135 research outputs found
A control problem with fuel constraint and Dawson-Watanabe superprocesses
We solve a class of control problems with fuel constraint by means of the
log-Laplace transforms of -functionals of Dawson-Watanabe superprocesses.
This solution is related to the superprocess solution of quasilinear parabolic
PDEs with singular terminal condition. For the probabilistic verification
proof, we develop sharp bounds on the blow-up behavior of log-Laplace
functionals of -functionals, which might be of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AAP908 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework
Assuming geometric Brownian motion as unaffected price process ,
Gatheral & Schied (2011) derived a strategy for optimal order execution that
reacts in a sensible manner on market changes but can still be computed in
closed form. Here we will investigate the robustness of this strategy with
respect to misspecification of the law of . We prove the surprising result
that the strategy remains optimal whenever is a square-integrable
martingale. We then analyze the optimization criterion of Gatheral & Schied
(2011) in the case in which is any square-integrable semimartingale and
we give a closed-form solution to this problem. As a corollary, we find an
explicit solution to the problem of minimizing the expected liquidation costs
when the unaffected price process is a square-integrable semimartingale. The
solutions to our problems are found by stochastically solving a finite-fuel
control problem without assumptions of Markovianity
On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals
Motivated by optimal investment problems in mathematical finance, we consider
a variational problem of Neyman-Pearson type for law-invariant robust utility
functionals and convex risk measures. Explicit solutions are found for
quantile-based coherent risk measures and related utility functionals.
Typically, these solutions exhibit a critical phenomenon: If the capital
constraint is below some critical value, then the solution will coincide with a
classical solution; above this critical value, the solution is a superposition
of a classical solution and a less risky or even risk-free investment. For
general risk measures and utility functionals, it is shown that there exists a
solution that can be written as a deterministic increasing function of the
price density
On a class of generalized Takagi functions with linear pathwise quadratic variation
We consider a class of continuous functions on that is
of interest from two different perspectives. First, it is closely related to
sets of functions that have been studied as generalizations of the Takagi
function. Second, each function in admits a linear pathwise
quadratic variation and can thus serve as an integrator in F\"ollmer's pathwise
It\=o calculus. We derive several uniform properties of the class
. For instance, we compute the overall pointwise maximum, the
uniform maximal oscillation, and the exact uniform modulus of continuity for
all functions in . Furthermore, we give an example of a pair
such that the quadratic variation of the sum does not
exist
Optimal Investments for Risk- and Ambiguity-Averse Preferences: A Duality Approach
Ambiguity, also called Knightian or model uncertainty, is a key feature in financial modeling. A recent paper by Maccheroni et al. (2004) characterizes investor preferences under aversion against both risk and ambiguity. Their result shows that these preferences can be numerically represented in terms of convex risk measures. In this paper we study the corresponding problem of optimal investment over a given time horizon, using a duality approach and building upon the results by Kramkov and Schachermayer (1999, 2001).Model uncertainty, ambiguity, convex risk measures, optimal investments, duality theory
Robust Optimal Control for a Consumption-investment Problem
We give an explicit PDE characterization for the solution of the problem of maximizing the utility of both terminal wealth and intertemporal consumption under model uncertainty. The underlying market model consists of a risky asset, whose volatility and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with risk aversion parameter 0Optimal Consumption, Robust Control, Model Uncertainty, Incomplete Markets, Stochastic Volatility, Coherent Risk Measures, Convex Duality
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