Prof. Dr. Dörte Kreher

Profil

• EXC 2046 AG Kreher AA4-4

  Quelle ↗

  Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 01/2019 - 04/2022 Projektleitung: Prof. Dr. Dörte Kreher

• EXC 2046: Berlin Mathematics Research Center (MATH+)

  Quelle ↗

  Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 01/2019 - 12/2024 Projektleitung: Prof. Dr. Caren Tischendorf, Prof. Dr. Michael Hintermüller, Prof. Dr. Max Klimm, Prof. Dr. Dörte Kreher, Prof. Chris Wendl, Prof. Dr. Bettina Rösken-Winter, Prof. Dr. rer. nat. Dr. h.c. Edda Klipp

• IGRK 2544/1: Stochastische Analysis in Interaktion

  Quelle ↗

  Förderer: DFG Graduiertenkolleg Zeitraum: 04/2020 - 09/2024 Projektleitung: Prof. Dr. Ulrich Horst, Prof. Dr. Markus Reiß, Prof. Dr. Dirk Becherer, Prof. Dr. Dörte Kreher

• IGRK 2544: Stochastische Analysis in Interaktion

  Quelle ↗

  Förderer: DFG Graduiertenkolleg Zeitraum: 04/2020 - 03/2029 Projektleitung: Prof. Dr. Peter Bank, Terry Lyons Ph.D.

• SFB/TRR 388/1: „Mikrostrukturelle Grundlagen rauer Volatilitätsmodelle“ (TP B02)

  Quelle ↗

  Förderer: DFG Sonderforschungsbereich Zeitraum: 10/2024 - 06/2028 Projektleitung: Prof. Dr. Ulrich Horst, Dr. Christian Bayer, Prof. Dr. Dörte Kreher

• RISC Software GmbH

  PT

  7 Treffer57.2%
  • EU: Scattering Amplitudes: From Geometry to EXperiment (SAGEX)
    P

    57.2%

• Science & Startups Berlin

  T

  5 Treffer55.5%
  • Zuwendung im Rahmen des Programms „exist – Existenzgründungen aus der Wissenschaft“ aus dem Bundeshaushalt, Einzelplan 09, Kapitel 02, Titel 68607, Haushaltsjahr 2026, sowie aus Mitteln des Europäischen Strukturfonds (hier Euro-päischer Sozialfonds Plus – ESF Plus) Förderperiode 2021-2027 – Kofinanzierung für das Vorhaben: „exist Women“
    T

    55.5%

• space application services

  PT

  7 Treffer52.4%
  • INTeractive RObotics Research Network
    P

    52.4%

• ROBOSOFT Services Robots

  PT

  8 Treffer52.4%
  • INTeractive RObotics Research Network
    P

    52.4%

• Dycotec Materials Limited

  P

  8 Treffer52.3%
  • EU: HIgh ACCuracy printed electronics to <1μm, for OLAE TFT and Display Applications (HI-ACCURACY)
    P

    52.3%

• Neudrive Limited Leading oft Technology

  P

  8 Treffer52.3%
  • EU: HIgh ACCuracy printed electronics to <1μm, for OLAE TFT and Display Applications (HI-ACCURACY)
    P

    52.3%

• Centro Ricerche Fiat SCPA

  P

  8 Treffer52.3%
  • EU: HIgh ACCuracy printed electronics to <1μm, for OLAE TFT and Display Applications (HI-ACCURACY)
    P

    52.3%

• Teknologian tutkimuskeskus

  P

  8 Treffer52.3%
  • EU: HIgh ACCuracy printed electronics to <1μm, for OLAE TFT and Display Applications (HI-ACCURACY)
    P

    52.3%

• Precision Varionic International Ltd

  P

  8 Treffer52.3%
  • EU: HIgh ACCuracy printed electronics to <1μm, for OLAE TFT and Display Applications (HI-ACCURACY)
    P

    52.3%

• Gesellschaft für angewandte Mathematik und Mechanik

  PT

  19 Treffer52.1%
  • Workshop Reliable Methods and Mathematical Modeling
    P

    52.1%

• Strict local martingales and bubbles

  2015

  The Annals of Applied Probability · 45 Zitationen · DOI

  This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the “default term” apparent in risk-neutral option prices if the underlying stock exhibits a bubble modeled by a strict local martingale. Results for certain path dependent options and last passage time formulas are given.

