Probability, Uncertainty and Quantitative Risk
June 2022 , Volume 7 , Issue 2
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In this short note we consider reflected backward stochastic differential equations (RBSDEs) with a Lipschitz driver and barrier processes that are optional and right lower semicontinuous. In this case, the barrier is represented as a nondecreasing limit of right continuous with left limit (RCLL) barriers. We combine some well-known existence results for RCLL barriers with comparison arguments for the control process to construct solutions. Finally, we highlight the connection of these RBSDEs with standard RCLL BSDEs.
In this note, we establish a compact law of the iterated logarithm under the upper capacity for independent and identically distributed random variables in a sub-linear expectation space. For showing the result, a self-normalized law of the iterated logarithm is established.
In this study, we are interested in stochastic differential equations driven by G-Lévy processes. We illustrate that a certain class of additive functionals of the equations of interest exhibits the path-independent property, generalizing a few known findings in the literature. The study is ended with many examples.
In this paper, the Harnack and shift Harnack inequalities for functional G-SDEs with degenerate noise are derived by the method of coupling by change of measure. Moreover, the gradient estimate for the associated nonlinear semigroup
It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearities converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is z-independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.
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