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In this paper, we establish an analytic framework for studying setvalued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of the Hukuhara difference between sets, in order to compensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental issues regarding generalized Aumann–Itô integrals, especially when it is connected to the martingale representation theorem. In particular, we propose some necessary extensions of the integral that can be used to represent set-valued martingales with nonsingleton initial values. This extension turns out to be essential for the study of set-valued BSDEs. 

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not subadditive in general, so it is beyond the scope of Peng’s framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time inconsistency, that is, the usual “tower property” fails. By localizing the probability distortion and restricting to a smaller class of random variables, we introduce a so-called distorted probability and construct a conditional expectation in such a way that it coincides with the original nonlinear expectation at time zero, but has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that in the continuous time model this conditional expectation corresponds to a parabolic differential equation whose coefficient involves the law of the underlying diffusion. This work is the first step toward a new understanding of nonlinear expectations under probability distortion and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems. 

Abstract We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. If the diffusion coefficient is semilinear (i.e, linear in the gradient of the solution and nonlinear in the solution; the drift can still be fully nonlinear), we establish a complete theory, including global existence and a comparison principle. 

Abstract. The uptake of carbonyl sulfide (COS) by terrestrial plants is linked tophotosynthetic uptake of CO2 as these gases partly share the sameuptake pathway. Applying COS as a photosynthesis tracer in models requires anaccurate representation of biosphere COS fluxes, but these models have notbeen extensively evaluated against field observations of COS fluxes. In thispaper, the COS flux as simulated by the Simple Biosphere Model, version 4(SiB4), is updated with the latest mechanistic insights and evaluated with siteobservations from different biomes: one evergreen needleleaf forest, twodeciduous broadleaf forests, three grasslands, and two crop fields spread overEurope and North America. We improved SiB4 in several ways to improve itsrepresentation of COS. To account for the effect of atmospheric COS molefractions on COS biosphere uptake, we replaced the fixed atmospheric COS molefraction boundary condition originally used in SiB4 with spatially andtemporally varying COS mole fraction fields. Seasonal amplitudes of COS molefractions are ∼50–200 ppt at the investigated sites with aminimum mole fraction in the late growing season. Incorporating seasonalvariability into the model reduces COS uptake rates in the late growingseason, allowing better agreement with observations. We also replaced theempirical soil COS uptake model in SiB4 with a mechanistic model thatrepresents both uptake and production of COS in soils, which improves thematch with observations over agricultural fields and fertilized grasslandsoils. The improved version of SiB4 was capable of simulating the diurnal andseasonal variation in COS fluxes in the boreal, temperate, and Mediterraneanregion. Nonetheless, the daytime vegetation COS flux is underestimated onaverage by 8±27 %, albeit with large variability across sites. On aglobal scale, our model modifications decreased the modeled COS terrestrialbiosphere sink from 922 Gg S yr−1 in the original SiB4 to753 Gg S yr−1 in the updated version. The largest decrease influxes was driven by lower atmospheric COS mole fractions over regions withhigh productivity, which highlights the importance of accounting forvariations in atmospheric COS mole fractions. The change to a different soilmodel, on the other hand, had a relatively small effect on the globalbiosphere COS sink. The secondary role of the modeled soil component in theglobal COS budget supports the use of COS as a global photosynthesis tracer. Amore accurate representation of COS uptake in SiB4 should allow for improvedapplication of atmospheric COS as a tracer of local- to global-scaleterrestrial photosynthesis.