• A diffusion approximation for limit order book models

  2018

  Stochastic Processes and their Applications · 6 Zitationen · DOI

• A Weak Law of Large Numbers for a Limit Order Book Model with Fully State Dependent Order Dynamics

  2017

  SIAM Journal on Financial Mathematics · 6 Zitationen · DOI

  This paper studies a limit order book model, in which the order dynamics depend on both, the current best available prices and the current volume density functions. For the joint dynamics of the best bid price, the best ask price, and the standing volume densities on both sides of the limit order book we derive a weak law of large numbers, which states that the limit order book model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the scaling limit the two volume densities each follow a nonlinear PDE coupled with two nonlinear ODEs that describe the best bid and ask price.

• A new kind of augmentation of filtrations suitable for a change of probability measure by a strict local martingale

  2015

  Statistics & Probability Letters · 5 Zitationen · DOI

• Red noise in continuous-time stochastic modelling

  2025

  Royal Society Open Science · 3 Zitationen · DOI

  The concept of time-correlated noise is important to applied stochastic modelling. Nevertheless, there is no generally agreed-upon definition of the term red noise in continuous-time stochastic modelling settings. We present here a rigorous argumentation for the Ornstein-Uhlenbeck process integrated against time ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>U</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mrow><mml:mi>d</mml:mi></mml:mrow> <mml:mi>t</mml:mi></mml:mrow> </mml:math> ) as a uniquely appropriate red noise implementation. We also identify the term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow> <mml:msub><mml:mi>U</mml:mi> <mml:mi>t</mml:mi></mml:msub> </mml:mrow> </mml:math> as an erroneous formulation of red noise commonly found in the applied literature. To this end, we prove a theorem linking properties of the power spectral density (PSD) to classes of Itô-differentials. The commonly ascribed red noise attribute of a PSD decaying as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> <mml:mo>∼</mml:mo></mml:mrow> <mml:mrow><mml:msup><mml:mi>ω</mml:mi> <mml:mrow><mml:mo>-</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:msup> </mml:mrow> </mml:math> restricts the range of possible Itô-differentials <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow> <mml:msub><mml:mi>Y</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mo>=</mml:mo></mml:mrow> <mml:mrow><mml:msub><mml:mi>α</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mrow><mml:mi>d</mml:mi></mml:mrow> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo></mml:mrow> <mml:mrow><mml:msub><mml:mi>β</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mrow><mml:mi>d</mml:mi></mml:mrow> <mml:msub><mml:mi>W</mml:mi> <mml:mi>t</mml:mi></mml:msub> </mml:mrow> </mml:math> . In particular, any such differential with continuous, square-integrable integrands must have a vanishing martingale part, i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow> <mml:msub><mml:mi>Y</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mo>=</mml:mo></mml:mrow> <mml:mrow><mml:msub><mml:mi>α</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:mrow><mml:mi>d</mml:mi></mml:mrow> <mml:mi>t</mml:mi></mml:mrow> </mml:math> for almost all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi> <mml:mo>≥</mml:mo></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:math> . We further point out that taking <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>α</mml:mi> <mml:mi>t</mml:mi></mml:msub> <mml:msub><mml:mo>)</mml:mo> <mml:mrow><mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn></mml:mrow> </mml:msub> </mml:mrow> </mml:math> to be an Ornstein-Uhlenbeck process constitutes a uniquely relevant model choice due to its Gauss-Markov property. The erroneous use of the noise term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow> <mml:msub><mml:mi>U</mml:mi> <mml:mi>t</mml:mi></mml:msub> </mml:mrow> </mml:math> as red noise and its consequences are discussed in two examples from the literature.

