GRK 1845: Stochastic Analysis with Applications in Biology, Finance and Physics
Final Report Abstract
The RTG research in the area stochastic finance had its highlights in the following topics. Model free arbitrage pricing, for example for basket options, was treated with novel methods based on a game theoretic formulation of mathematical finance, paracontrolled distributions and rough paths. It produced scaling limits for superreplication prices under market friction. Studies of hedging and pricing under model uncertainty were done via representation results for increasing convex functions. Illiquid financial market models or multiplicative transient price impact models were investigated, in particular for optimal order execution problems. Methods employed focussed on optimal control, asymptotic methods therefore, and convex analysis. A constructive solution for the variational HJB inequality for two-dimensional free boundary in a non-convex stochastic singular control problem for random liquidity was derived. For backward stochastic differential equations (BSDE), a crucial tool for stochastic optimization, RTG students dealt with discrete and continuous time BSDE driven by general semimartingales (with jumps), the stability of solutions and their approximation by regression schemes. New approaches of convex BSDE went via duality of minimal supersolutions. Second order BSDE and G-Lévy processes were investigated. Utility optimization was studied in financial market models with asymmetric information. In the research area stochastic analysis and statistical inference for stochastic dynamics one of the main highlights was work with high impact on the foundations of rough paths, paracontrolled distributions and regularity structures, and a deep link between the two, that led to novel insights on (singular) stochastic partial differential equations (SPDE) and ordinary differential equations (ODE) with rough signals. It gave access to a systematic analysis of singular SPDE, and a characterization of SPDE as universal scaling limits, for instance in quantum field theory. (Forward) backward stochastic differential equations ((F)BSDE) were treated by the efficient tool of decoupling fields, leading to a new approach of Skorokhod’s embedding for diffusion processes. Work on the asymptotics of random dynamical systems highlighted in a study on the connectedness of random attractors, dealt primarily with the strong completeness of flows, and the long-time behavior of Brownian flows. Ergodic behavior was tackled by means of generalized asymptotic couplings, e. g. for stochastic delay equations. Stabilization by noise was on the agenda, and work with impact done on the novel topic of synchronization of random dynamical systems. The research on statistical inference for diffusion processes and SDE emphasized adaptation and efficiency in statistics. One focus was on the statistics for integrated volatility estimation for semi-martingales, another one on high-dimensional covariance matrix estimation, and statistical inverse problems. In the area of stochastic processes in biology and physics one of the foci was given by random walks and diffusions in random environments and effective interface models. Remarkable work was on random bridges of jump processes in connection with Schrödinger’s problem and optimal transport. RTG students dealt with spectral concentration of two types of random operators (random Schrödinger and random Laplace operator), did an extreme-value analysis of concentration sites, and studied Anderson localization at the edge of the spectrum. A methodological highlight was a combination of large deviations techniques with homogenization, where a transition between homogenized and localized behavior was studied. Duality formulas on path spaces were investigated, skew Brownian motion with several barriers, and a generalized rejection sampling method. Work on stochastic processes in neuroscience dealt with the analysis of neural activity in the brain on all scales, the numerical approximation of stochastic nerve axon equations, and stochastic neural field equations. One highlight was the multiscale analysis of stochastic neural field equations and stochastic reaction diffusion equations w.r.t. travelling waves as well as diffusion limits of discrete neural population models. Data assimilation problems in neuroscience and related stochastic filtering equations were treated. Successful work was done on stochastic models in population genetics and evolution. It highlighted in results on the duality theory for generalized Wright-Fisher models with frequency dependent selection as well as with geometric seed bank component. SPDE arising in interacting species models, in particular the symbiotic branching model, were studied. Many of the RTG students obtained their PhD degrees with summa cum laude, several were awarded local and national prizes (e. g. Tiburtius prize 2014, DMV student conference prize 2015, Michelson prize of U Potsdam 2016, publication prize of Leibniz-Kolleg Potsdam 2017, Förderpreis der Fachgruppe Stochastik 2016, Rollo-Davidson prize 2018, Heinz-Maier-Leibnitz prize 2019), and many obtained very good international positions in- and outside academia.
