I'll try in this talk to present the main ideas on zero-sum continuous time games where one of the two players has some private information (for instance when only one player observes a Brownian motion): how to formalize these games, the associated Hamilton-Jacobi-Isaacs-equation and the analyse of the optimal revelation in terms of an optimization problem over a set of martingales. In a second time I'll present the last developments in this area.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

In this talk, we propose a novel approach to the modelling of multiple yield curves. Adopting the HJM philosophy, we model term structures of forward rate agreements (FRA) and OIS bonds. Our approach embeds most of the existing approaches and additionally allows for stochastic discontinuities. In particular, this last feature has an important motivation in interest rate markets, which are affected by political events and decisions occurring at predictable times. We study absence of arbitrage using results from the recent literature on large financial markets and discuss special cases and examples. This talk is based on joint work with Zorana Grbac, Sandrine Gümbel and Thorsten Schmidt.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

Insurance contracts are efficient risk management techniques to operate and reduce losses. However, very often, the underlying probability model for losses - on the basis of which premium is computed - is not completely known. Furthermore, in the case of extreme climatic events, the lack of data increases the epistemic uncertainty of the model. In this talk we propose a method to incorporate ambiguity into the design of an optimal insurance contract. Due to coverage limitations in this market, we focus on the limited stop-loss contract, given by $I(x)=min(max(x-d_1),d_2)$, with deductible $d_1$ and cap $d_2$. Therefore, we formulate an optimization problem for finding the optimal balance between the contract parameters that minimize some risk functional of the final wealth. To compensate for possible model misspecification, the optimal decision is taken with respect to a set of non-parametric models. The ambiguity set is built using a modified version of the well-known Wasserstein distance, which results to be more sensitive to deviations in the tail of distributions. The optimization problem is solved using a distributionally robust optimization setup. We examine the dependence of the objective function as well as the deductible and cap levels of the insurance contract on the tolerance level change. Numerical simulations illustrate the procedure.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

TBA

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

In this talk I will explain what is a Variable Annuities (VA) contract and how we can find the pricing formula of VA when the financial market is hybrid in the sense introduced by Eberlein.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

Probability-valued jump-diffusions provide useful approximations of large stochastic systems in finance, such as large sets of equity returns, or particle systems with mean-field interaction. The dynamics of a probability-valued jump-diffusion is governed by an integro-differential operator of Levy type, expressed using a notion of derivative that is well-known from the superprocesses literature. General and easy-to-use existence criteria for probability-valued jump-diffusions are derived using new optimality conditions for functions of probability arguments. In general, we consider the space of probability measures as endowed with the topology of weak convergence. For jump-diffusions taking value on a specific subset of the probability measures, it can however be useful to work with a stronger notion of convergence. Think for instance at the well-known Wasserstein spaces. This change of topology permits to include in the theory a larger class of generators, and hence, a larger class of probability-valued jump-diffusions. We derive general and easy-to-use existence criteria for jump-diffusions valued in those spaces.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

A market of financial securities with restricted participation is considered. Agents are heterogeneous in beliefs, risk tolerance and endowments, and may not have access to the trade of all securities. The market is assumed thin: agents may influence the market and strategically trade against their price impacts. Existence and uniqueness of the equilibrium is shown, and an efficient algorithm is provided to numerically obtain the equilibrium prices and allocations given market’s inputs. (Based on joint work with M. Anthropelos.)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companys aim to minimise their costs, and deterministic Black-Scholes-type PDEs are also highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model. For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In this talk we introduce of a class of new stochastic approximation algorithms for high-dimensional nonlinear PDEs. We prove that these algorithms do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE with a nonlinearity depending on the PDE solutiothe approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

When we are ill, most of us would prefer to receive treatment that was supported by scientific evidence, rather than anecdotal tradition or superstition. When a nation's economy is ill, policy-makers turn to economists for advice, but how well is their advice supported by evidence? This talk critiques the value of economic theory in practice, and tries to suggest ways of increasing the practical relevance of the subject.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

We will discuss non-exponential versions of well known limit theorems, specialising on the case of Brownian motion. The proofs will partially rely on the theory of BSDEs and their convex dual formulations, and an application to (stochastic) optimal transport will be provided.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

We introduce a new method to price American options based on Chebyshev interpolation. The key advantage of this approach is that it allows to shift the model-dependent computations into an offline phase prior to the time-stepping. This leads to a highly efficient online phase. The model-dependent part can be solved with any computational method such as solving a PDE, using Fourier integration or Monte Carlo simulation.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

Motivated by recent advances in rough volatility modeling, we introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classica affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. Nonetheless, their Fourier-Laplace functionals admit exponential-affine representations in terms of solutions of associated deterministic integral equations, extending the well-known Riccati equations for classical affine diffusions. Our findings generalize and clarify recent results in the literature on rough volatility.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

In a joint work with Walter Schachermayer (still in progress), we investigate the optimal strategy of an economic agent trading a fractional asset in presence of transaction costs. A fascinating conjecture by us asserts that, contrary to the Bronwnian case, such an optimal trading would be fully discrete, only involving countably many trading times. What we can already prove is that only certain specific times, which we call "potential trading times", may involve trading, regardless of the agent's porfolio (this shall be explained more in detail). An idea towards our conjecture (though unsuccessful yet) would be to bound above the fractal dimension of the set of potential trading times. The nice point with this approach is that, contrary to the optimal strategy, this fractal dimension can be computed numerically: the goal of my talk will be to explain how one can do so. The method I propose involves quasi-stationary distributions, that is, killed Markov processes conditioned by long-time survival: which is rather surprising, as this concept has a priori nothing to do with fractal dimension ...

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set $Theta$ of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programming principle we link this non-linear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in $Theta$. This non-linear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity. We then develop an appropriate Ito-formula, the respective term-structure equations and study the non-linear versions of the Vasicek and the Cox-Ingersoll-Ross (CIR) model. Thereafter we introduce the non-linear Vasicek-CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence the approach solves this modelling issue arising with negative interest rates. Joint work with Tolulope Fadina and Ariel Neufeld.

• Thematic program: Stochastics and Finance (2018/2019)
• Event: Stochastic Finance Workshop (2018)