A distorted expectation is a Choquet expectation with respect to the capacity induced by a concave probability distortion. Distorted expectations are encountered in various static settings, in risk theory, mathematical finance and mathematical economics. There are a number of different ways to extend a distorted expectation to a multi-period setting, which are not all time-consistent. One time-consistent extension is to define the non-linear expectation by backward recursion, applying the distorted expectation stepwise, over single periods. In a multinomial random walk model we show that this non-linear expectation is stable when the number of intermediate periods increases to infinity: Under a suitable scaling of the probability distortions and provided that the tick-size and time step-size converge to zero in such a way that the multinomial random walks converge to a Levy process , we show that values of random variables under the multi-period distorted expectations converge to the values under a continuous-time non-linear expectation operator, which may be identified with a certain type of Peng's g-expectation. A coupling argument is given to show that this operator reduces to a classical linear expectation when restricted to the set of pathwise increasing claims. Our results also show that a certain class of g-expectations driven by a Brownian motion and a Poisson random measure may be computed numerically by recursively defined distorted expectations .
In this paper we explore a novel way to combine the dynamic notion of time-consistency with the static notion of quantile-based coherent risk-measure or spectral risk measure, of which Expected Shortfall is a prime example. We introduce a class of dynamic risk measures in terms of a certain family of g-expectations driven by Wiener and Poisson point processes. In analogy with the static case , we show that these risk measures, which we label dynamic spectral risk measures, are locally law-invariant and additive on the set of pathwise increasing random variables. We substantiate the link between dynamic spectral risk measures and their static counterparts by establishing a limit theorem for general path-functionals which shows that such dynamic risk measures arise as limits under vanishing time-step of iterated spectral risk measures driven by approximating lattice random walks. This involves a certain non-standard scaling of the corresponding spectral weight-measures that we identify explicitly .
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A distorted expectation is a Choquet expectation with respect to the capacity induced by a concave probability distortion. Distorted expectations are encountered in various static settings, in risk theory, mathematical finance and mathematical economics. There are a number of different ways to extend a distorted expectation to a multi-period setting, which are not all time-consistent. One time-consistent extension is to define the non-linear expectation by backward recursion, applying the distorted expectation stepwise, over single periods. In a multinomial random walk model we show that this non-linear expectation is stable when the number of intermediate periods increases to infinity: Under a suitable scaling of the probability distortions and provided that the tick-size and time step-size converge to zero in such a way that the multinomial random walks converge to a Levy process
In this paper we explore a novel way to combine the dynamic notion of time-consistency with the static notion of quantile-based coherent risk-measure or spectral risk measure, of which Expected Shortfall is a prime example. We introduce a class of dynamic risk measures in terms of a certain family of g-expectations driven by Wiener and Poisson point processes. In analogy with the static case
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values of random variables under the multi-period distorted expectations converge to the values under a continuous-time non-linear expectation operator, which may be identified with a certain type of Peng's g-expectation. A coupling argument is given to show that this operator reduces to a classical linear expectation when restricted to
these risk measures, which we label dynamic spectral risk measures, are locally law-invariant and additive on
R
claims. Our results also show that a certain class of g-expectations driven by a Brownian motion and a Poisson random measure may be computed numerically by recursively defined distorted expectations
random variables. We substantiate the link between dynamic spectral risk measures and their static counterparts by establishing a limit theorem for general path-functionals which shows that such dynamic risk measures arise as limits under vanishing time-step of iterated spectral risk measures driven by approximating lattice random walks. This involves a certain non-standard scaling of the corresponding spectral weight-measures that we identify explicitly