## Publications

17 results found

Mueller C, Neumann E, 2023, The effective radius of self repelling elastic manifolds, *Annals of Applied Probability*, Vol: 33, Pages: 5668-5692, ISSN: 1050-5164

We study elastic manifolds with self-repelling terms and estimate theireffective radius. This class of manifolds is modelled by a self-repellingvector-valued Gaussian free field with Neumann boundary conditions overthe domain [−N,N]d ∩ Zd , that takes values in Rd . Our main result statesthat in two dimensions (d = 2), the effective radius RN of the manifold isapproximately N. This verifies the conjecture of Kantor, Kardar and Nelson(Phys. Rev. Lett. 58 (1987) 1289–1292) up to a logarithmic correction. Our results in d ≥ 3 give a similar lower bound on RN and an upper of order Nd/2.This result implies that self-repelling elastic manifolds undergo a substantialstretching at any dimension.

Micheli A, Muhle-Karbe J, Neumann E, 2023, Closed-loop nash competition for liquidity, *Mathematical Finance*, Vol: 33, Pages: 1082-1118, ISSN: 0960-1627

We study a multiplayer stochastic differential game, where agents interact through their joint price impact on an asset that they trade to exploit a common trading signal. In this context, we prove that a closed-loop Nash equilibrium exists if the price impact parameter is small enough. Compared to the corresponding open-loop Nash equilibrium, both the agents' optimal trading rates and their performance move towards the central-planner solution, in that excessive trading due to lack of coordination is reduced. However, the size of this effect is modest for plausible parameter values.

Neumann E, Voss M, 2023, Trading with the crowd, *Mathematical Finance*, Vol: 33, Pages: 548-617, ISSN: 0960-1627

We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact, along with taking into account a common general price predicting signal. The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal to the aggregated transient price impact induced by all other agents. This unfolds a quantitative relation between trading signals and the order flow in crowded markets. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents. We prove that the equilibrium trading speed and the value function of an agent in the finite N-player game converges to the corresponding trading speed and value function in the mean field game at rate O(N⁻²). In addition, we prove that the mean field optimal strategy provides an approximate Nash-equilibrium for the finite-player game.

Mueller C, Neumann E, 2022, Scaling properties of a moving polymer, *Annals of Applied Probability*, Vol: 32, Pages: 4251-4278, ISSN: 1050-5164

We set up an SPDE model for a moving, weakly self-avoiding polymer with intrinsic length J taking values in (0, ∞). Our main result states that the effective radius of the polymer is approximately J5/3; evidently for large J the polymer undergoes stretching. This contrasts with the equilibrium situation without the time variable, where many earlier results show that the effective radius is approximately J.For such a moving polymer taking values in R2, we offer a conjecture that the effective radius is approximately J5/4.

Hager P, Neumann E, 2022, The multiplicative chaos of H=0 fractional Brownian fields, *Annals of Applied Probability*, Vol: 32, Pages: 2139-2179, ISSN: 1050-5164

We consider a family of fractional Brownian fields {BH}H∈(0,1) on Rd, whereH denotes their Hurst parameter. We first define a rich class of normalizingkernels ψ and we rescale the normalised field by the square-root of the gammafunction Γ(H), such that the covariance ofXH(x) = Γ(H)12 BH(x) −ZRdBH(u)ψ(u, x)du ,converges to the covariance of a log-correlated Gaussian field when H ↓ 0.We then use Berestycki’s “good points” approach [11] in order to derive theconvergence of the exponential measure of the fractional Brownian fieldMHγ(dx) = eγXH(x)−γ22E[XH(x)2]dx,towards a Gaussian multiplicative chaos, as H ↓ 0 for all γ ∈ (0, γ∗(d)], whereγ∗(d) >q74d. As a corollary we establish the L2convergence of MHγ over thesets of “good points”, where the field XH has a typical behaviour. As a byproduct of the convergence result, we prove that for log-normal rough volatilitymodels with small Hurst parameter, the volatility process is supported on thesets of “good points” with probability close to 1. Moreover, on these sets thevolatility converges in L2to the volatility of multifractal random walks.

