arXiv subjects

Ulrich Horst

Publications and source records attributed to Ulrich Horst.

At least 19 recordsLinked to original sources

arXiv2026

Multidimensional quadratic BSDEs with weak interactions and their applications in mean-field games of controls

The well-posedness of multidimensional quadratic backward stochastic differential equations (qBSDEs) remains one of the central open problems in BSDE theory. Motivated by a mean-field utility maximization model with price impact, we introduce a new class of multidimensional qBSDEs that lies beyond the scope of existing well-posedness results (see (2.15) for the generator). In order to study the limit as the number of players tends to infinity, we establish existence and uniqueness for a large class of such qBSDEs under suitable smallness conditions imposed on each individual dynamics. A key feature of our approach is that the smallness condition is independent of the dimension of the BSDE system. In particular, the system itself is not confined to a small neighborhood, which allows us to analyze the mean-field limit of the underlying utility maximization problem. Such a condition is also natural in view of the well-known fact that general multidimensional qBSDEs may fail to admit solutions in the absence of suitable smallness assumptions. In addition, we derive a stability result for this class of equations based on the application of Picard iterations. Finally, using this stability result, we establish quantitative convergence rates toward the corresponding mean-field equilibria in two settings: Nash and Radner equilibria.

math.OC↗

arXiv2026

Microstructural Foundation of Rough Log-Normal Volatility Models

We establish a microstructural foundation of the rough Bergomi model. Specifically, we consider a sequence of order driven financial market models where orders to buy or sell an asset arrive according to a Poisson process and have a long lasting impact on volatility. Using a recently established C-tightness result for c\`adl\`ag processes we establish the weak convergence of the price-volatility process to a log-normal rough volatility model. Our weak convergence result is accompanied by weak error rates that employ a recently established Clark-Ocone formula for Poisson processes and turn our microstructure model into viable alternative to classical simulation schemes. The weak error rates strongly hinge on Poisson arrival dynamics and are novel to the rough microstructure literature.

q-fin.MF↗

arXiv2026

Mean-field games with unbounded controls: a weak formulation approach to global solutions

We establish an existence of equilibrium result for a class of non-Markovian mean-field games with unbounded control space in weak formulation. Our result is based on new existence and stability results for quadratic-growth generalized McKean-Vlasov BSDEs. Unlike earlier approaches, our approach does not require boundedness assumptions on the model parameters or time horizons and allows for running costs that are quadratic in the control variable.

math.OC↗

arXiv2026

Mean-field games with rough common noise: the linear-quadratic case

Motivated by mean-field games (MFG) with common noise on the one hand and pathwise stochastic control theory on the other, we formulate here a linear-quadratic (LQ) MFG with rough common noise, along with a satisfactory well-posedness theory for the linear-quadratic case. A novel Volterra-type (or mild) formulation allows to keep technical (rough-stochastic) consideration to a minimum. We derive a characterization of the optimal state and optimal control through a rough forward-backward SDE (rough FBSDE), and provide an existence and uniqueness result under the usual assumptions. Our theory is accompanied by stability estimates with respect to initial data and common noise while we also establish continuity of what we call the It\^o-Lions-Lyons map for rough mean-field games. Finally, we discuss randomization of the rough common noise under appropriate conditions on the coefficients. When the latter is given by the Stratonovich lift of a Brownian motion independent of the idiosyncratic noise, we show that solutions of the rough LQ MFG coincide with those obtained by conditioning on the common noise.

math.PR↗

arXiv2025

Mean Field Portfolio Games with Epstein-Zin Preferences

We study mean field portfolio games under Epstein-Zin preferences, which naturally encompass the classical time-additive power utility as a special case. In a general non-Markovian framework, we establish a uniqueness result by proving a one-to-one correspondence between Nash equilibria and the solutions to a class of BSDEs. A key ingredient in our approach is a necessary local stochastic maximum principle, applied to log-wealth, tailored to Epstein-Zin utility, and a nonlinear transformation. In the deterministic setting, we further derive an explicit closed-form solution for equilibrium investment and consumption policies. The strength of our approach is further illustrated by two special cases: (i) in the power utility setting without consumption, we obtain the same one-to-one correspondence as in Fu and Zhou [22] under exactly the same assumption, but without invoking the dynamic programming principle in Espinosa and Touzi [16]; and (ii) in the power utility setting with both investment and consumption, we strengthen the correspondence result of Fu [18], by proving a genuine one-to-one relation in the BMO space, where both the equilibrium strategy and the associated BSDE components belong to BMO.

q-fin.MF↗

arXiv2025

Pontryagin Maximum Principle for rough stochastic systems and pathwise stochastic control

