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Documents  Schweizer, Martin | enregistrements trouvés : 12

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Research talks;Probability & Statistics

In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model
with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.
This new distance $\widetilde{W}^{2}$ is defined similarly to the classical Wasserstein distance $\widetilde{W}^{2}$ but the set of couplings is restricted to the set of laws of solutions of 2$d$-dimensional stochastic differential equations. We prove that this new distance $\widetilde{W}^{2}$ metrizes the weak topology. Furthermore this distance $\widetilde{W}^{2}$ is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.
In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model
with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional ...

91B70 ; 60H30 ; 60H15 ; 60J60 ; 93E20

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Research talks;Analysis and its Applications;Computer Science;Control Theory and Optimization;Mathematics in Science and Technology;Probability and Statistics

I discuss some recent developments related to the robust framework for pricing and hedging in discrete time. I introduce pointwise approach based on pathspace restrictions and compare it with the quasi-sure setting of Bouchard and Nutz (2015), and show that their versions of the Fundamental Theorem of Asset Pricing and the Pricing-Hedging duality may be deduced one from the other via a construction of a suitable set of paths which represents a given set of measures. I show that the setup with statically traded hedging instruments can be naturally lifted to a setup with only dynamically traded assets without changing the superhedging prices. This allows one to deduce, in particular, a pricing-hedging duality for American options. Subsequently, I focus on the superhedging problem and discuss the choice of a trading strategy amongst all feasible super-hedging strategies. First, I establish existence of a minimal superhedging strategy and characterise its value via a concave envelope construction. Then I introduce a secondary problem of maximisation of expected utility of consumption. Building on Nutz (2014) and Blanchard and Carassus (2017) I provide suitable assumptions under which an optimal strategy exists and is unique. Finally, I also explain how additional information can be seen as a further restriction of the pathspace. This allows one to quantify to value of such a new information. The talk is based on a number of recent works (see references) as well as ongoing research with Johannes Wiesel. I discuss some recent developments related to the robust framework for pricing and hedging in discrete time. I introduce pointwise approach based on pathspace restrictions and compare it with the quasi-sure setting of Bouchard and Nutz (2015), and show that their versions of the Fundamental Theorem of Asset Pricing and the Pricing-Hedging duality may be deduced one from the other via a construction of a suitable set of paths which represents a ...

91G20 ; 91B70 ; 60G40 ; 60G42 ; 90C46 ; 90C47 ; 28A05 ; 49N15

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Research talks;Probablity & Statistics

Non-convex random sets of admissible positions naturally arise in the setting of fixed transaction costs or when only a finite range of possible transactions is considered. The talk defines set-valued risk measures in such cases and explores the situations when they return convex result, namely, when Lyapunov's theorem applies. The case of fixed transaction costs is analysed in greater details.
Joint work with Andreas Haier (FINMA, Switzerland).
Non-convex random sets of admissible positions naturally arise in the setting of fixed transaction costs or when only a finite range of possible transactions is considered. The talk defines set-valued risk measures in such cases and explores the situations when they return convex result, namely, when Lyapunov's theorem applies. The case of fixed transaction costs is analysed in greater details.
Joint work with Andreas Haier (FINMA, Switzerland).

91G70 ; 91G10

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Research talks;Probablity & Statistics

For large financial markets as introduced in Kramkov and Kabanov 94, there are several existing absence-of-arbitrage conditions in the literature. They all have in common that they depend in a crucial way on the discounting factor. We introduce a new concept, generalizing NAA1 (K&K 94) and NAA (Rokhlin 08), which is invariant with respect to discounting. We derive a dual characterization by a contiguity property (FTAP).We investigate connections to the in finite time horizon framework (as for example in Karatzas and Kardaras 07) and illustrate negative result by counterexamples. Based on joint work with M. Schweizer. For large financial markets as introduced in Kramkov and Kabanov 94, there are several existing absence-of-arbitrage conditions in the literature. They all have in common that they depend in a crucial way on the discounting factor. We introduce a new concept, generalizing NAA1 (K&K 94) and NAA (Rokhlin 08), which is invariant with respect to discounting. We derive a dual characterization by a contiguity property (FTAP).We investigate connections ...

91C99 ; 91B02 ; 60G48

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Research talks;Probablity & Statistics

Some questions in mathematics are not answered for quite some time, but just sidestepped. One of those questions is the following: What is the quantile of a multi-dimensional random variable? The "sidestepping" in this case produced so-called depth functions and depth regions, and the most prominent among them is the halfspace depth invented by Tukey in 1975, a very popular tool in statistics. When it comes to the definition of multivariate quantiles, depth functions replace cummulative distribution functions, and depth regions provide potential candidates for quantile vectors. However, Tukey depth functions, for example, do not share all features with (univariate) cdf's and do not even generalize them.
On the other hand, the naive definition of quantiles via the joint distribution function turned out to be not very helpful for statistical purposes, although it is still in use to define multivariate V@Rs (Embrechts and others) as well as stochastic dominance orders (Muller/Stoyan and others).
The crucial point and an obstacle for substantial progress for a long time is the missing (total) order for the values of a multi-dimensional random variable. On the other hand, (non-total) orders appear quite natural in financial models with proportional transaction costs (a.k.a. the Kabanov market) in form of solvency cones.
We propose new concepts for multivariate ranking functions with features very close to univariate cdf's and for set-valued quantile functions which, at the same time, generalize univariate quantiles as well as Tukey's halfspace depth regions. Our constructions are designed to deal with general vector orders for the values of random variables, and they produce unambigious lower and upper multivariate quantiles, multivariate V@Rs as well as a multivariate first order stochastic dominance relation. Financial applications to markets with frictions are discussed as well as many other examples and pictures which show the interesting geometric features of the new quantile sets.
The talk is based on: AH Hamel, D Kostner, Cone distribution functions and quantiles for multivariate random variables , J. Multivariate Analysis 167, 2018
Some questions in mathematics are not answered for quite some time, but just sidestepped. One of those questions is the following: What is the quantile of a multi-dimensional random variable? The "sidestepping" in this case produced so-called depth functions and depth regions, and the most prominent among them is the halfspace depth invented by Tukey in 1975, a very popular tool in statistics. When it comes to the definition of multivariate ...

