51 research outputs found
Malliavin Calculus for regularity structures: the case of gPAM
Malliavin calculus is implemented in the context of [M. Hairer, A theory of
regularity structures, Invent. Math. 2014]. This involves some constructions of
independent interest, notably an extension of the structure which accomodates a
robust, and purely deterministic, translation operator, in -directions,
between "models". In the concrete context of the generalized parabolic Anderson
model in 2D - one of the singular SPDEs discussed in the afore-mentioned
article - we establish existence of a density at positive times.Comment: Minor revision of [v1]. This version published in Journal of
Functional Analysis, Volume 272, Issue 1, 1 January 2017, Pages 363-41
Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation
We consider a utility maximization problem for an investment-consumption
portfolio when the current utility depends also on the wealth process. Such
kind of problems arise, e.g., in portfolio optimization with random horizon or
with random trading times. To overcome the difficulties of the problem we use
the dual approach. We define a dual problem and treat it by means of dynamic
programming, showing that the viscosity solutions of the associated
Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth
functions. This allows to define a smooth solution of the primal
Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique
in a suitable class and coincides with the value function of the primal
problem. Some financial applications of the results are provided
Investment/consumption problem in illiquid markets with regime-switching
We consider an illiquid financial market with different regimes modeled by a
continuous-time finite-state Markov chain. The investor can trade a stock only
at the discrete arrival times of a Cox process with intensity depending on the
market regime. Moreover, the risky asset price is subject to liquidity shocks,
which change its rate of return and volatility, and induce jumps on its
dynamics. In this setting, we study the problem of an economic agent optimizing
her expected utility from consumption under a non-bankruptcy constraint. By
using the dynamic programming method, we provide the characterization of the
value function of this stochastic control problem in terms of the unique
viscosity solution to a system of integro-partial differential equations. We
next focus on the popular case of CRRA utility functions, for which we can
prove smoothness results for the value function. As an important
byproduct, this allows us to get the existence of optimal
investment/consumption strategies characterized in feedback forms. We analyze a
convergent numerical scheme for the resolution to our stochastic control
problem, and we illustrate finally with some numerical experiments the effects
of liquidity regimes in the investor's optimal decision
A Free Boundary Characterisation of the Root Barrier for Markov Processes
We study the existence, optimality, and construction of non-randomised
stopping times that solve the Skorokhod embedding problem (SEP) for Markov
processes which satisfy a duality assumption. These stopping times are hitting
times of space-time subsets, so-called Root barriers. Our main result is,
besides the existence and optimality, a potential-theoretic characterisation of
this Root barrier as a free boundary. If the generator of the Markov process is
sufficiently regular, this reduces to an obstacle PDE that has the Root barrier
as free boundary and thereby generalises previous results from one-dimensional
diffusions to Markov processes. However, our characterisation always applies
and allows, at least in principle, to compute the Root barrier by dynamic
programming, even when the well-posedness of the informally associated obstacle
PDE is not clear. Finally, we demonstrate the flexibility of our method by
replacing time by an additive functional in Root's construction. Already for
multi-dimensional Brownian motion this leads to new class of constructive
solutions of (SEP).Comment: 31 pages, 14 figure
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