Neumann E, Moritz V, 2022, Optimal signal-adaptive trading with temporary and transient price impact, *SIAM Journal on Financial Mathematics*, Vol: 13, Pages: 551-575, ISSN: 1945-497X

We study optimal liquidation in the presence of linear temporary and transient price impact along with taking into account a general price predictingfinite-variation signal. We formulate this problem as minimization of a cost-riskfunctional over a class of absolutely continuous and signal-adaptive strategies.The stochastic control problem is solved by following a probabilistic and convexanalytic approach. We show that the optimal trading strategy is given by asystem of four coupled forward-backward SDEs, which can be solved explicitly.Our results reveal how the induced transient price distortion provides togetherwith the predictive signal an additional predictor about future price changes.As a consequence, the optimal signal-adaptive trading rate trades off exploitingthe predictive signal against incurring the transient displacement of the execution price from its unaffected level. This answers an open question from Lehalleand Neuman [29] as we show how to derive the unique optimal signal-adaptiveliquidation strategy when price impact is not only temporary but also transient.

Neumann E, Schied A, 2022, Protecting pegged currency markets from speculative investors, *Mathematical Finance*, Vol: 32, Pages: 405-420, ISSN: 0960-1627

We consider a stochastic game between a trader and a central bank in a target zone market with a lower currency peg. This currency peg is maintained by the central bank through the generation of permanent price impact, thereby aggregating an ever-increasing risky position in foreign reserves. We describe this situation mathematically by means of two coupled singular control problems, where the common singularity arises from a local time along a random curve. Our first result identifies a certain local time as that central bank strategy for which this risk position is minimized. We then consider the worst-case situation the central bank may face by identifying that strategy of the strategic investor that maximizes the expected inventory of the central bank under a cost criterion, thus establishing a Stackelberg equilibrium in our model.

Brigo D, Graceffa F, Neumann E, 2021, Price impact on term structure, *Quantitative Finance*, Vol: 22, Pages: 171-195, ISSN: 1469-7688

We introduce a rst theory of price impact in presence of an interest-rates termstructure. We explain how one can formulate instantaneous and transient price impacton zero-coupon bonds with di erent maturities, including a cross price impact that isendogenous to the term structure. We connect the introduced impact to classic noarbitrage theory for interest rate markets, showing that impact can be embedded in thepricing measure and that no-arbitrage can be preserved. We extend the price impactsetup to coupon-bearing bonds and further show how to implement price impact in aHJM framework. We present pricing examples in presence of price impact and numericalexamples of how impact changes the shape of the term structure. Finally, we show thatour approach is applicable by solving an optimal execution problem in interest ratemarkets with the type of price impact we developed in the paper.

Micheli A, Muhle-Karbe J, Neuman E, 2021, Closed-Loop Nash Competition for Liquidity, Publisher: Elsevier BV

Bellani C, Brigo D, Done A,
et al., 2021, Optimal trading: the importance of being adaptive, *International Journal of Financial Engineering*, Vol: 8, ISSN: 2424-7863

We compare optimal static and dynamic solutions in trade execution. An optimal trade execution problem is considered where a trader is looking at a short-term price predictive signal while trading. When the trader creates an instantaneous market impact, it is shown that transaction costs of optimal adaptive strategies are substantially lower than the corresponding costs of the optimal static strategy. In the same spirit, in the case of transient impact, it is shown that strategies that observe the signal a finite number of times can dramatically reduce the transaction costs and improve the performance of the optimal static strategy.

Neumann E, Xinghua Z, 2021, On the maximal displacement of near-critical branching random walks, *Probability Theory and Related Fields*, Vol: 180, Pages: 199-232, ISSN: 0178-8051

We consider a branching random walk on Z started by n particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring 1+θ/n. For t≥0, we study Mnt, the rightmost position reached by the branching random walk up to generation [nt]. Under certain moment assumptions on the branching law, we prove that Mnt/n−−√ converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of Mnt. We also confirm that when θ>0, the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky (Ann Probab 23(4):1748–1754, 1995). The rightmost position over all generations, M:=suptMnt, is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Gumbel distribution when θ<0.

Neumann E, Mueller C, Lee JJ, 2020, Hitting probabilities of a Brownian flow with Radial Drift, *Annals of Probability*, Vol: 48, Pages: 646-671, ISSN: 0091-1798

We consider a stochastic flowφt(x,ω) inRnwith ini-tial pointφ0(x,ω) =x, driven by a singlen-dimensional Brownianmotion, and with an outward radial drift of magnitudeF(‖φt(x)‖)‖φt(x)‖,withFnonnegative, bounded and Lipschitz. We consider initialpointsxlying in a set of positive distance from the origin. Weshow that there exist constantsC∗,c∗>0 not depending onn,such that ifF > C∗nthen the image of the initial set under theflow has probability 0 of hitting the origin. If 0≤F≤c∗n3/4, andif the initial set has nonempty interior, then the image of the sethas positive probability of hitting the origin.

