an investor encounters proportional transaction costs and trades only when a Poisson process triggers trading opportunities. We show that the optimal trading strategy is described by a no‐trade region. We introduce a novel asymptotic framework applicable when both transaction costs and search frictions are small. Using this framework, we derive explicit asymptotics for the no‐trade region and the value function along a specific parametric curve. This approach unifies existing asymptotic results for models dealing exclusively with either transaction costs or search frictions.