Publications
- On large deviations for small noise itô processes. Adv. in Appl. Probab., Vol. 46 , no. 4, 1126–1147, 2014
A. Chiarini and M. Fischer
(See online at https://doi.org/10.1017/s0001867800007576) - A note on the extremal process of the supercritical Gaussian Free Field. Electron. Commun. Probab., Vol 20, no. 74, pp. 1–10, 2015
A. Chiarini, A. Cipriani and R. S. Hazra
(See online at https://doi.org/10.1214/ecp.v20-4332) - An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift. Electron. J. Probab., Vol. 20, no. 127, pp. 1-38, 2015
A. Fromm, P. Imkeller and D.J. Prömel
(See online at https://doi.org/10.1214/ejp.v20-3758) - Existence of Lévy’s area and pathwise integration. Commun. Stoch. Anal., Vol. 9, no. 1, pp. 93-111, 2015
P. Imkeller and D.J. Prömel
(See online at https://doi.org/10.31390/cosa.9.1.06) - Functional stable limit theorems for quasi-efficient spectral covolatility estimators. Stochastic Processes and Applications, Vol. 125, no. 12, pp. 4556-4600, 2015
R. Altmeyer and M. Bibinger
(See online at https://doi.org/10.1016/j.spa.2015.07.009) - Invariance principle for diffusions in degenerate and unbounded random environment. Doctoral thesis. 2015
A. Chiarini
(See online at https://dx.doi.org/10.14279/depositonce-4723) - Local Central Limit Theorem for diffusions in a degenerate and unbounded random medium. Electron. J. Probab., Vol 20, no. 112, pp. 1–30, 2015
A. Chiarini and J.-D. Deuschel
(See online at https://doi.org/10.1214/ejp.v20-4190) - Local times for typical price paths and pathwise Tanaka formulas. Electron. J. Probab., Vol. 20, no. 46, pp. 1-15, 2015
N. Perkowski and D.J. Prömel
(See online at https://doi.org/10.1214/ejp.v20-3534) - Singular-degenerate multivalued stochastic fast diffusion equations. SIAM J. Math. Anal., Vol. 47, no. 5, pp. 4058–4090, 2015
B. Gess and M. Röckner
(See online at https://doi.org/10.1137/151003726) - A Stefan-type stochastic moving boundary problem. Stochastics and Partial Differential Equations: Analysis and Computations, Vol. 4, no. 4, pp. 746-790, 2016
M. Keller-Ressel and M.S. Müller
(See online at https://doi.org/10.1007/s40072-016-0076-z) - An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers. Monte Carlo Methods Appl., Vol. 22, no. 1, pp. 1-23, 2016
D. Dereudre, S. Roelly, and S. Mazzonetto
(See online at https://doi.org/10.1515/mcma-2016-0100) - Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations. SIAM J. Math. Anal. (SIMA), Vol. 48, no. 6, pp. 4094-4154, 2016
B. Gess and J. M. Tölle
(See online at https://doi.org/10.1137/15m1049774) - Extremes of some Gaussian random interfaces. Journal of Statistical Physics, Vol 165, no. 3, pp. 521-544, 2016
A. Chiarini, A. Cipriani and R. S. Hazra
(See online at https://doi.org/10.1007/s10955-016-1634-5) - Extremes of the supercritical Gaussian Free Field. ALEA, Lat. Am. J. Probab. Math. Stat., Vol 13, no. 2, pp. 711–724, 2016
A. Chiarini, A. Cipriani and R. S. Hazra
(See online at https://doi.org/10.30757/alea.v13-28) - Invariance Principle for symmetric Diffusions in a degenerate and unbounded stationary and ergodic random medium. Ann. de l’Inst. Henri Poincaré, Vol 52, no. 4, pp. 1535- 1563, 2016
A. Chiarini and J.-D. Deuschel
(See online at https://doi.org/10.1214/15-aihp688) - On the speed of convergence of Newton’s method for complex polynomials. AMS Math. Comp., Vol. 85, pp. 693–705, 2016
T. Bilarev, M. Aspenberg and D. Schleicher
(See online at https://doi.org/10.1090/mcom/2985) - Pathwise stochastic integrals for model free finance. Bernoulli, Vol. 22, no. 4, pp. 2486-2520, 2016
