Probability, Uncertainty and Quantitative Risk

January 2020 , Volume 5

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Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions
Rainer Buckdahn, Christian Keller, Jin Ma and Jianfeng Zhang
2020, 5: 7 doi: 10.1186/s41546-020-00049-8 +[Abstract](43) +[HTML](0) +[PDF](1100.23KB)
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.
Efficient hedging under ambiguity in continuous time
Ludovic Tangpi
2020, 5: 6 doi: 10.1186/s41546-020-00048-9 +[Abstract](92) +[HTML](34) +[PDF](454.83KB)
Abstract:
It is well known that the minimal superhedging price of a contingent claim is too high for practical use. In a continuous-time model uncertainty framework, we consider a relaxed hedging criterion based on acceptable shortfall risks. Combining existing aggregation and convex dual representation theorems, we derive duality results for the minimal price on the set of upper semicontinuous discounted claims.
Convergence of the deep BSDE method for coupled FBSDEs
Jiequn Han and Jihao Long
2020, 5: 5 doi: 10.1186/s41546-020-00047-w +[Abstract](94) +[HTML](35) +[PDF](734.36KB)
Abstract:
The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs.
Uncertainty and filtering of hidden Markov models in discrete time
Samuel N. Cohen
2020, 5: 4 doi: 10.1186/s41546-020-00046-x +[Abstract](107) +[HTML](39) +[PDF](604.43KB)
Abstract:
We consider the problem of filtering an unseen Markov chain from noisy observations, in the presence of uncertainty regarding the parameters of the processes involved. Using the theory of nonlinear expectations, we describe the uncertainty in terms of a penalty function, which can be propagated forward in time in the place of the filter. We also investigate a simple control problem in this context.
Upper risk bounds in internal factor models with constrained specification sets
Jonathan Ansari and Ludger Rüschendorf
2020, 5: 3 doi: 10.1186/s41546-020-00045-y +[Abstract](76) +[HTML](37) +[PDF](926.7KB)
Abstract:
For the class of (partially specified) internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models. This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor. The proof of our main comparison result is not standard. It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers. An application to real market data shows considerable improvement over the standard method.
Moderate deviation for maximum likelihood estimators from single server queues
Saroja Kumar Singh
2020, 5: 2 doi: 10.1186/s41546-020-00044-z +[Abstract](87) +[HTML](37) +[PDF](335.83KB)
Abstract:
Consider a single server queueing model which is observed over a continuous time interval (0,T], where T is determined by a suitable stopping rule. Let θ be the unknown parameter for the arrival process and $\hat {\theta }_{T}$ be the maximum likelihood estimator of θ. The main goal of this paper is to obtain a moderate deviation result of the maximum likelihood estimator for the single server queueing model under certain regular conditions.
Limit behaviour of the minimal solution of a BSDE with singular terminal condition in the non Markovian setting
Dmytro Marushkevych and Alexandre Popier
2020, 5: 1 doi: 10.1186/s41546-020-0043-5 +[Abstract](100) +[HTML](33) +[PDF](526.71KB)
Abstract:
We use the functional Itô calculus to prove that the solution of a BSDE with singular terminal condition verifies at the terminal time: lim inftT Y (t) = ξ = Y (T). Hence, we extend known results for a non-Markovian terminal condition.

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