Exponential stability for stochastic differential equation driven by G-Brownian motion

AbstractConsider a stochastic differential equation driven by G-Brownian motion dX(t)=AX(t)dt+σ(t,X(t))dBt which might be regarded as a stochastic perturbed system of dX(t)=AX(t)dt. Suppose the second equation is quasi surely exponentially stable. In this paper, we investigate the sufficient conditions under which the first equation is still quasi surely exponentially stable

Proofs of the Ethier and Lee slot machine conjectures

Suppose a gambler pays one coin per coup to play a two-armed Futurity slot machine, an antique casinos, and two coins are refunded for every two consecutive gambler losses. This payoff is called the Futurity award. The casino owner honestly advertises that each arm on his/her two-armed machine is fair in the sense that the asymptotic expected profit of both gambler and dealer is 0 if the gambler only plays either arm. The gambler is allowed to play either arm on each coup alternatively in some deterministic order or at random. For almost 90 years, since Futurity slot machines is designed in 1936, an open problem that has not been solved for a long time is whether the slot machine will obey the so-called "long bet will lose" phenomenon so common to casino games. Ethier and Lee [Ann. Appl. Proba. 20(2010), pp.1098-1125] conjectured that a player will also definitely lose in the long run by applying any non-random-mixture strategy. In this paper, we shall prove Ethier and Lee's conjecture. Our result with Ethier and Lee's conclusion straightforwardly demonstrates that players decide to use either random or non-random two-arm strategies before playing and then repeated without interruption, the casino owners are always profitable even when the Futurity award is taken into account. The contribution of this work is that it helps complete the demystification of casino profitability. Moreover, it paves the way for casino owners to improve casino game design and for players to participate effectively in gambling.Comment: 47 page

Ambiguity, risk and asset returns in continuous time

Existing models in stochastic continuous-time settings assume that beliefs are represented by a probability measure. As illustrated by the Ellsberg Paradox, this feature rules out a priori any concern with ambiguity. This paper formulates a continuous-time intertemporal version of multiple-priors utility, where aversion to ambiguity is admissible. When applied to a representative agent asset market setting, the model delivers restrictions on excess returns that admit interpretations reflecting a premium for risk and a seperate premium for ambiguity.ambiguity, risk, continuous-time, asset returns, Knightian uncertainty, backward stochastic differential equation

A Central Limit Theorem, Loss Aversion and Multi-Armed Bandits

This paper establishes a central limit theorem under the assumption that conditional variances can vary in a largely unstructured history-dependent way across experiments subject only to the restriction that they lie in a fixed interval. Limits take a novel and tractable form, and are expressed in terms of oscillating Brownian motion. A second contribution is application of this result to a class of multi-armed bandit problems where the decision-maker is loss averse

Explicit solutions for a class of nonlinear backward stochastic differential equations and their nodal sets

In this paper, we investigate a class of nonlinear backward stochastic differential equations (BSDEs) arising from financial economics, and give specific information about the nodal sets of the related solutions. As applications, we are able to obtain the explicit solutions to an interesting class of nonlinear BSDEs including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing