# Austrian Stochastic Days 2018

## Program of the conference

### Thursday, September 13

 930 Registration

Session I Chair: Birgit Rudloff (WU Vienna)
0945-1030 Wolfgang Trutschnig (University of Salzburg)
Stochastic, dynamical and topological aspects of copulas
1030-1045 Short break
1045-1110 Sándor Guzmics (University of Vienna)
Copula orderings in a modified exponential lifetimel model
1110-1135 Daniel Bartl (University of Vienna)
Wasserstein distances and explicit formulas for robust risk measures
1135-1200 Jana Hlavinová (WU Vienna)
Elicitability and identifiability of measures of systemic risk
1200-1330 Lunch

Session II Chair: Gunther Leobacher (University of Graz)
1330-1355 Gaurav Dhariwal (TU Vienna)
Global martingale solutions for a stochastic population cross-diffusion system
1355-1420 Christoph Gerstenecker (TU Vienna)
Moment explosions in the rough Heston model
1420-1435 Short break
1435-1500 Gabriela Kováčová (WU Vienna)
Time Consistency of the Mean-Risk Problem
1500-1525 Timo Welti (ETH Zurich)
Deep Optimal Stopping: Solving High-dimensional Optimal Stopping Problems with Deep Learning
1525-1555 Coffee break

Session III Chair: Stefan Thonhauser (TU Graz)
1555-1620 Leopold Sögner (IHS)
Optimal High-Risk Investment
1620-1645 Josef Strini (TU Graz)
A dividend problem with random capital supply
1645-1710 Michael Preischl (TU Graz)
Optimal reinsurance for gerber-Shiu Functions in the Cramer-Lundberg Model

1830 Conference dinner in restaurant "Schöne Perle"

### Friday, September 14

Session IV Chair: Michaela Szölgyenyi (ETH Zurich)
0930-1015 Larisa Yaroslavtseva (ETH Zurich)
On sub-polynomial lower error bounds for strong approximation of SDEs
1015-1040 Diyora Salimova (ETH Zurich)
Numerical approximations of nonlinear stochastic differential equations
1040-1110 Coffee break
1110-1135 Stefan Thonhauser (TU Graz)
Approximation methods for PDMPs and applications in risk theory
1135-1200 Tijana Levajkovic (University of Innsbruck)
The stochastic linear quadratic regulator problem: Theory and numerical approximation
1200-1215 Short break
1215-1240 Alexander Steinicke (Montanuniversität Leoben)
Finite Activity Approximations and Comparison Theorems for Lévy Driven BSDEs
1240-1305 Stefan Kremsner (University of Graz)
Lp Solutions of BSDEs
1305-1310 Closing

## Titles and abstracts

1. Wolfgang Trutschnig (University of Salzburg)
Title: Stochastic, dynamical and topological aspects of copulas

Abstract: Copulas are not only important from the statistical point of view due to their wide domain of applicability, they also provide several interesting problems in basic mathematical research (the solution of which might in turn also trigger novel applied approaches). Main objective of the talk is to illustrate this fact by several examples and recent developements. After recalling basic properties of copulas we will first sketch the one-to-one-to-one correspondence of copulas, Markov kernels and Markov operators and some of its consequences, then discuss some geometric and topological properties of the convex and compact set of all copulas, and finally show that supposedly smooth objects like doubly stochastic measures may exhibit surprisingly singular, even fractal behaviour.

2. Sándor Guzmics (University of Vienna)
Title: Copula orderings in a modified exponential lifetimel model

Abstract: We introduce a multivariate joint lifetime model, where the whole dependence structure is based on modifications of univariate exponential distributions. Namely, we assume that the original lifetime parameters $\lambda_1, \lambda_2, \ldots, \lambda_n$ of the institutions will be modified as $\lambda_i +a_{ji}$ $(i=1,\ldots ,n, i\neq j)$, when the default of institution $j$ occurs. This model enables a very flexible dependence modelling due to the fact, that for each pair $(i,j)$ there is a parameter $a_{ij}$ defined.

Our work is partly motivated by the well-known Marshall-Olkin model, which is also established on modifications of exponential distributions. However, while the Marshall-Olkin distribution allows common shocks for subsets of entities (which is not typical in banking systems), in our suggested model we deal with cascading effects, where the default of a particular institution affects the remaining lifetime of (some) other institutions. Our model also includes the Marshall-Olkin type models as a special case.

We present the fundamentals of our lifetime model as well as some interesting properties of the multivariate copula which stems from the model.

