Probability Colloqium
Date
Time
17:15
Location:
WIAS Berlin
Eduardo Abi Jaber (École Polytechnique )

Some path-dependent processes from signatures

We provide explicit series expansions to certain stochastic path-dependent in- tegral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index H in (0, 1).

Our expressions allow to disentangle an infinite dimensional Markovian struc- ture. In addition they open the door to: (i) straightforward and simple approxima- tion schemes that we illustrate numerically, (ii) representations of certain Fourier- Laplace transforms in terms of a non-standard infinite dimensional Riccati equa- tion with important applications for pricing and hedging in quantitative finance.

Based on joint works with Louis-Amand Gérard and Yuxing Huang.

Mathematical Finance Seminar
Date
Time
17:15
Location:
TU; MA042
Shige Peng (Shandong University)

Solving probability measure uncertainty by nonlinear expectations

In 1921, economist Frank Knight published his famous ”Uncertainty, Risk and Profit”in which his challenging is still largely open. In this talk we explain why nonlinear expectation theory provides a powerful and fundamentally important mathematical tool to this century problem.

Mathematical Finance Seminar
Date
Time
16:15
Location:
TUB; MA042
Marko Weber (National University of Singapore)

General Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents

We examine the implications of unhedgeable fundamental risk, combined with agents’ hete- rogeneous 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 hete- rogeneous 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.

 

Probability Colloqium
Date
Time
17:15
Location:
WIAS Berlin
Shige Peng (Shandong University)

Space-time white noises in a nonlinear expectation space

Under the framework of nonlinear expectation, we introduce a new type of random fields, which contains a type of space-time white noise as a special case. Based on this result, we also introduce a space white noise. Different from the case of linear expectation, in which the probability measure is given and fixed.

Under the uncertainty of probability measures, space white noises are intrinsi- cally different from the space cases, which is generalized from G-Gaussian processes which are different from a G-Brownian motion (joint work with Xiaojun JI).

Mathematical Finance Seminar
Date
Time
16:15
Location:
TUB; MA042
Sina Dahms, Matthias Drees, and Lea Fernandez (B & W Deloitte GmbH, Berlin)

Extreme value theory in the insurance sector

Probability Colloqium
Date
Time
16:15
Location:
WIAS Berlin
Adrián González-Casanova (UC Berkeley)

Sample Duality

Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x".

Implicitly, this technique can be traced back to the work of Blaise Pascal. Explic- itly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique proves to be useful, including applications to the Simple Exclusion Process and (work in progress) a universality result for the FKPP equation.

Mathematical Finance Seminar
Date
Time
17:15
Location:
TU Berlin; MA042
Katharina Oberpriller (Ludwig-Maximilians-Universität München)

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

We introduce a sublinear conditional operator with respect to a family of possibly non- dominated probability measures in presence of multiple ordered default times. In this way we generalize the results in [3] where a consistent reduced-form framework under model uncertainty for a single default is developed. Moreover, we present a probabilistic construction Rd-valued non-linear affine processes with jumps, which allows to model intensities in a reduced-form framework. This yields a tractable model for Knightian uncertainty for which the sublinear expectation of a Markovian functional can be calculated via a partial integro-differential equation. This talk is based on [1] and [2].

[1] Francesca Biagini, Georg Bollweg, and Katharina Oberpriller. Non-linear affine processes with jumps. Probability, Uncertainty and Quantitative Risk, 8(3):235–266, 2023.
[2] Francesca Biagini, Andrea Mazzon, and Katharina Oberpriller. Reduced-form framework for multiple default times under model uncertainty. Stochastic Processes and Their Applications, 156:1–43, 2023.

[3] Francesca Biagini and Yinglin Zhang. Reduced-form framework under model uncertainty. The Annals of Applied Probability, 29(4):2481–2522, 2019

Mathematical Finance Seminar
Date
Time
17:15
Location:
TUB; MA 042
Nils Detering (Düsseldorf)

Local Volatility Models for Commodity Forwards

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)

Mathematical Finance Seminar
Date
Time
16:15
Location:
TUB; MA 042
Alexandre Pannier (Paris)

A path-dependent PDE solver based on signature kernels

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).

Mathematical Finance Seminar
Date
Time
16:oo
Location:
HUB; RUD 25; 1.115
Sergio Pulido (Paris)

Polynomial Volterra processes

Recent studies have extended the theory of affine processes to the stochastic Volterra equati- ons framework. In this talk, I will describe how the theory of polynomial processes extends to the Volterra setting. In particular, I will explain the moment formula and an interesting stochastic invariance result in this context. This is joint work with Eduardo Abi Jaber, Christa Cuchiero, Luca Pelizzari and Sara Svaluto-Ferro.