N. Perkowski and D.J. Prömel
(See online at https://doi.org/10.3150/15-bej735) - Regularization and well-posedness by noise for ordinary and partial differential equations, Springer Proceedings in Mathematics & Statistics, SPDERF: International Conference on Stochastic Partial Differential Equations and Related Fields, In Honor of Michael Röckner SPDERF, Bielefeld, Germany, October 10 -14, 2016, pp. 43-67
B. Gess
(See online at https://doi.org/10.1007/978-3-319-74929-7_3) - Rough differential equations driven by signals in Besov spaces. J. Differential Equations, Vol. 260, no. 6, pp. 4973-5025, 2016
D.J. Prömel and M. Trabs
(See online at https://doi.org/10.1016/j.jde.2015.12.012) - Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes. SIAM Journal on Numerical Analysis, Vol. 54, no. 4, pp. 2184-2209, 2016
B. Gess, B. Perthame and P. E. Souganidis
(See online at https://doi.org/10.1137/15m1053670) - Stability of solutions to stochastic partial differential equations. J. Differential Equations, Vol. 260, no. 6, pp. 4973–5025, 2016
B. Gess and J. Tölle
(See online at https://doi.org/10.1016/j.jde.2015.11.039) - Stochastic scalar conservation laws driven by rough paths. Annales de l’Institut Henri Poincaré, Analyse Non Linéaire (AIHP), Vol. 33, no. 4, pp. 933-963, 2016
P. Friz and B. Gess
(See online at https://doi.org/10.1016/j.anihpc.2015.01.009) - Weak synchronization for isotropic flows. AMS, Discrete and Continuous Dynamical Systems – Series B (DCDS-B) Vol. 21, no. 9, pp. 3003-3014, 2016
M. Cranston, B. Gess and M. Scheutzow
(See online at https://doi.org/10.3934/dcdsb.2016084) - Exact simulation of Brownian diffusions with drift admitting jumps. SIAM J. Sci. Comput., Vol. 39, no. 3, pp. A711-A740, 2017
D. Dereudre, S. Roelly, and S. Mazzonetto
(See online at https://doi.org/10.1137/16m107699x) - Game Options in an Imperfect Market with Default. Journal of Optimization Theory and Applications, Vol. 8, no. 1, 2017
R. Dumitrescu, M-C. Quenez and A. Sulem
(See online at https://doi.org/10.1137/16m1109102) - Improved Fréchet–Hoeffding bounds for d-copulas and applications in model-free finance. Annals of Applied Probability, Vol. 27, pp. 3633–3671, 2017
T. Lux and A. Papapantoleon
(See online at https://doi.org/10.1214/17-aap1292) - Long-time behavior, invariant measures and regularizing effects for stochastic scalar conservation laws. Comm. on Pure and Applied Mathematics, Vol. 70, no. 8, pp. 1562-1597, 2017
B. Gess and P.E. Souganidis
(See online at https://doi.org/10.1002/cpa.21646) - On Skorokhod Embeddings and Poisson Equations. Ann. Appl. Probab., 2017
L. Döring, L. Gonon, D.J. Prömel and O. Reichmann
(See online at https://doi.org/10.1214/18-aap1454) - Pathwise super-replication via Vovk’s outer measure. Finance Stoch., Vol. 21, no. 4, pp. 1141-1166, 2017
M. Beiglböck, A.M.G. Cox, M. Huesmann, N. Perkowski and D.J.Prömel
(See online at https://doi.org/10.1007/s00780-017-0338-2) - Sobolev regularity for the porous medium equation with a force. J. European Mathematical Society, 2017
B. Gess
(See online at https://doi.org/10.48550/arXiv.1708.04408) - Stochastic non-isotropic degenerate parabolic-hyperbolic equations. Stochastic Process. Appl., 2017
B. Gess and P. E. Souganidis
(See online at https://doi.org/10.1016/j.spa.2017.01.005) - Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations. Trans.Amer.Math.Soc., Vol. 369, no. 5, pp. 3017-3045, 2017
B. Gess and M. Röckner