62H99 ; 90B50

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Research talks;Probablity & Statistics

For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. Here, we propose a new approach based on convex duality instead of martingale measures duality: our prices will be expressed using Fenchel conjugate and biconjugate.
This naturally leads to a weak condition of absence of arbitrage opportunity, called Absence of Immediate Profit (AIP), which asserts that the price of the zero claim should be zero. We study the link between (AIP), (NA) and the no-free lunch condition. We show in a one step model that, under (AIP), the super-hedging cost is just the payoff's concave envelop and that (AIP) is equivalent to the non-negativity of the super-hedging prices of some call option.
In the multiple-period case, for a particular, but still general setup, we propose a recursive scheme for the computation of a the super-hedging cost of a convex option. We also give some numerical illustrations.
For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. Here, we propose a new approach based on convex duality instead of martingale measures duality: our prices will be expressed using Fenchel conjugate and biconjugate.
This naturally leads to a weak condition of absence of arbitrage opportunity, called Absence of Immediate Profit (AIP), which asserts ...

60G42 ; 91G10 ; 49N15 ; 90C15

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Research talks;Mathematics in Science and Technology;Probability and Statistics

We study duality and asymptotic of super-replication with market delay. Our main result is the link between scaling limits of delayed markets and the $G$-expectation of Peng.

91G10 ; 91G20 ; 60F05

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Research talks;Mathematics in Science and Technology;Probability and Statistics

We provide a general framework to study viability and arbitrage in models for financial markets. Viability is intended as the existence of a preference relation with the following properties: It is consistent with a set of preferences representing all the plausible agents trading in the market; An agent with such a preference is in equilibrium, namely, he or she prefers to stay at the initial endowment respect to trade. We extend the original framework of Kreps ('79) and Harrison-Kreps ('79) to accommodate for Knightian Uncertainty: preferences of plausible agents are not necessarily determined by a single probability measure. The relations between arbitrage, viability, and existence of (non-)linear pricing rules are investigated.
This is a joint work with Frank Riedel and Mete Soner.
We provide a general framework to study viability and arbitrage in models for financial markets. Viability is intended as the existence of a preference relation with the following properties: It is consistent with a set of preferences representing all the plausible agents trading in the market; An agent with such a preference is in equilibrium, namely, he or she prefers to stay at the initial endowment respect to trade. We extend the original ...

91B02 ; 91B52 ; 60H30

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Research talks;Control Theory and Optimization;Partial Differential Equations;Mathematics in Science and Technology

Following the seminal work by Benamou and Brenier on the time continuous formulation of the optimal transport problem, we show how optimal transport techniques can be used in various areas, ranging from "the reconstruction problem" cosmology to a problem of volatility calibration in finance.

65K10 ; 85A30 ; 85A40 ; 35Q35

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Research talks;Analysis and its Applications

We prove a new stochastic Fubini theorem in a setting where we stochastically integrate a mixture of parametrised integrands, with the mixture taken with respect to a stochastic kernel instead of a fixed measure on the parameter space. To that end, we introduce a notion of measure-valued stochastic integration with respect to a multidimensional semimartingale. As an application, we show how one can handle a class of quite general stochastic Volterra semimartingales.
The talk is based on joint work with Tahir Choulli (University of Alberta, Edmonton).
We prove a new stochastic Fubini theorem in a setting where we stochastically integrate a mixture of parametrised integrands, with the mixture taken with respect to a stochastic kernel instead of a fixed measure on the parameter space. To that end, we introduce a notion of measure-valued stochastic integration with respect to a multidimensional semimartingale. As an application, we show how one can handle a class of quite general stochastic ...

60H05 ; 28A35

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Research talks;Mathematics in Science and Technology;Probability and Statistics

Motivated by recent advances in rough volatility modeling, we introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semi-martingales, 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 simplify recent results in the literature on rough volatility. Motivated by recent advances in rough volatility modeling, we introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semi-martingales, nor Markov processes in general. Nonetheless, their Fourier-Laplace functionals admit exponential-affine representations in terms of solutions ...

91G10 ; 60J60 ; 91G20 ; 65R20

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Research talks;Mathematics in Science and Technology;Probability and Statistics

We represent Hawkes process and their Volterra long term limits, which have recently been used as rough variance processes, as functionals of infinite dimensional affine Markov processes. The representations lead to several new views on affine Volterra processes considered by Abi-Jaber, Larsson and Pulido. We also discuss possible extensions to rough covariance modeling via Volterra Wishart processes.
The talk is based on joint work with Josef Teichmann.
We represent Hawkes process and their Volterra long term limits, which have recently been used as rough variance processes, as functionals of infinite dimensional affine Markov processes. The representations lead to several new views on affine Volterra processes considered by Abi-Jaber, Larsson and Pulido. We also discuss possible extensions to rough covariance modeling via Volterra Wishart processes.
The talk is based on joint work with Josef ...

60J25 ; 91B70

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