• Jump Diffusion Approximation for the Price Dynamics of a Fully State Dependent Limit Order Book Model

  2023

  SIAM Journal on Financial Mathematics · 3 Zitationen · DOI

  We study a microscopic limit order book model, in which the order dynamics depend on the current best bid and ask price and the current volume density functions, simultaneously, and derive its macroscopic high-frequency dynamics. As opposed to the existing literature on scaling limits for limit order book models, we include price changes which do not scale with the tick size in our model to account for large price movement, for example, being triggered by highly unforeseen events. Our main result states that, when the size of an individual limit order and the tick size tend to zero, while the order arrival rate tends to infinity, the microscopic limit order book model dynamics converge to two one dimensional jump diffusion processes describing the prices coupled with two infinite dimensional fluid processes describing the standing volumes at the buy and sell side.

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

  2023

  arXiv (Cornell University) · 1 Zitationen · DOI

  We establish a first and second-order approximation for an infinite dimensional limit order book model (LOB) 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-order and second-order approximations for the price and volume dynamics. While the first-order approximation is given by a standard coupled ODE-PDE system, the second-order approximation is non-standard and 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 LOB models converges weakly to the solution of the evolution equation. The proof uses a non-standard martingale problem. We calibrate a simplified version of our model to market data and show that the model accurately captures correlations between price and volume fluctuations.

• Red noise in continuous-time stochastic modelling

  2022

  arXiv (Cornell University) · 1 Zitationen · DOI

  The concept of time-correlated noise is important to applied stochastic modelling. Nevertheless, there is no generally agreed-upon definition of the term red noise in continuous-time stochastic modelling settings. We present here a rigorous argumentation for the Ornstein-Uhlenbeck process integrated against time ($U_t \mathrm{d} t$) as a uniquely appropriate red noise implementation. We also identify the term $\mathrm{d}U_t$ as an erroneous formulation of red noise commonly found in the applied literature. To this end, we prove a theorem linking properties of the power spectral density (PSD) to classes of Itô-differentials. The commonly ascribed red noise attribute of a PSD decaying as $S(ω)\simω^{-2}$ restricts the range of possible Itô-differentials $\mathrm{d}Y_t=α_t\mathrm{d} t+β_t\mathrm{d} W_t$. In particular, any such differential with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $\mathrm{d}Y_t=α_t\mathrm{d} t$ for almost all $t\geq 0$. We further point out that taking $(α_t)_{t\geq 0}$ to be an Ornstein-Uhlenbeck process constitutes a uniquely relevant model choice due to its Gauss-Markov property. The erroneous use of the noise term $\mathrm{d} U_t$ as red noise and its consequences are discussed in two examples from the literature.

• Change of measure up to a random time: Details

  2016

  Stochastic Processes and their Applications · 1 Zitationen · DOI

• Strict Local Martingales, Random Times and Non-Standard Changes of Probability Measure in Financial Mathematics

  2014

  Zurich Open Repository and Archive (University of Zurich) · 1 Zitationen · DOI

  Die vorliegende Arbeit befasst sich mit nicht gewhnlichen Wechseln von Wahrscheinlichkeitsmaen in Verbindung mit strikt lokalen Martingalen oder beliebigen Zufallszeiten, die durch Probleme in der Finanzmathematik motiviert sind.Zuerst untersuchen wir Finanzblasen in Aktienkursen, die durch strikt lokale Martingale modelliert werden.Um den Einfluss von Finanzblasen auf die Bewertung von Derivaten zu bestimmen, konstruieren wir zunchst ein neues Wahrscheinlichkeitsma mit Hilfe eines strikt positiven, strikt lokalen Martingales mit cdlg Pfaden.Anschlieend leiten wir Zerlegungsformeln her fr die Preise bestimmter Klassen von pfadabhngigen europischen Optionen und Last-Passage-Time-Formeln fr europische und amerikanische Tauschoptionen, wenn die Basiswerte der Optionen Finanzblasen aufweisen.Auerdem schlagen wir eine neue Art der Filtrationsvervollstndigung entlang einer Folge von Stoppzeiten vor, die fr Mawechsel mit Hilfe von strikt lokalen Martingalen geeignet ist.Danach untersuchen wir Mawechsel

• A Cross‐Border Market Model with Limited Transmission Capacities

  2025

  Mathematical Finance · DOI

  ABSTRACT We 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.