Lehalle C-A, Neuman E, 2019, Incorporating signals into optimal trading, *Finance and Stochastics*, ISSN: 1432-1122

Optimal trading is a recent field of research which was initiated by Almgren, Chriss, Bertsimas and Lo in the late 90's. Its main application is slicing large trading orders, in the interest of minimizing trading costs and potential perturbations of price dynamics due to liquidity shocks. The initial optimization frameworks were based on mean-variance minimization for the trading costs. In the past 15 years, finer modelling of price dynamics, more realistic control variables and different cost functionals were developed. The inclusion of signals (i.e. short term predictors of price dynamics) in optimal trading is a recent development and it is also the subject of this work.We incorporate a Markovian signal in the optimal trading framework which was initially proposed by Gatheral, Schied, and Slynko [21] and provide results on the existence and uniqueness of an optimal trading strategy. Moreover, we derive an explicit singular optimal strategy for the special case of an Ornstein-Uhlenbeck signal and an exponentially decaying transient market impact. The combination of a mean-reverting signal along with a market impact decay is of special interest, since they affect the short term price variations in opposite directions.Later, we show that in the asymptotic limit were the transient market impact becomes instantaneous, the optimal strategy becomes continuous. This result is compatible with the optimal trading framework which was proposed by Cartea and Jaimungal [10].In order to support our models, we analyse nine months of tick by tick data on 13 European stocks from the NASDAQ OMX exchange. We show that orderbook imbalance is a predictor of the future price move and it has some mean-reverting properties. From this data we show that market participants, especially high frequency traders, use this signal in their trading strategies.

Neuman E, 2018, Pathwise uniqueness of the stochastic heat equation with spatially inhomogeneous white noise, *The Annals of Probability*, Vol: 46, Pages: 3090-3187, ISSN: 0091-1798

We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is Hölder continuous of index γ>1−η2(η+1). Here η∈(0,1) is a constant that defines the spatial regularity of the noise.

Neuman E, Zheng X, 2017, On the maximal displacement of subcritical branching random walks, *Probability Theory and Related Fields*, Vol: 167, Pages: 1137-1164, ISSN: 0178-8051

We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each ∈ℕ, let be the rightmost position reached by the branching random walk up to generation n. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists >1 such that the function ( , ):= ( ≥ ),for each >0 and ∈ℕ,satisfies the following properties: there exist 0< ⎯⎯≤ ⎯⎯⎯<∞ such that if < ⎯⎯, then0<liminf →∞ ( , )≤limsup →∞ ( , )≤1,while if > ⎯⎯⎯, thenlim →∞ ( , )=0.Moreover, if the jump distribution has a finite right range R, then ⎯⎯⎯< . If furthermore the jump distribution is “nearly right-continuous”, then there exists ∈(0,1] such that lim →∞ ( , )= for all < ⎯⎯. We also show that the tail distribution of :=sup ≥0 , namely, the rightmost position ever reached by the branching random walk, has a similar exponential decay (without the cutoff at ⎯⎯). Finally, by duality, these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior.

Neuman E, Schied A, 2016, Optimal portfolio liquidation in target zone models and catalytic superprocesses, *Finance and Stochastics*, Vol: 20, Pages: 495-509, ISSN: 0949-2984

We study optimal buying and selling strategies in target zone models. In these models, the price is modelled by a diffusion process which is reflected at one or more barriers. Such models arise, for example, when a currency exchange rate is kept above a certain threshold due to central bank interventions. We consider the optimal portfolio liquidation problem for an investor for whom prices are optimal at the barrier and who creates temporary price impact. This problem is formulated as the minimization of a cost–risk functional over strategies that only trade when the price process is located at the barrier. We solve the corresponding singular stochastic control problem by means of a scaling limit of critical branching particle systems, which is known as a catalytic superprocess. In this setting, the catalyst is given by the barriers of the price process. For the cases in which the unaffected price process is a reflected arithmetic or geometric Brownian motion with drift, we moreover give a detailed financial justification of our cost functional by means of an approximation with discrete-time models.

Mytnik L, Neuman E, 2015, Pathwise uniqueness for the stochastic heat equation with Holder continuous drift and noise coefficients, *STOCHASTIC PROCESSES AND THEIR APPLICATIONS*, Vol: 125, Pages: 3355-3372, ISSN: 0304-4149

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