Probability, Uncertainty and Quantitative Risk

September 2021 , Volume 6 , Issue 3

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Reduced-form setting under model uncertainty with non-linear affine intensities
Francesca Biagini and Katharina Oberpriller
2021, 6(3): 159-188 doi: 10.3934/puqr.2021008 +[Abstract](153) +[HTML](66) +[PDF](1377.35KB)
Abstract:

In this paper we extend the reduced-form setting under model uncertainty introduced in [5] to include intensities following an affine process under parameter uncertainty, as defined in [15]. This framework allows us to introduce a longevity bond under model uncertainty in a way consistent with the classical case under one prior and to compute its valuation numerically. Moreover, we price a contingent claim with the sublinear conditional operator such that the extended market is still arbitrage-free in the sense of “no arbitrage of the first kind” as in [6].

Optimal unbiased estimation for maximal distribution
Hanqing Jin and Shige Peng
2021, 6(3): 189-198 doi: 10.3934/puqr.2021009 +[Abstract](149) +[HTML](62) +[PDF](446.56KB)
Abstract:

Unbiased estimation for parameters of maximal distribution is a fundamental problem in the statistical theory of sublinear expectations. In this paper, we proved that the maximum estimator is the largest unbiased estimator for the upper mean and the minimum estimator is the smallest unbiased estimator for the lower mean.

Stein’s method for the law of large numbers under sublinear expectations
Yongsheng Song
2021, 6(3): 199-212 doi: 10.3934/puqr.2021010 +[Abstract](162) +[HTML](70) +[PDF](502.64KB)
Abstract:

Peng, S. [6] proved the law of large numbers under a sublinear expectation. In this paper, we give its error estimates by Stein’s method.

Stochastic maximum principle for systems driven by local martingales with spatial parameters
Jian Song and Meng Wang
2021, 6(3): 213-236 doi: 10.3934/puqr.2021011 +[Abstract](126) +[HTML](60) +[PDF](643.93KB)
Abstract:

We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter. Assuming the convexity of the control domain, we obtain the stochastic maximum principle as the necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. The stochastic linear quadratic problem in this setting is also discussed.

An FBSDE approach to market impact games with stochastic parameters
Samuel Drapeau, Peng Luo, Alexander Schied and Dewen Xiong
2021, 6(3): 237-260 doi: 10.3934/puqr.2021012 +[Abstract](131) +[HTML](66) +[PDF](693.14KB)
Abstract:

In this study, we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting in a unique Nash equilibrium.

Convergence rate of Peng’s law of large numbers under sublinear expectations
Mingshang Hu, Xiaojuan Li and Xinpeng Li
2021, 6(3): 261-266 doi: 10.3934/puqr.2021013 +[Abstract](166) +[HTML](77) +[PDF](302.13KB)
Abstract:

This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations, which improves the results presented by Song [15] and Fang et al. [3].

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