We investigate -- under symmetric and asymmetric parameter setting -- the monotonicity of our copula in certain copula orderings, especially in the upper orthant order, in the convex order and in the increasing convex order, as the parameter(s) of the copula varies. Then we relate our copula model to the quantification of systemic risk in financial systems.

Joint work with Georg Ch. Pflug.

3. Daniel Bartl (University of Vienna)
Title: Wasserstein distances and explicit formulas for robust risk measures

Abstract: We briefly discuss Wasserstein distances and show that the infinite dimensional problem of computing the maximal expectation under all probabilities within a Wasserstein neighborhood around some baseline distribution has a finite dimensional dual problem. This is applied to some risk measures (e.g.~shortfall risk, or average value at risk) under Knightian uncertainty modeled by Wasserstein distances. In some cases closed-form formulas are presented.

Joint work with Samuel Drapeau and Ludovic Tangpi.

4. Jana Hlavinová (WU Vienna)
Title: Elicitability and identifiability of measures of systemic risk

Abstract: Estimating different risk measures, such as Value at Risk or Expected Shortfall, for reporting as well as testing purposes is a common task in various financial institutions. The question of evaluating and comparing these estimates is closely related to two concepts already well known in the literature: elicitability and identifiability. A statistical functional, e.g. a risk measure, is called elicitable if there is a strictly consistent scoring function for it, i.e. a function of two arguments, a forecast and a realization of a random variable, such that its expectation with respect to the second argument is minimized only by the correct forecast. It is called identifiable, if there is a strict identification function, i.e. again a function of two arguments such that the root of its expectation with respect to the second argument is exactly the correct forecast. We introduce these concepts for systemic risk measures defined by Feinstein, Rudloff and Weber (2016). A banking system with n participants is represented by a random vector Y and the quantity of interest is its aggregated outcome, using some nondecreasing aggregation function. The measure of systemic risk is defined as the set of n-dimensional capital allocation vectors k such that the aggregated outcome of Y+k is acceptable under a given scalar risk measure. We establish the link between the elicitability and/or identifiability of the systemic risk measure and the underlying scalar risk measure.

Joint work with Tobias Fissler and Birgit Rudloff.

5. Gaurav Dhariwal (TU Vienna)
Title: Global martingale solutions for a stochastic population cross-diffusion system

Abstract: The existence of global nonnegative martingale solutions to a stochastic cross-diffusion system for an arbitrary but finite number of interacting population species is shown. The random influence of the environment is modeled by a multiplicative noise term. The diffusion matrix is generally neither symmetric nor positive definite, but it possesses a quadratic entropy structure. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzezniak and co-workers, and Jakubowski\s generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia\s truncation method due to Chekroun, Park, and Temam.

Joint work with Ansgar Jüngel and Nicola Zamponi.

6. Christoph Gerstenecker (TU Vienna)
Title: Moment explosions in the rough Heston model

Abstract: We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arXiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions). This algorithm is then applied to computing the critical moments, which are shown to be finite for all maturities.

Joint work with Gerhold Stefan and Pinter Arpad.

7. Gabriela Kováčová (WU Vienna)
Title: Time Consistency of the Mean-Risk Problem

Abstract: The mean-risk problem is a well known and extensively studied problem in Mathematical Finance. Its aim is to identify portfolios that maximize the expected terminal value and at the same time minimize the risk. The usual approach in the literature is to combine the two to obtain a problem with a single objective. This scalarization, however, comes at the cost of time inconsistency.

In this work we show that these difficulties disappear by considering the problem in its natural form, that is, as a vector optimization problem. As such the mean-risk problem can be shown to satisfy under mild assumptions an appropriate notion of time consistency. Additionally, the upper images, whose boundaries are the efficient frontiers, recurse backwards in time. We argue that this represents a Bellman's principle appropriate for a vector optimization problem: a set-valued Bellman's principle. Furthermore, we provide conditions under which this recursion can be directly used to compute the efficient frontiers backwards in time.

Joint work with Birgit Rudloff.

