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## Probability, Uncertainty and Quantitative Risk

January 2018 , Volume 3

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2018, 3: 9
doi: 10.1186/s41546-018-0034-y

*+*[Abstract](656)*+*[HTML](315)*+*[PDF](748.09KB)**Abstract:**

We show that the comparison results for a backward SDE with jumps established in Royer (Stoch. Process. Appl 116:1358-1376, 2006) and Yin and Mao (J. Math. Anal. Appl 346:345-358, 2008) hold under more simplified conditions. Moreover, we prove existence and uniqueness allowing the coefficients in the linear growth- and monotonicity-condition for the generator to be random and time-dependent. In the

*L*

^{2}-case with linear growth, this also generalizes the results of Kruse and Popier (Stochastics 88:491-539, 2016). For the proof of the comparison result, we introduce an approximation technique:Given a BSDE driven by Brownian motion and Poisson random measure, we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/

*n*.

2018, 3: 8
doi: 10.1186/s41546-018-0033-z

*+*[Abstract](870)*+*[HTML](313)*+*[PDF](599.27KB)**Abstract:**

In this paper, we study strongly robust optimal control problems under volatility uncertainty. In the

*G*-framework, we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.

2018, 3: 7
doi: 10.1186/s41546-018-0028-9

*+*[Abstract](812)*+*[HTML](303)*+*[PDF](507.64KB)**Abstract:**

In this paper, we provide a valuation formula for different classes of actuarial and financial contracts which depend on a general loss process by using Malliavin calculus. Similar to the celebrated Black-Scholes formula, we aim to express the expected cash flow in terms of a building block. The former is related to the loss process which is a cumulated sum indexed by a doubly stochastic Poisson process of claims allowed to be dependent on the intensity and the jump times of the counting process. For example, in the context of stop-loss contracts, the building block is given by the distribution function of the terminal cumulated loss taken at the Value at Risk when computing the expected shortfall risk measure.

2018, 3: 6
doi: 10.1186/s41546-018-0032-0

*+*[Abstract](702)*+*[HTML](370)*+*[PDF](917.26KB)**Abstract:**

The main aim of this paper is to introduce the notion of risk excess measure, to analyze its properties, and to describe some basic construction methods. To compare the risk excess of one distribution

*Q*w.r.t. a given risk distribution

*P*, we apply the concept of hemi-metrics on the space of probability measures. This view of risk comparison has a natural basis in the extension of orderings and hemi-metrics on the underlying space to the level of probability measures. Basic examples of these kind of extensions are induced by mass transportation and by function class induced orderings. Our view towards measuring risk excess adds to the usually considered method to compare risks of

*Q*and

*P*by the values

*ρ*(

*Q*),

*ρ*(

*P*) of a risk measure

*ρ*. We argue that the difference

*ρ*(

*Q*)-

*ρ*(

*P*) neglects relevant aspects of the risk excess which are adequately described by the new notion of risk excess measure. We derive various concrete classes of risk excess measures and discuss corresponding ordering and measure extension properties.

2018, 3: 5
doi: 10.1186/s41546-018-0031-1

*+*[Abstract](763)*+*[HTML](399)*+*[PDF](3636.41KB)**Abstract:**

Asset returns are modeled by locally bilateral gamma processes with zero covariations. Covariances are then observed to be consequences of randomness in variations. Support vector machine regressions on prices are employed to model the implied randomness. The contributions of support vector machine regressions are evaluated using reductions in the economic cost of exposure to prediction residuals. Both local and global mean reversion and momentum are represented by drift dependence on price levels. Optimal portfolios maximize conservative portfolio values calculated as distorted expectations of portfolio returns observed on simulated path spaces. They are also shown to outperform classical alternatives.

2018, 3: 4
doi: 10.1186/s41546-018-0030-2

*+*[Abstract](1017)*+*[HTML](624)*+*[PDF](783.0KB)**Abstract:**

We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations (BSVIEs) with jumps, where pathdependence means the dependence of the free term and generator of a path of a càdlàg process. Furthermore, we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation (FSVIE) with jumps and a linear path-dependent BSVIE with jumps. As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.

2018, 3: 3
doi: 10.1186/s41546-018-0029-8

*+*[Abstract](672)*+*[HTML](304)*+*[PDF](583.16KB)**Abstract:**

We study an optimal investment problem under default risk where related information such as loss or recovery at default is considered as an exogenous random mark added at default time. Two types of agents who have different levels of information are considered. We first make precise the insider's information flow by using the theory of enlargement of filtrations and then obtain explicit logarithmic utility maximization results to compare optimal wealth for the insider and the ordinary agent.

2018, 3: 2
doi: 10.1186/s41546-018-0027-x

*+*[Abstract](936)*+*[HTML](583)*+*[PDF](1028.49KB)**Abstract:**

The objective of this paper is to provide a comprehensive study of the no-arbitrage pricing of financial derivatives in the presence of funding costs, the counterparty credit risk and market frictions affecting the trading mechanism, such as collateralization and capital requirements. To achieve our goals, we extend in several respects the nonlinear pricing approach developed in (El Karoui and Quenez 1997) and (El Karoui et al. 1997), which was subsequently continued in (Bielecki and Rutkowski 2015).

2018, 3: 1
doi: 10.1186/s41546-017-0025-4

*+*[Abstract](565)*+*[HTML](314)*+*[PDF](1600.36KB)**Abstract:**

We consider the class of affine LIBOR models with multiple curves, which is an analytically tractable class of discrete tenor models that easily accommodates positive or negative interest rates and positive spreads. By introducing an interpolating function, we extend the affine LIBOR models to a continuous tenor and derive expressions for the instantaneous forward rate and the short rate. We show that the continuous tenor model is arbitrage-free, that the analytical tractability is retained under the spot martingale measure, and that under mild conditions an interpolating function can be found such that the extended model fits any initial forward curve. This allows us to compute value adjustments (i.e. XVAs) consistently, by solving the corresponding ‘pre-default’ BSDE. As an application, we compute the price and value adjustments for a basis swap, and study the model risk associated to different interpolating functions.

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