• Author response for "Red noise in continuous-time stochastic modelling"

  2025

  DOI

• A Cross‐Border Market Model with Limited Transmission Capacities

  2025

  edoc Publication server (Humboldt University of Berlin) · DOI

  We 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.

• Non-negative Martingale Solutions to the Stochastic Porous Medium Equation with Sticky Behavior

  2024

  arXiv (Cornell University) · DOI

  We construct non-negative martingale solutions to the stochastic porous medium equation in one dimension with homogeneous Dirichlet boundary conditions which exhibit a type of sticky behavior at zero. The construction uses the stochastic Faedo--Galerkin method via spatial semidiscretization, so that the pre-limiting system is given by a finite-dimensional diffusion with Wentzell boundary condition. We derive uniform moment estimates for the discrete systems by an Aubin-Lions-type interpolation argument, which enables us to implement a general weak convergence approach for the construction of martingale solutions of an SPDE using a Skorokhod representation-type result for non-metrizable spaces. We rely on a stochastic argument based on the occupation time formula for continuous semimartingales for the identification of the diffusion coefficient in the presence of an indicator function.

• Bubbles in discrete-time models

  2022

  Finance and Stochastics · DOI

  Abstract We introduce a new definition of bubbles in discrete-time models based on the discounted stock price losing mass under an equivalent martingale measure at some finite drawdown. We provide equivalent probabilistic characterisations of this definition and give examples of discrete-time martingales that are bubbles and others that are not. In the Markovian case, we provide sufficient analytic conditions for the presence of bubbles. We also show that the existence of bubbles is directly linked to the existence of a non-trivial solution to a linear Volterra integral equation of the second kind involving the Markov kernel. Finally, we show that our definition of bubbles in discrete time is consistent with the strict local martingale definition of bubbles in continuous time in the sense that a properly discretised strict local martingale in continuous time is a bubble in discrete time.

• Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model

  2020

  arXiv (Cornell University) · DOI

  We study a microscopic limit order book model, in which the order dynamics depend on the current best bid and ask price and the current volume density functions, simultaneously, and derive its macroscopic high-frequency dynamics. As opposed to the existing literature on scaling limits for limit order book models, we include price changes which do not scale with the tick size in our model to account for large price movement, being for example triggered by highly unforeseen events. Our main result states that, when the size of an individual limit order and the tick size tend to zero while the order arrival rate tends to infinity, the microscopic limit order book model dynamics converge to two one-dimensional jump diffusion processes describing the prices coupled with two infinite dimensional fluid processes describing the standing volumes at the buy and sell side.

• Second order approximations for limit order books

  2018

  Finance and Stochastics · DOI

• A FUNCTIONAL CONVERGENCE THEOREM FOR INTERPOLATED MARKOV CHAINS TO AN INFINITE DIMENSIONAL DIFFUSION

  2016

  arXiv (Cornell University)

  This paper is concerned with the derivation of a functional scaling limit theorem for a certain class of discrete time Markov chains, each consisting of a one dimensional reference process and an L 2-valued volume process, for which the conditional probability distributions of the increments are assumed to depend only on the current value of the reference process and on the volumes standing to the left of it. Such dynamics appear for example in the modeling of state dependent limit order books. It is shown that under suitable assumptions the sequence of interpolated discrete time models is relatively compact in a localized sense and that any limit point satisfies a certain infinite dimensional SDE. Under additional assumptions on the dependence structure we then construct two classes of models, which fit in the general framework, such that the limiting SDE admits a unique solution and thus the discrete dynamics converge to a diffusion limit in a localized sense. 1. Motivation and setup In this paper we study a sequence of discrete time R+ × L 2 (R+;R)-valued Markov chains e