8. Timo Welti (ETH Zurich)
Title: Deep Optimal Stopping: Solving High-dimensional Optimal Stopping Problems with Deep Learning

Abstract: Optimal stopping problems generally suffer from the curse of dimensionality. This talk concerns a new deep learning based method for solving high-dimensional optimal stopping problems. Among several possible application scenarios, the introduced algorithm can, in particular, be employed for the pricing of American options with a large number of underlyings. The presented algorithm is in fact applicable to a very broad class of financial derivatives (such as path-dependent American, Bermudan, and Asian options) as well as multifactor underlying models (such as Heston-type stochastic volatility models, local volatility models, and jump-diffusion models). Moreover, the algorithm allows to approximatively compute both the price as well as an optimal exercise strategy for the American option claim. Numerical results on benchmark problems are presented which suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

Joint work with Sebastian Becker, Patrick Cheridito and Arnulf Jentzen.

9. Leopold Sögner (IHS)
Title: Optimal High-Risk Investment

Abstract: This article extends the investment model of Bruss and Ferguson (2002), where an investor observes a sequence of T investment alternatives, each endowed with a random quality characteristic. The information available at any period is the current and all prior quality characteristics. The investor has to decide whether to invest in the same period the project shows up. Finally, after the last investment alternative has shown up, those n projects with highest realized quality characteristics generate positive gross-returns which depend on their relative ranking, while the payoffs of all other projects are zero.

Under these assumptions this article derives the value functions and optimal investment rules for risk-neutral or risk-averse investors. A simulation study demonstrates how optimal investment decisions are affected by the time horizon and by the attitudes towards risk. In addition, we provide sufficient conditions under that the value function is non-increasing in the number of periods T.

Joint work with Martin Meier.

10. Josef Strini (TU Graz)
Title: A dividend problem with random capital supply

Abstract: The problem considered in this talk is an extension of the classical dividend maximization problem in risk theory. Here we model the surplus process according to a compound Poisson process, extend it by dividend payments and introduce a novel additional control opportunity.

Namely, we allow the search for outside investors. The search is supposed to be successful at random times, given by jump times of an additional jump process, the associated funding height is the subject of the additional control. Our analysis of this problem for exponentially distributed claim amounts led to an interesting form of optimal strategy - contrasting the optimal one in a related diffusion setting. Besides our theoretical results we will present some illustrative numerical examples.

Joint work with Stefan Thonhauser.

11. Michael Preischl (TU Graz)
Title: Optimal reinsurance for gerber-Shiu Functions in the Cramer-Lundberg Model

Abstract: We want to minimize expected discounted utility functions (or Gerber-Shiu functions) in a Cramer-Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modelled as time dependant control functions, which leads to a setting from the theory of optimal stochastic control and ultimately to the problems Hamilton-Jacobi-Bellman equation.Though several authors have already contributed literature on stochastic optimal control, the focus has not been on Gerber Shiu functions so far.

Joint work with Stefan Thonhauser.

12. Larisa Yaroslavtseva (ETH Zurich)
Title: On sub-polynomial lower error bounds for strong approximation of SDEs

Abstract: We consider the problem of strong approximation of the solution of a stochastic differential equation (SDE) at the final time based on finitely many evaluations of the driving Brownian motion $W$. While the majority of results for this problem deals with equations that have globally Lipschitz continuous coefficients, such assumptions are typically not met for real world applications. In recent years a number of positive results for this problem has been established under substantially weaker assumptions on the coefficients such as global monotonicity conditions: new types of algorithms have been constructed that are easy to implement and still achieve a polynomial rate of convergence under these weaker assumptions.

In our talk we present negative results for this problem.

First we show that there exist SDEs with bounded smooth coefficients such that their solutions can not be approximated by means of any kind of adaptive method with a polynomial rate of convergence. Even worse, we show that for any sequence $(a_n)_{n \in \mathbb N}\subset (0, \infty)$, which may converge to zero arbitrarily slowly, there exists an SDE with bounded smooth coefficients such that no approximation method based on $n$ adaptively chosen evaluations of $W$ on average can achieve a smaller absolute mean error than the given number $a_n$.

While the diffusion coefficients of these pathological SDEs are globally Lipschitz continuous, the first order partial derivatives of the drift coefficients are, essentially, of exponential growth. In the second part of the talk we show that sub-polynomial rates of convergence may happen even when the first order partial derivatives of the coefficients have at most polynomial growth, which is one of the typical assumptions in the literature on numerical approximation of SDEs with globally monotone coefficients.

Joint work with Arnulf Jentzen, Benno Kuckuck and Thomas Müller-Gronbach.