(See online at https://doi.org/10.1090/tran/6981) - Synchronization by noise for order-preserving random dynamical systems. Ann. Probab. Vol. 45, no. 2 , pp. 1325-1350, 2017
F. Flandoli, B. Gess and M. Scheutzow
(See online at https://doi.org/10.1214/16-aop1088) - Synchronization by noise. Probability Theory and Related Fields, Vol. 168, no. 3–4, pp. 511–556, 2017
F. Flandoli, B. Gess and M. Scheutzow
(See online at https://doi.org/10.1007/s00440-016-0716-2) - A Liouville theorem for stationary and ergodic ensembles of parabolic systems. Probab. Theory Relat. Fields, 2018
P. Bella, A. Chiarini and B. Fehrman
(See online at https://doi.org/10.1007/s00440-018-0843-z) - A stochastic Stefan-type problem under first-order boundary conditions. The Annals of Applied Probability, Vol. 28, no. 4, pp. 2335-2369, 2018
M.S. Müller
(See online at https://doi.org/10.1214/17-aap1359) - A Superhedging Approach to Stochastic Integration. Stoch. Process. Appl., Vol. 128, no. 12, pp. 4078-4103, 2018
R.M. Lochowski, N. Perkowski and D.J. Prömel
(See online at https://doi.org/10.1016/j.spa.2018.01.009) - Approximating diffusion reflections at elastic boundaries. Electron. Commun. Probab., Vol. 23, no. 40, p. 12, 2018
D. Becherer, T. Bilarev and P. Frentrup
(See online at https://doi.org/10.1214/18-ecp141) - Connectedness of random set attractors. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., Vol. 29, no. 4, pp. 607-617, 2018
M. Scheutzow and I. Vorkastner
(See online at https://doi.org/10.4171/rlm/824) - Control for Mean-Field Stochastic Partial Differential Equations with Jumps. Journal of Optimization Theory and Applications, Vol. 176, no. 3, pp. 549-584, 2018
R. Dumitrescu, B. Oksendal and A. Sulem
(See online at https://doi.org/10.1007/s10957-018-1243-3) - Duality and fixation in Ξ- Wright-Fisher processes with frequency-dependent selection. Annals of Applied Probability, Vol. 28, no. 1, pp. 250-284, 2018
A. González Casanova and D. Spanò
(See online at https://doi.org/10.1214/17-aap1305) - Estimating error for occupation time functionals of stationary Markov processes. Stochastic Processes and Applications, Vol. 128, no. 6, pp. 1830-1848, 2018
R. Altmeyer and J. Chorowski
(See online at https://doi.org/10.1016/j.spa.2017.08.013) - Examples of Itô càdlàg rough paths. Proc.Amer. Math. Soc., Vol. 146, no. 11, pp. 4937-4950, 2018
C. Liu and D.J. Prömel
(See online at https://doi.org/10.1090/proc/14142) - Existence and uniqueness results for BSDE with jumps: The whole nine yards. Electronic Journal of Probability, Vol. 23, no. 121, pp. 1–68, 2018
A. Papapantoleon, D. Possamaï and A. Saplaouras
(See online at https://doi.org/10.1214/18-ejp240) - Noise dependent synchronization of a degenerate SDE. Stoch. Dyn., Vol. 18, no.1 :1850007, p. 21, 2018
I. Vorkastner
(See online at https://doi.org/10.1142/s0219493718500077) - Optimal Asset Liquidation with Multiplicative Transient Price Impact. Appl. Math. Optim., Vol. 78, no. 3, pp. 643–676, 2018
D. Becherer, T. Bilarev and P. Frentrup
(See online at https://doi.org/10.1007/s00245-017-9418-0) - Optimal Liquidation under Stochastic Liquidity. Finance Stoch., Vol. 22, no. 1, pp. 39–68, 2018
D. Becherer, T. Bilarev and P. Frentrup
(See online at https://doi.org/10.1007/s00780-017-0346-2) - Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques, 2018
J. Martin and N. Perkowski
(See online at https://doi.org/10.48550/arXiv.1704.08653) - Quenched invariance principle for random walks with time-dependent ergodic degenerate weights. Annals of Probability, Vol 46, no. 1, pp. 302-336, 2018
