74 research outputs found
Superreplication under Model Uncertainty in Discrete Time
We study the superreplication of contingent claims under model uncertainty in
discrete time. We show that optimal superreplicating strategies exist in a
general measure-theoretic setting; moreover, we characterize the minimal
superreplication price as the supremum over all continuous linear pricing
functionals on a suitable Banach space. The main ingredient is a closedness
result for the set of claims which can be superreplicated from zero capital;
its proof relies on medial limits.Comment: 14 pages; forthcoming in 'Finance and Stochastics
Robust Superhedging with Jumps and Diffusion
We establish a nondominated version of the optional decomposition theorem in
a setting that includes jump processes with nonvanishing diffusion as well as
general continuous processes. This result is used to derive a robust
superhedging duality and the existence of an optimal superhedging strategy for
general contingent claims. We illustrate the main results in the framework of
nonlinear L\'evy processes.Comment: Forthcoming in 'Stochastic Processes and their Applications
Utility Maximization under Model Uncertainty in Discrete Time
We give a general formulation of the utility maximization problem under
nondominated model uncertainty in discrete time and show that an optimal
portfolio exists for any utility function that is bounded from above. In the
unbounded case, integrability conditions are needed as nonexistence may arise
even if the value function is finite.Comment: 18 page
The Opportunity Process for Optimal Consumption and Investment with Power Utility
We study the utility maximization problem for power utility random fields in
a semimartingale financial market, with and without intermediate consumption.
The notion of an opportunity process is introduced as a reduced form of the
value process of the resulting stochastic control problem. We show how the
opportunity process describes the key objects: optimal strategy, value
function, and dual problem. The results are applied to obtain monotonicity
properties of the optimal consumption.Comment: 24 pages, forthcoming in 'Mathematics and Financial Economics
Random G-expectations
We construct a time-consistent sublinear expectation in the setting of
volatility uncertainty. This mapping extends Peng's G-expectation by allowing
the range of the volatility uncertainty to be stochastic. Our construction is
purely probabilistic and based on an optimal control formulation with
path-dependent control sets.Comment: Published in at http://dx.doi.org/10.1214/12-AAP885 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Risk Aversion Asymptotics for Power Utility Maximization
We consider the economic problem of optimal consumption and investment with
power utility. We study the optimal strategy as the relative risk aversion
tends to infinity or to one. The convergence of the optimal consumption is
obtained for general semimartingale models while the convergence of the optimal
trading strategy is obtained for continuous models. The limits are related to
exponential and logarithmic utility. To derive these results, we combine
approaches from optimal control, convex analysis and backward stochastic
differential equations (BSDEs).Comment: 45 page
The Bellman equation for power utility maximization with semimartingales
We study utility maximization for power utility random fields with and
without intermediate consumption in a general semimartingale model with closed
portfolio constraints. We show that any optimal strategy leads to a solution of
the corresponding Bellman equation. The optimal strategies are described
pointwise in terms of the opportunity process, which is characterized as the
minimal solution of the Bellman equation. We also give verification theorems
for this equation.Comment: Published in at http://dx.doi.org/10.1214/11-AAP776 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Superreplication under Volatility Uncertainty for Measurable Claims
We establish the duality-formula for the superreplication price in a setting
of volatility uncertainty which includes the example of "random G-expectation."
In contrast to previous results, the contingent claim is not assumed to be
quasi-continuous.Comment: 16 pages; forthcoming in 'Electronic Journal of Probability
Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
We study a class of stochastic target games where one player tries to find a
strategy such that the state process almost-surely reaches a given target, no
matter which action is chosen by the opponent. Our main result is a geometric
dynamic programming principle which allows us to characterize the value
function as the viscosity solution of a non-linear partial differential
equation. Because abstract mea-surable selection arguments cannot be used in
this context, the main obstacle is the construction of measurable
almost-optimal strategies. We propose a novel approach where smooth
supersolutions are used to define almost-optimal strategies of Markovian type,
similarly as in ver-ification arguments for classical solutions of
Hamilton--Jacobi--Bellman equations. The smooth supersolutions are constructed
by an exten-sion of Krylov's method of shaken coefficients. We apply our
results to a problem of option pricing under model uncertainty with different
interest rates for borrowing and lending.Comment: To appear in MO
Optimal stopping under adverse nonlinear expectation and related games
We study the existence of optimal actions in a zero-sum game
between a stopper and a controller choosing a
probability measure. This includes the optimal stopping problem
for a class of sublinear expectations
such as the -expectation. We show that the game has a
value. Moreover, exploiting the theory of sublinear expectations, we define a
nonlinear Snell envelope and prove that the first hitting time
is an optimal stopping time. The existence of a saddle
point is shown under a compactness condition. Finally, the results are applied
to the subhedging of American options under volatility uncertainty.Comment: Published at http://dx.doi.org/10.1214/14-AAP1054 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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