50 research outputs found
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
Approximate optimality and the risk/reward tradeoff in a class of bandit problems
This paper studies a sequential decision problem where payoff distributions
are known and where the riskiness of payoffs matters. Equivalently, it studies
sequential choice from a repeated set of independent lotteries. The
decision-maker is assumed to pursue strategies that are approximately optimal
for large horizons. By exploiting the tractability afforded by asymptotics,
conditions are derived characterizing when specialization in one action or
lottery throughout is asymptotically optimal and when optimality requires
intertemporal diversification. The key is the constancy or variability of risk
attitude. The main technical tool is a new central limit theorem