Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

18.04.2024

16. Uhr c.t.

Alexandre Pannier (Université Paris Cité)

A path-dependent PDE solver based on signature kernels

Abstract: We develop a provably convergent kernel-based solver for path-dependent PDEs (PPDEs). Our numerical scheme leverages signature kernels, a recently introduced class of kernels on path-space. Specifically, we solve an optimal recovery problem by approximating the solution of a PPDE with an element of minimal norm in the signature reproducing kernel Hilbert space (RKHS) constrained to satisfy the PPDE at a finite collection of collocation paths. In the linear case, we show that the optimisation has a unique closed-form solution expressed in terms of signature kernel evaluations at the collocation paths. We prove consistency of the proposed scheme, guaranteeing convergence to the PPDE solution as the number of collocation points increases. Finally, several numerical examples are presented, in particular in the context of option pricing under rough volatility. Our numerical scheme constitutes a valid alternative to the ubiquitous Monte Carlo methods. Joint work with Cristopher Salvi (Imperial College London).

18.04.2024

17 Uhr c.t.

Nils Detering (Universität Düsseldorf)

Local Volatility Models for Commodity Forwards

Abstract: 

We present a dynamic model for forward curves in commodity markets, which is defined as the solution to a stochastic partial differential equation (SPDE) with state-dependent coefficients, taking values in a Hilbert space H of real valued functions. The model can be seen as an infinite dimensional counterpart of the classical local volatility model frequently used in equity markets. We first investigate a class of point-wise operators on H, which we then use to define the coefficients of the SPDE. Next, we derive growth and Lipchitz conditions for coefficients resulting from this class of operators to establish existence and uniqueness of solutions.  We also derive conditions that ensure positivity of the entire forward curve. Finally, we study the existence of an equivalent measure under which related traded, 1-dimensional projections of the forward curve are martingales.

Our approach encompasses a wide range of specifications, including a Hilbert-space valued counterpart of a constant elasticity of variance (CEV) model, an exponential model, and a spline specification which can resemble the S shaped local volatility function that well reproduces the volatility smile in equity markets. A particularly pleasant property of our model class is that the one-dimensional projections of the curve can be expressed as one dimensional stochastic differential equation. This provides a link to models for forwards with a fixed delivery time for which formulas and numerical techniques exist. In a first numerical case study we observe that a spline based, S shaped local volatility function can calibrate the volatility surface in electricity markets. 

Joint work with Silvia Lavagnini (BI Norwegian Business School)

02.05.2024

17 Uhr c.t.

Katharina Oberpriller (Universität München) 

Reduced-form framework and affine processes with jumps under model uncertainty

Abstract: 

02.05.2024

N.N.

16.05.2024

16 Uhr c.t.

Sina Dahms, Matthias Drees, and Lea Fernandez

(B & W Deloitte GmbH, Berlin)

Extrem value theory in the insurance sector

Abstract: 

Dusty insurance industry or buzzword bingo? Not with us! We work both in the world of insurance industry and management consulting, which means for us no two days are the same. Our practice supports many of the world’s leading organizations by using modern data analytics and complex mathematical models. Thereby we quantify the risks of the insurance industry, making risks visible and
manageable. We work together with our clients to assess their strategic priorities, increase economic value, optimize capital, and drive organizational performance. Sina Dahms, Matthias Drees, and Lea Fernandez from Deloitte will talk about extreme value theory and its applications in insurance and will give exclusive insights into the day-to-day work of an actuarial consultant. Sina has a PhD in financial mathematics from HU Berlin, Matthias holds degrees in mathematics and physics from universities in Munich, Cambridge and Tokyo, and Lea recently finished her studies of mathematics and physics at the TU Berlin

16.05.2024

N.N. 

30.05.2024

16 Uhr c.t.

Marko Weber (National University of Singapore)

Abstract: 

We examine the implications of unhedgeable fundamental risk, combined with agents' heterogeneous preferences and wealth allocations, on dynamic asset pricing and portfolio choice. We solve in closed form a continuous-time general equilibrium model in which unhedgeable fundamental risk affects aggregate consumption dynamics, rendering the market incomplete. Several long-lived agents with heterogeneous risk-aversion and time-preference make consumption and investment decisions, trading risky assets and borrowing from and lending to each other. We find that a representative agent does not exist. Agents trade assets dynamically. Their consumption rates depend on the history of unhedgeable shocks. Consumption volatility is higher for agents with preferences and wealth allocations deviating more from the average. Unhedgeable risk reduces the equilibrium interest rate only through agents' heterogeneity and proportionally to the cross-sectional variance of agents' preferences and allocations.

30.05.2024

17 Uhr c.t.

Shige Peng (Shandong University)

Solving probability measure uncertainty by nonlinear expectations

Abstract: 

13.06.2024

16 Uhr c.t.

Eduardo Abi Jaber (École Polytechnique, Palaiseau)

Stochastic Fredholm equations: a passe-partout for propagator models with cross-impact, constraints and mean-field interactions

Abstract: 

17 Uhr c.t.

Likai Jiao (HU Berlin)

An infinite-dimensional price impact model

Abstract: 

In this talk, we introduce an infinite-dimensional price impact process as a kind of Markovian lift of non-Markovian 1-dimensional price impact processes with completely monotone decay kernels. In an additive price impact scenario, the related optimal control problem is extended and transformed into a linear-quadratic framework. The optimal strategy is characterized by an operator-valued Riccati equation and a linear backward stochastic evolution equation (BSEE). By incorporating stochastic in-flow, the BSEE is simplified into an infinite-dimensional ODE. With appropriate penalizations, the well-posedness of the Riccati equation is well-known. This is a joint work with Prof. Dirk Becherer and Prof. Christoph Reisinger.

27.06.2024

16 Uhr c.t.

Abstract: 

We study the upper and lower bounds for prices of European and American style options with the possibility of an external termination, meaning that the contract may be terminated at some random time. Under the assumption that the underlying market model is incomplete and frictionless, we obtain duality results linking the upper price of a vulnerable European option with the price of an American option whose exercise times are constrained to times at which the external termination can happen with a non-zero probability. Similarly, the upper and lower prices for a vulnerable American option are linked to the price of an American option and a game option, respectively. In particular, the minimizer of the game option is only allowed to stop at times which the external termination may occur with a non-zero probability.

27.06.2024

17 Uhr c.t.

Benjamin Jourdan (University of Paris-Est)

Convexity propagation and convex ordering of one-dimensional stochastic differential equations

Abstract: 

We consider driftless one-dimensional stochastic differential equations. We first recall how they propagate convexity at the level of single marginals. We show that some spatial convexity of the diffusion coefficient is needed to obtain more general convexity propagation and obtain functional convexity propagation under a slight reinforcement of this necessary condition. Such conditions are not needed for directional convexity..
This is a joint work with Gilles Pagès.

11.07.2024

16 Uhr c.t.

Tiziano de Angelis (University of Turin)

Linear-quadratic stochastic control with state constraints on finite-time horizon

Abstract: 

We obtain a probabilistic solution to linear-quadratic optimal control problems with state
constraints. Given a closed set D ⊆ [0, T ] × Rd, a diffusion X in Rd must be linearly controlled in order to
keep the time-space process (t, Xt) inside the set C := ([0, T ] × Rd) \ D, while at the same time minimising
an expected cost that depends on the state (t, Xt) and it is quadratic in the speed of the control exerted.
We find an explicit probabilistic representation for the value function and the optimal control under a set
of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of
the set C. Fully explicit formulae are presented in some relevant examples.
(Joint work with Erik Ekstr¨om, University of Uppsala, Sweden)

11.07.2024

17 Uhr c.t.

Jianniao Qui (University of Calgary) 

Consensus-based optimization for equilibrium points of games

Abstract: 

In this talk, we will introduce Consensus-Based Optimization (CBO) for min-max problems,
a novel multi-particle, derivative-free optimization method that can provably identify global equilibrium
points. This paradigm facilitates the transition to the mean-field limit, making the method amenable to
theoretical analysis and providing rigorous convergence guarantees under reasonable assumptions about
the initialization and the objective function, including nonconvex-nonconcave objectives. Additionally,
numerical evidence will be presented to demonstrate the algorithm’s effectiveness.
This talk is based on joint works with Giacomo Borghi, Enis Chenchene, Hui Huang, and Konstantin Riedl.