We analyze a novel class of rough stochastic control problems that allows for a convenient approach to solving pathwise stochastic control problems with both non-anticipative and anticipative controls. We first establish the well-posedness of a class of controlled rough SDEs with affine rough driver and establish the continuity of the solution w.r.t.~the driving rough path. This allows us to define pathwise stochastic control problems with anticipative controls. Subsequently, we apply a flow transformation argument to establish a necessary and sufficient maximum principle to identify and characterize optimal strategies for rough and hence pathwise stochastic control problems. We show that the rough and the corresponding pathwise stochastic control problems share the same value function. For the benchmark case of linear-quadratic problems with bounded controls a similar result is shown for optimal controls.

math.OC↗

arXiv2024

Path-dependent Fractional Volterra Equations and the Microstructure of Rough Volatility Models driven by Poisson Random Measures

We consider a microstructure foundation for rough volatility models driven by Poisson random measures. In our model the volatility is driven by self-exciting arrivals of market orders as well as self-exciting arrivals of limit orders and cancellations. The impact of market order on future order arrivals is captured by a Hawkes kernel with power law decay, and is hence persistent. The impact of limit orders on future order arrivals is temporary, yet possibly long-lived. After suitable scaling the volatility process converges to a fractional Heston model driven by an additional Poisson random measure. The random measure generates occasional spikes and clusters of spikes in the volatility process. Our results are based on novel existence and uniqueness of solutions results for stochastic path-dependent Volterra equations driven by Poisson random measures.

math.PR↗

arXiv2024

A Mean-Field Game of Market Entry: Portfolio Liquidation with Trading Constraints

We consider both $N$-player and mean-field games of optimal portfolio liquidation in which the players are not allowed to change the direction of trading. Players with an initially short position of stocks are only allowed to buy while players with an initially long position are only allowed to sell the stock. Under suitable conditions on the model parameters we show that the games are equivalent to games of timing where the players need to determine the optimal times of market entry and exit. We identify the equilibrium entry and exit times and prove that equilibrium mean-trading rates can be characterized in terms of the solutions to a highly non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium both in the mean-field and the $N$-player game.

q-fin.MF↗

arXiv2024

Extended mean-field games with multi-dimensional singular controls and non-linear jump impact

We establish a probabilistic framework for analysing extended mean-field games with multi-dimensional singular controls and state-dependent jump dynamics and costs. Two key challenges arise when analysing such games: the state dynamics may not depend continuously on the control and the reward function may not be u.s.c.~Both problems can be overcome by restricting the set of admissible singular controls to controls that can be approximated by continuous ones. We prove that the corresponding set of admissible weak controls is given by the weak solutions to a Marcus-type SDE and provide an explicit characterisation of the reward function. The reward function will in general only be u.s.c.~To address the lack of continuity we introduce a novel class of MFGs with a broader set of admissible controls, called MFGs of parametrisations. Parametrisations are laws of state/control processes that continuously interpolate jumps. We prove that the reward functional is continuous on the set of parametrisations, establish the existence of equilibria in MFGs of parametrisations, and show that the set of Nash equilibria in MFGs of parametrisations and in the underlying MFG with singular controls coincide. This shows that MFGs of parametrisations provide a canonical framework for analysing MFGs with singular controls and non-linear jump impact.

math.OC↗

arXiv2024

Functional Limit Theorems for Hawkes Processes

We prove that the long-run behavior of Hawkes processes is fully determined by the average number and the dispersion of child events. For subcritical processes we provide FLLNs and FCLTs under minimal conditions on the kernel of the process with the precise form of the limit theorems depending strongly on the dispersion of child events. For a critical Hawkes process with weakly dispersed child events, functional central limit theorems do not hold. Instead, we prove that the rescaled intensity processes and rescaled Hawkes processes behave like CIR-processes without mean-reversion, respectively integrated CIR-processes. We provide the rate of convergence by establishing an upper bound on the Wasserstein distance between the distributions of rescaled Hawkes process and the corresponding limit process. By contrast, critical Hawkes process with heavily dispersed child events share many properties of subcritical ones. In particular, functional limit theorems hold. However, unlike subcritical processes critical ones with heavily dispersed child events display long-range dependencies.

math.PR↗

arXiv2023

Convergence of Heavy-Tailed Hawkes Processes and the Microstructure of Rough Volatility

We establish the weak convergence of the intensity of a nearly-unstable Hawkes process with heavy-tailed kernel. Our result is used to derive a scaling limit for a financial market model where orders to buy or sell an asset arrive according to a Hawkes process with power-law kernel. After suitable rescaling the price-volatility process converges weakly to a rough Heston model. Our convergence result is stronger than previously established ones that have either focused on light-tailed kernels or the convergence of integrated volatility process. The key is to establish the tightness of the family of rescaled volatility processes. This is achieved by introducing a new methods to establish the $C$-tightness of c\`adl\`ag processes based on the classical Kolmogorov-Chentsov tightness criterion for continuous processes.

q-fin.MF↗

arXiv2023

Second-Order Regular Variation and Second-Order Approximation of Hawkes Processes

