1,398 research outputs found
Data and uncertainty in extreme risks - a nonlinear expectations approach
Estimation of tail quantities, such as expected shortfall or Value at Risk,
is a difficult problem. We show how the theory of nonlinear expectations, in
particular the Data-robust expectation introduced in [5], can assist in the
quantification of statistical uncertainty for these problems. However, when we
are in a heavy-tailed context (in particular when our data are described by a
Pareto distribution, as is common in much of extreme value theory), the theory
of [5] is insufficient, and requires an additional regularization step which we
introduce. By asking whether this regularization is possible, we obtain a
qualitative requirement for reliable estimation of tail quantities and risk
measures, in a Pareto setting
Representing filtration consistent nonlinear expectations as -expectations in general probability spaces
We consider filtration consistent nonlinear expectations in probability
spaces satisfying only the usual conditions and separability. Under a
domination assumption, we demonstrate that these nonlinear expectations can be
expressed as the solutions to Backward Stochastic Differential Equations with
Lipschitz continuous drivers, where both the martingale and the driver terms
are permitted to jump, and the martingale representation is infinite
dimensional. To establish this result, we show that this domination condition
is sufficient to guarantee that the comparison theorem for BSDEs will hold, and
we generalise the nonlinear Doob-Meyer decomposition of Peng to a general
context
Ergodic BSDEs with jumps and time dependence
In this paper we look at ergodic BSDEs in the case where the forward dynamics
are given by the solution to a non-autonomous (time-periodic coefficients)
Ornstein-Uhlenbeck SDE with L\'evy noise, taking values in a separable Hilbert
space. We establish the existence of a unique bounded solution to an infinite
horizon discounted BSDE. We then use the vanishing discount approach, together
with coupling techniques, to obtain a Markovian solution to the EBSDE. We also
prove uniqueness under certain growth conditions. Applications are then given,
in particular to risk-averse ergodic optimal control and power plant evaluation
under uncertainty
Nash equilibria for non zero-sum ergodic stochastic differential games
In this paper we consider non zero-sum games where multiple players control
the drift of a process, and their payoffs depend on its ergodic behaviour. We
establish their connection with systems of Ergodic BSDEs, and prove the
existence of a Nash equilibrium under the generalised Isaac's conditions. We
also study the case of interacting players of different type
Solutions of Backward Stochastic Differential Equations on Markov Chains
We consider backward stochastic differential equations (BSDEs) related to
finite state, continuous time Markov chains. We show that appropriate solutions
exist for arbitrary terminal conditions, and are unique up to sets of measure
zero. We do not require the generating functions to be monotonic, instead using
only an appropriate Lipschitz continuity condition.Comment: To appear in Communications on Stochastic Analysis, August 200
Filters and smoothers for self-exciting Markov modulated counting processes
We consider a self-exciting counting process, the parameters of which depend
on a hidden finite-state Markov chain. We derive the optimal filter and
smoother for the hidden chain based on observation of the jump process. This
filter is in closed form and is finite dimensional. We demonstrate the
performance of this filter both with simulated data, and by analysing the
`flash crash' of 6th May 2010 in this framework
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