Titles and Abstracts of Invited Speakers
Eduardo Abi Jaber (University of Paris, France)
LinearQuadratic control of stochastic Volterra equations
Abstract: We treat LinearQuadratic control problems for a class of stochastic Volterra equations of convolution type. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than 1 = 2 as a special case. We prove that the value function is of linear quadratic form with a linear optimal feedback control, depending on nonstandard infinite dimensional Riccati equations, for which we provide generic existence and uniqueness results. Furthermore, we show that the stochastic Volterra optimization problem can be approximated by conventional finite dimensional Markovian Linear Quadratic problems, which is of crucial importance for numerical implementation. Joint work with Enzo Miller and Huyên Pham.
Reference
E. Abi Jaber, E. Miller, H. Pham: LinearQuadratic control for a class of stochastic Volterra equations: solvability and approximation, arXiv:1911.01900.
Vladimir Bogachev (Lomonosov Moscow State University and Higher School of Economics, Russia)
Fractional Sobolev classes on infinitedimensional spaces
Abstract:
We discuss fractional Sobolev classes on infinitedimensional spaces with probability measures, in particular, some analogs of Besov and NikolskiiBesov classes and BV classes of functions of bounded variation. In the case of a Gaussian reference measure such classes can be defined by means of the OrnsteinUhlenbeck semigroup or ChebyshevHermite expansions, however, there are other interesting options based on certain "nonlinear integration by parts formulas". There are also reasonable similar constructions for more general probability measures.
Reference
Slides of the presentation (pdf)
Damir Filipović (EPF Lausanne and Swiss Finance Institute, Switzerland)
Machine Learning With Kernels for Portfolio Valuation and Risk Management
Abstract:
We introduce a simulation method for dynamic portfolio valuation and risk management building on machine learning with kernels. We learn the dynamic value process of a portfolio from a finite sample of its cumulative cash flow. The learned value process is given in closed form thanks to a suitable choice of the kernel. We show asymptotic consistency and derive finite sample error bounds under conditions that are suitable for finance applications. Numerical experiments show good results in large dimensions for a moderate training sample size.
Reference
L. Boudabsa, D. Filipović: Machine Learning With Kernels for Portfolio Valuation and Risk Management, Swiss Finance Institute Research, Paper No. 1934.
Martin Grothaus (University of Kaiserslautern, Germany)
An improved characterization theorem  its interpretation in terms of Mallivain calculus and applications to SPDEs
Abstract: We consider spaces of test and regular generalised functions of white noise. These spaces 20 years ago were characterized by holomorphy on infinite dimensional spaces together with an integrability condition. We, instead, give a characterisation in terms of Ufunctionals, i.e., classic holomorphic functions on the one dimensional field of complex numbers, together with the same integrability condition. The characterisation of regular generalised functions is useful for solving singular SPDEs. Whereas, the characterisation of test functions is useful for showing smoothness of solutions to SPDEs in the sense of Mallivain calculus. We present concrete examples confirming the usefulness in both cases.
Reference
Slides of the presentation (pdf)
Grothaus, M., Müller, J. & Nonnenmacher, A. An improved characterisation of regular generalised functions of white noise and an application to singular SPDEs. Stoch PDE: Anal Comp (2021).
https://doi.org/10.1007/s40072021002002 (pdf)
M. Grothaus; J. Müller; A. Nonnenmacher (2021). An improved characterisation of regular generalised functions of white noise and an application to singular SPDEs. Accepted for publication in Stochastics and Partial Differential Equations: Analysis and Computations.
Rajat Subhra Hazra (Indian Statistical Institute, Kolkata & Leiden University, Netherlands)
Scaling limit of some random interface models
Abstract:
In this talk we will discuss two examples of interface models, namely the Gaussian free field and Membrane model. We will make a comparison between the two models and discuss some properties of the membrane model. We will discuss some results on the scaling limit of the Membrane model and some of its variants on the lattice. We will display that the techniques from PDE help us prove the scaling limit. This talk is based on joint works with Alessandra Cipriani (TU Delft) and Biltu Dan (IISc).
Reference
Slides of the presentation (pdf)
Alessandra Cipriani, Biltu Dan, Rajat Subhra Hazra. Scaling limit of semiflexible polymers: a phase transition. Communications in Mathematical Physics. 377, 15051544, 2020.
Alessandra Cipriani,Biltu Dan, Rajat Subhra Hazra. The scaling limit of
the membrane model. Annals of Probability. Vol. 47, No. 6, 39634001, 2019.
Alessandra Cipriani, Biltu Dan, Rajat Subhra Hazra. The scaling limit of the (∇+Δ)model. Available at arXiv:1808.02676, 2020. To appear in the Journal of Statistical Physics.