13. Diyora Salimova (ETH Zurich)
Title: Numerical approximations of nonlinear stochastic differential equations

Abstract: In this talk we present a few recent results on regularity properties and numerical approximations for stochastic differential equations. We propose an explicit and easily implementable full-discrete numerical approximation scheme and prove that the suggested scheme converges both in the strong and numerically weak sense for a large class of additive noise driven stochastic evolution equations with superlinearly growing nonlinearities. In particular, we establish strong and numerically weak convergence of the proposed scheme in the case of stochastic Burgers equations, stochastic Kuramoto-Sivashinsky equations, and stochastic Allen-Cahn equations.

Joint work with Máté Gerencsér, Martin Hutzenthaler, Arnulf Jentzen and Timo Welti.

14. Stefan Thonhauser (TU Graz)
Title: Approximation methods for PDMPs and applications in risk theory

Abstract: Many models in risk theory can be formulated in terms of piecewise deterministic Markov processes. In this context, one is interested in computing certain quantities of interest such as the probability of ruin of an insurance company, or the insurance company’s value, defined as the expected discounted future dividend payments until the time of ruin. Instead of explicitly solving associated integro-differential equations, we adapt the problem in a way that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules.

For the derivation of error bounds for the numerical procedure we need to introduce a smoothing technique which is applied to the integrands involved. On the analytical side, we prove a convergence result for our PDMP approximation, which is of independent interest as it justifies phase-type approximations on the process level for example. Finally, we illustrate the smoothing technique for a risk-theoretic example, and provide a numerical example.

Joint work with Peter Kritzer, Gunther Leobacher and Michaela Szölgyenyi.

15. Tijana Levajkovic (University of Innsbruck)
Title: The stochastic linear quadratic regulator problem: Theory and numerical approximation

Abstract: The stochastic linear quadratic regulator (SLQR) problem arise naturally in science and engineering, e.g. electronics circuits and mathematical finance. We study the infinite dimensional SLQR problem and show that the optimal control is given in feedback form in terms of a Riccati equation. We investigate the numerical approximation of the problem, in particular, the convergence of Riccati operators and the numerical solution of the state equation. In addition, we provide a numerical framework for solving this problem using a polynomial chaos expansion approach. Numerical experiments of specific applications show the performance of the proposed method.

16. Alexander Steinicke (Montanuniversität Leoben)
Title: Finite Activity Approximations and Comparison Theorems for Lévy Driven BSDEs

Abstract: We consider backward stochastic differential equations driven by Levy processes: $$Y_t=\xi+\int_t^T f(s,Y_s,Z_s,U_s)ds-\int_t^T Z_sdW_s-\int_{]t,T]\times{\mathbb{R}\setminus\{0\}}}U_s(x) \tilde{N}(ds,dx),$$ $0\leq t\leq T$, where $\xi \in L^2$, $W$ is a Brownian motion and $\tilde{N}$ is the compensated Poisson random measure of a Levy process $X$. The solution of such a BSDE is a triple of processes $(Y,Z,U)$.The regularity of the generator function $f$ in its $(y,z,u)$-variables we assume is given by an extended monotonicity condition with linear growth. This setting exceeds Lipschitz assumptions for $f$. In the talk we discuss that solutions to such equations as above can be obtained as limits of solutions to BSDEs with a driving Levy process of finite-activity type (this means that $\mathbb{E}\tilde{N}([0,1]\times\mathbb{R})<\infty$, which is not the case for a general Levy process $X$).\\

Beside existence and uniqueness results, we show that this approximation procedure can be used to prove comparison theorems for BSDEs. Comparison theorems are an important tool in BSDE theory (e.g. to show existence of certain BSDE-related PDEs) and basically state, that if $\xi^1\leq \xi^2$ and $f^1\leq f^2$, then, for the respective solutions $Y^1$ and $Y^2$, we have $Y^1_t\leq Y^2_t$, $0\leq t\leq T$, a.s.

Joint work with Christel Geiss.

17. Stefan Kremsner (University of Graz)
Title: $L^p$ Solutions of BSDEs

Abstract: We consider Backward Stochastic Differential Equations (BSDEs) with Lev\`y jumps in a general setting considering a monotonicity condition on the driver. Based on the work of Kruse and Popier we further investigate existence and uniqueness results of $L^p$ solutions.

First, we show existence and uniqueness for solutions of BSDEs, following typical proof techniques in $L^p$ for $p > 2$. Then we discuss the more delicate case for $1 < p < 2$, where non-differentiability of $|x|^p$ is an issue. Finally we treat the case $p = 1$, where a counterexample shows that there can not be an $L^1$ solution. This is proven using tools from functional analysis.

Joint work with Alexander Steinicke.