• A diffusion approximation for limit order book models

  2016

  arXiv (Cornell University) · DOI

  This paper derives a diffusion approximation for a sequence of discrete-time one-sided limit order book models with non-linear state dependent order arrival and cancellation dynamics. The discrete time sequences are specified in terms of an $\R_+$-valued best bid price process and an $L^2_{loc}$-valued volume process. It is shown that under suitable assumptions the sequence of interpolated discrete time models is relatively compact in a localized sense and that any limit point satisfies a certain infinite dimensional SDE. Under additional assumptions on the dependence structure we construct two classes of models, which fit in the general framework, such that the limiting SDE admits a unique solution and thus the discrete dynamics converge to a diffusion limit in a localized sense.

• A functional convergence theorem for interpolated Markov chains to an infinite dimensional diffusion with application to limit order books

  2016

  arXiv (Cornell University)

  This paper is concerned with the derivation of a functional scaling limit theorem for a certain class of discrete time Markov chains, each consisting of a one dimensional reference process and an $L^2_{loc}$-valued volume process, for which the conditional probability distributions of the increments are assumed to depend only on the current value of the reference process and on the volumes standing to the left of it. Such dynamics appear for example in the modeling of state dependent limit order books. It is shown that under suitable assumptions the sequence of interpolated discrete time models is relatively compact in a localized sense and that any limit point satisfies a certain infinite dimensional SDE. Under additional assumptions on the dependence structure we then construct two classes of models, which fit in the general framework, such that the limiting SDE admits a unique solution and thus the discrete dynamics converge to a diffusion limit in a localized sense.

• A weak law of large numbers for a limit order book model with fully\n state dependent order dynamics

  2015

  arXiv (Cornell University) · DOI

  This paper studies a limit order book (LOB) model, in which the order\ndynamics depend on both, the current best available prices and the current\nvolume density functions. For the joint dynamics of the best bid price, the\nbest ask price, and the standing volume densities on both sides of the LOB we\nderive a weak law of large numbers, which states that the LOB model converges\nto a continuous-time limit when the size of an individual order as well as the\ntick size tend to zero and the order arrival rate tends to infinity. In the\nscaling limit the two volume densities follow each a non-linear PDE coupled\nwith two non-linear ODEs that describe the best bid and ask price.\n

• Strict local martingales, random times, and non-standart changes of probability measure in financial mathematicss

  2014

  Zurich Open Repository and Archive (University of Zurich)

• Strict local martingales and bubbles

  2011

  London School of Economics and Political Science Research Online (London School of Economics and Political Science) · DOI

  This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the "default term" apparent in risk-neutral option prices if the underlying stock exhibits a bubble modeled by a strict local martingale. Results for certain path dependent options and last passage time formulas are given.

• Freie Universität Berlin

  IGRK 2544/1: Stochastische Analysis in Interaktion

  university

• Technische Universität Berlin

  IGRK 2544: Stochastische Analysis in Interaktion

  university

• Universität Oxford

  IGRK 2544: Stochastische Analysis in Interaktion

  university

• Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin

  SFB/TRR 388/1: „Mikrostrukturelle Grundlagen rauer Volatilitätsmodelle“ (TP B02)

  other

• Zuse-Institut Berlin

  EXC 2046: Berlin Mathematics Research Center (MATH+)

  research_institute

Name

Prof. Dr. Dörte Kreher

Titel

Prof. Dr.

Fakultät

Mathematisch-Naturwissenschaftliche Fakultät

Institut

Institut für Mathematik

Arbeitsgruppe

Angewandte Stochastische Analysis (J)

Telefon

+49 30 2093-45455

HU-FIS-Profil

Quelle ↗

Zuletzt gescrapt

26.4.2026, 01:08:05