Conditional coherent risk measures and regime-switching conic pricing

<p style='text-indent:20px;'>This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function. A model is then developed for the bid and ask prices of a European-type asset by a conic formulation. The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. The bid and ask prices of a European-type asset are then characterized using conic quantization.</p>

General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition

<p style='text-indent:20px;'>This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator <inline-formula> <tex-math id="M1">\begin{document}$ g $\end{document}</tex-math> </inline-formula> satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable <inline-formula> <tex-math id="M2">\begin{document}$ y $\end{document}</tex-math> </inline-formula>, and a stochastic-Lipschitz condition in the state variable <inline-formula> <tex-math id="M3">\begin{document}$ z $\end{document}</tex-math> </inline-formula>. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [<xref ref-type="bibr" rid="b25">25</xref>] and Liu et al. [<xref ref-type="bibr" rid="b15">15</xref>]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities. </p>

Reduced-form setting under model uncertainty with non-linear affine intensities

<p style='text-indent:20px;'>In this paper we extend the reduced-form setting under model uncertainty introduced in [<xref ref-type="bibr" rid="b5">5</xref>] to include intensities following an affine process under parameter uncertainty, as defined in [<xref ref-type="bibr" rid="b15">15</xref>]. 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 [<xref ref-type="bibr" rid="b6">6</xref>]. </p>

Convergence rate of Peng’s law of large numbers under sublinear expectations

<p style='text-indent:20px;'>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 [<xref ref-type="bibr" rid="b15">15</xref>] and Fang et al. [<xref ref-type="bibr" rid="b3">3</xref>]. </p>

Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales

<p style='text-indent:20px;'>The existence, uniqueness, and strict comparison for solutions to a BSDE driven by a multi-dimensional RCLL martingale are developed. The goal is to develop a general multi-asset framework encompassing a wide spectrum of non-linear financial models with jumps, including as particular cases, the setups studied by Peng and Xu [<xref ref-type="bibr" rid="b27">27</xref>, <xref ref-type="bibr" rid="b28">28</xref>] and Dumitrescu et al. [<xref ref-type="bibr" rid="b7">7</xref>] who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump. </p>

On the laws of the iterated logarithm under sub-linear expectations

<p style='text-indent:20px;'>In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities.</p>

Extended conditional G-expectations and related stopping times

<p style='text-indent:20px;'>In this paper, we extend the definition of conditional <inline-formula> <tex-math id="M2">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula> to a larger space on which the dynamical consistency still holds. We can consistently define, by taking the limit, the conditional <inline-formula> <tex-math id="M3">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula> for each random variable <inline-formula> <tex-math id="M4">\begin{document}$ X $\end{document}</tex-math> </inline-formula>, which is the downward limit (respectively, upward limit) of a monotone sequence <inline-formula> <tex-math id="M5">\begin{document}$ \{X_{i}\} $\end{document}</tex-math> </inline-formula> in <inline-formula> <tex-math id="M6">\begin{document}$ L_{G}^{1}(\Omega) $\end{document}</tex-math> </inline-formula>. To accomplish this procedure, some careful analysis is needed. Moreover, we present a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional <inline-formula> <tex-math id="M7">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula>. </p>