This paper provides and extends second-order versions of several fundamental theorems on first-order regularly varying functions such as Karamata's theorem/representation and Tauberian's theorem. Our results are used to establish second-order approximations for the mean and variance of Hawkes processes with general kernels. Our approximations provide novel insights into the asymptotic behavior of Hawkes processes. They are also of key importance when establishing functional limit theorems for Hawkes processes.

math.PR↗

arXiv2023

A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategies

We consider a mean-field control problem with càdlàg semimartingale strategies arising in portfolio liquidation models with transient market impact and self-exciting order flow. We show that the value function depends on the state process only through its law, and that it is of linear-quadratic form and that its coefficients satisfy a coupled system of non-standard Riccati-type equations. The Riccati equations are obtained heuristically by passing to the continuous-time limit from a sequence of discrete-time models. A sophisticated transformation shows that the system can be brought into standard Riccati form from which we deduce the existence of a global solution. Our analysis shows that the optimal strategy jumps only at the beginning and the end of the trading period.

q-fin.MF↗

arXiv2023

Mean-Field Liquidation Games with Market Drop-out

We consider a novel class of portfolio liquidation games with market drop-out ("absorption"). More precisely, we consider mean-field and finite player liquidation games where a player drops out of the market when her position hits zero. In particular round-trips are not admissible. This can be viewed as a no statistical arbitrage condition. In a model with only sellers we prove that the absorption condition is equivalent to a short selling constraint. We prove that equilibria (both in the mean-field and the finite player game) are given as solutions to a non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium in the MFG and the existence of a unique equilibrium in the $N$-player game. We establish the convergence of the equilibria in the finite player games to the obtained mean-field equilibrium and illustrate the impact of the drop-out constraint on equilibrium trading rates.

q-fin.MF↗

arXiv2023

Mean-field control problems with multi-dimensional singular controls

We consider extended mean-field control problems with multi-dimensional singular controls. A key challenge when analysing singular controls are jump costs. When controls are one-dimensional, jump costs are most naturally computed by linear interpolation. When the controls are multi-dimensional the situation is more complex, especially when the model parameters depend on an additional mean-field interaction term, in which case one needs to "jointly" and "consistently" interpolate jumps both on a distributional and a pathwise level. This is achieved by introducing the novel concept of two-layer parametrisations of stochastic processes. Two-layer parametrisations allow us to equivalently rewrite rewards in terms of continuous functions of parametrisations of the control process and to derive an explicit representation of rewards in terms of minimal jump costs. From this we derive a DPP for extended mean-field control problems with multi-dimensional singular controls. Under the additional assumption that the value function is continuous we characterise the value function as the minimal super-solution to a certain quasi-variational inequality in the Wasserstein space.

q-fin.MF↗

arXiv2023

Second-Order Approximation of Limit Order Books in a Single-Scale Regime

We establish a first- and second-order approximation for an infinite dimensional limit order book model in a single (critical) scaling regime where market and limit orders arrive at a common time scale. With our choice of scaling we obtain non-degenerate first- and second-order approximations for the price and volume dynamics. While the first-order approximation is given by a coupled ODE-PDE system, the second-order approximation is described in terms of an infinite-dimensional stochastic evolution equation driven by a cylindrical Brownian motion. The driving noise processes exhibit a non-trivial correlation in terms of the model parameters. We prove that the evolution equation has a unique solution and that the sequence of standardized limit order book models converges weakly to the solution of the evolution equation. The proof uses a non-standard martingale problem. We calibrate a linearized model to market data and explain how our model can be used for deriving confidence intervals of portfolio liquidation values.

q-fin.MF↗

arXiv2023

Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies

We consider an optimal liquidation problem with instantaneous price impact and stochastic resilience for small instantaneous impact factors. Within our modelling framework, the optimal portfolio process converges to the solution of an optimal liquidation problem with general semimartingale controls when the instantaneous impact factor converges to zero. Our results provide a unified framework within which to embed the two most commonly used modelling frameworks in the liquidation literature and provide a microscopic foundation for the use of semimartingale liquidation strategies and the use of portfolio processes of unbounded variation. Our convergence results are based on novel convergence results for BSDEs with singular terminal conditions and novel representation results of BSDEs in terms of uniformly continuous functions of forward processes.

q-fin.MF↗

arXiv2021

A Maximum Principle approach to deterministic Mean Field Games of Control with Absorption

We study a class of deterministic mean field games on finite and infinite time horizons arising in models of optimal exploitation of exhaustible resources. The main characteristic of our game is an absorption constraint on the players' state process. As a result of the state constraint the optimal time of absorption becomes part of the equilibrium. This requires a novel approach when applying Pontyagin's maximum principle. We prove the existence and uniqueness of equilibria and solve the infinite horizon models in closed form. As players may drop out of the game over time, equilibrium production rates need not be monotone nor smooth.

econ.GN↗