Bernt Øksendal (University of Oslo, Norway)
SPDEs with space interactions  a model for optimal control of epidemics
Abstract:
We consider optimal control of a new type of nonlocal stochastic partial differential equations (SPDEs). The SPDEs have space interactions, in the sense that the dynamics of the system at time t and position in space x also depend on the spacemean of values at neighbouring points. This is a model with many applications, e.g. to population growth studies and epidemiology. We prove the existence and uniqueness of solutions of such SPDEs with space interactions, and using white noise theory we show that, under some conditions, the solutions are positive for all times if the initial values are. Sufficient and necessary maximum principles for the optimal control of such systems are derived. Finally, we apply the results to study an optimal vaccine strategy problem for an epidemic by modelling the population density as a spacemean stochastic reactiondiffusion equation. The talk is based on joint work with Nacira Agram, Astrid Hilbert and Khouloud Makhlouf.
Szymon Peszat (Jagiellonian University Cracow, Poland)
Heat equations with white noise Dirichlet boundary conditions
Abstract:
The talk in based on a joint work with Ben Goldys (University of Sydney). We study inhomogeneous Dirichlet boundary problems associated to heat equations on bounded and unbounded domains with white noise boundary data. We prove the existence of Markovian solutions living in weighted spaces of pintegrable functions where the weight is a proper power of the distance from the boundary.
Michael Röckner (University of Bielefeld, Germany)
Nonlinear FokkerPlanck equations with measures as initial data and McKeanVlasov equations
Abstract: This talk is about joint work with Viorel Barbu. We consider a class of nonlinear FokkerPlanck (Kolmogorov) equations (FPEs) of type
∂
/
∂t
u( t, x)  Δ _{x} β( u( t, x)) + div( D( x) b( u( t, x)) u( t, x)) = 0, u(0,·) = μ
where (t,x) ∈ [0,∞) × ℝ^{d}, d ≥ 3 and μ is a signed Borel measure on ℝ^{d} of bounded variation.
We shall explain how to construct a solution to the above PDE based on classical nonlinear operator semigroup theory on
L^{1}(ℝ^{d}) and new results on L^{1} − L^{∞} regularization of the solution semigroups in our case.
To motivate the study of such FPEs we shall first present a general result about the correspondence of nonlinear FPEs and McKeanVlasov type SDEs. In particular, it is shown that if one can solve the nonlinear FPE, then one can always construct a weak solution to the corresponding McKeanVlasov SDE. We would like to emphasize that this, in particular, applies to the singular case, where the coefficients depend “Nemytskitype” on the timemarginallaw law of the solution process, hence the coefficients are not continuous in the measurevariable with respect to the weak topology on probability measures. This is in contrast to the literature in which the latter is standardly assumed. Hence we can cover nonlinear FPEs as the ones above, which are PDEs for the marginal law densities, realizing an old vision of McKean.
(pdf)
Reference
V. Barbu, M. Röckner: From nonlinear FokkerPlanck equations to solutions of distribution dependent SDE, Ann. Prob. 48 (2020), no. 4, 19021920.
V. Barbu, M. Röckner: Solutions for nonlinear FokkerPlanck equations with measures as initial data and McKeanVlasov equations, J. Funct. Anal. 280 (2021), no. 7, 108926.
Sivaguru S. Sritharan (Applied Optimization, Inc., USA)
NavierStokes Equations: Ergodicity, Large Deviations, Control, Filtering and Malliavin Calculus
Abstract:
In this talk we will give an overview of several rigorous studies and results over the past number of years on compressible and incompressible NavierStokes and Euler equations including martingale, pathwise and strong solutions, invariant measures and ergodicity, large deviations of small noise (FreidlinWentzell) and large time (DonskerVaradhan) type, filtering, control and Malliavin calculus and related infinite dimensional partial differential equations, optimal stopping and impulse control and related infinite dimensional variational and quasivariational inequalities. We will also outline developments in related subjects such as magnetohydrodynamics as well as PDEs in physics such as the Einstein field equation, MaxwellDirac equation and the nonlinear Schrodinger equation, all subject to Gaussian and Lévy noise.
Reference
Slides of the presentation (pdf)
Josef Teichmann (ETH Zurich, Switzerland)
Gaussian processes, Signatures and Kernelizations
Abstract:
We shall introduce a functional analytic setting in the spirit of RöcknerSobol for signatures on path space, where kernelization techniques can then be applied. This allows for efficient training of approximations of path space functionals given through data. Applications from Finance are shown. Joint work with Christa Cuchiero and Philipp Schmocker.