S. Andres, A. Chiarini, J.-D. Deuschel and M. Slowik
(See online at https://doi.org/10.1214/17-aop1186) - Regularization by noise for stochastic Hamilton- Jacobi equations. Probab. Theory Relat. Fields, 2018
P. Gassiat and B. Gess
(See online at https://doi.org/10.1007/s00440-018-0848-7) - Rough path metrics on a Besov–Nikolskii type scale. Trans. Amer. Math. Soc, Vol. 370, no. 12, pp. 8521-8550, 2018
P.K. Friz and D.J. Prömel
(See online at https://doi.org/10.1090/tran/7264) - Singular weighted Sobolev spaces and diffusion processes: an example (due to V.V. Zhikov). Special issue in memory of V.V. Zhikov, Appl. Analysis pp. 439-457, 2018
A. Chiarini and P. Mathieu
(See online at https://doi.org/10.1080/00036811.2018.1484912) - Synchronization, Lyapunov exponents and stable manifolds for random dynamical systems. In Stochastic partial differential equations and related fields, pp. 359–366, Springer Proc. Math. Stat., 229, Springer, Cham, 2018
M. Scheutzow and I. Vorkastner
(See online at https://doi.org/10.1007/978-3-319-74929-7_23) - Value-at-Risk bounds with two-sided dependence information. Mathematical Finance, 2018
T. Lux and L. Rüschendorf
(See online at https://doi.org/10.1111/mafi.12192, 2018) - Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE. The Annals of Probability, Vol. 46, no. 5, pp. 2495-2544, 2018
B. Gess and M. Hofmanova
(See online at https://doi.org/10.1214/17-aop1231) - Approximation of the interface condition for stochastic Stefan-type problems. Discrete and Continuous Dynamical Systems: Series B, 2019
M.S. Müller
(See online at https://doi.org/10.3934/dcdsb.2019121) - Entropy solutions for stochastic porous media equations. J. Differential Equations, Vol. 266, no. 6, pp. 3732-3763, 2019
K. Dareiotis, M. Gerencsér, and B. Gess
(See online at https://doi.org/10.1016/j.jde.2018.09.012) - Model-free bounds on Value-at-Risk using extreme value information and statistical distances. Insurance: Mathematics and Economics, Vol 86, pp. 73-83, 2019
T. Lux and A. Papapantoleon
(See online at https://doi.org/10.1016/j.insmatheco.2019.01.007) - Regularity of solutions to scalar conservation laws with a force. Annales de l’Institut Henri Poincaré, Analyse Non Linéaire (AIHP), Vol. 36, no. 2, pp. 505-521, 2019
B. Gess and M. Lamyi
(See online at https://doi.org/10.1016/j.anihpc.2018.07.002) - Stability for gains from large investors’ strategies in M1/J1 topologies. Bernoulli., Vol. 25, no. 2, pp. 1105-1140, 2019
D. Becherer, T. Bilarev and P. Frentrup
(See online at https://doi.org/10.3150/17-bej1014) - Stochastic continuity equations with conservative noise. J. Math. Pures Appl. (JMPA), 2019
B. Gess and S. Smith
(See online at https://doi.org/10.1016/j.matpur.2019.02.002) - Uniform convergence of proliferating particles to the FKPP equation. Journal of Math. Anal. and Appl., Vol. 473, no. 1, pp. 27-52, 2019
F. Flandoli, M. Leimbach and Ch. Olivera
(See online at https://doi.org/10.1016/j.jmaa.2018.12.013) - Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations. Zeitschrift für Angewandte Mathematik und Physik, Vol. 70, no. 1, p. 16., 2019
P. Harms and M.S. Müller
(See online at https://doi.org/10.1007/s00033-018-1060-4) - Well-posedness of nonlinear diffusion equations with nonlinear, conservative noise. Archive for Rational Mechanics and Analysis, pp. 1-74, 2019
B. Fehrman and B. Gess
(See online at https://doi.org/10.1007/s00205-019-01357-w)
