75 research outputs found
Stochastic differential games involving impulse controls and double-obstacle quasi-variational inequalities
We study a two-player zero-sum stochastic differential game with both players
adopting impulse controls, on a finite time horizon. The
Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation of the game
turns out to be a double-obstacle quasi-variational inequality, therefore the
two obstacles are implicitly given. We prove that the upper and lower value
functions coincide, indeed we show, by means of the dynamic programming
principle for the stochastic differential game, that they are the unique
viscosity solution to the HJBI equation, therefore proving that the game admits
a value
Functional it{\^o} versus banach space stochastic calculus and strict solutions of semilinear path-dependent equations
Functional It\^o calculus was introduced in order to expand a functional
depending on time , past and present values of
the process . Another possibility to expand
consists in considering the path as an
element of the Banach space of continuous functions on and to use
Banach space stochastic calculus. The aim of this paper is threefold. 1) To
reformulate functional It\^o calculus, separating time and past, making use of
the regularization procedures which matches more naturally the notion of
horizontal derivative which is one of the tools of that calculus. 2) To exploit
this reformulation in order to discuss the (not obvious) relation between the
functional and the Banach space approaches. 3) To study existence and
uniqueness of smooth solutions to path-dependent partial differential equations
which naturally arise in the study of functional It\^o calculus. More
precisely, we study a path-dependent equation of Kolmogorov type which is
related to the window process of the solution to an It\^o stochastic
differential equation with path-dependent coefficients. We also study a
semilinear version of that equation.Comment: This paper is a substantial improvement with additional research
material of the first part of the unpublished paper arXiv:1401.503
A regularization approach to functional It\^o calculus and strong-viscosity solutions to path-dependent PDEs
First, we revisit functional It\^o/path-dependent calculus started by B.
Dupire, R. Cont and D.-A. Fourni\'e, using the formulation of calculus via
regularization. Relations with the corresponding Banach space valued calculus
introduced by C. Di Girolami and the second named author are explored. The
second part of the paper is devoted to the study of the Kolmogorov type
equation associated with the so called window Brownian motion, called
path-dependent heat equation, for which well-posedness at the level of
classical solutions is established. Then, a notion of strong approximating
solution, called strong-viscosity solution, is introduced which is supposed to
be a substitution tool to the viscosity solution. For that kind of solution, we
also prove existence and uniqueness. The notion of strong-viscosity solution
motivates the last part of the paper which is devoted to explore this new
concept of solution for general semilinear PDEs in the finite dimensional case.
We prove an equivalence result between the classical viscosity solution and the
new one. The definition of strong-viscosity solution for semilinear PDEs is
inspired by the notion of "good" solution, and it is based again on an
approximating procedure
Strong-viscosity Solutions: Semilinear Parabolic PDEs and Path-dependent PDEs
The aim of the present work is the introduction of a viscosity type solution,
called strong-viscosity solution to distinguish it from the classical one, with
the following peculiarities: it is a purely analytic object; it can be easily
adapted to more general equations than classical partial differential
equations. First, we introduce the notion of strong-viscosity solution for
semilinear parabolic partial differential equations, defining it, in a few
words, as the pointwise limit of classical solutions to perturbed semilinear
parabolic partial differential equations; we compare it with the standard
definition of viscosity solution. Afterwards, we extend the concept of
strong-viscosity solution to the case of semilinear parabolic path-dependent
partial differential equations, providing an existence and uniqueness result.Comment: arXiv admin note: text overlap with arXiv:1401.503
Backward SDE Representation for Stochastic Control Problems with Non Dominated Controlled Intensity
We are interested in stochastic control problems coming from mathematical
finance and, in particular, related to model uncertainty, where the uncertainty
affects both volatility and intensity. This kind of stochastic control problems
is associated to a fully nonlinear integro-partial differential equation, which
has the peculiarity that the measure characterizing the
jump part is not fixed but depends on a parameter which lives in a compact
set of some Euclidean space . We do not assume that the family
is dominated. Moreover, the diffusive part can be
degenerate. Our aim is to give a BSDE representation, known as nonlinear
Feynman-Kac formula, for the value function associated to these control
problems. For this reason, we introduce a class of backward stochastic
differential equations with jumps and partially constrained diffusive part. We
look for the minimal solution to this family of BSDEs, for which we prove
uniqueness and existence by means of a penalization argument. We then show that
the minimal solution to our BSDE provides the unique viscosity solution to our
fully nonlinear integro-partial differential equation.Comment: arXiv admin note: text overlap with arXiv:1212.2000 by other author
BSDEs with diffusion constraint and viscous Hamilton-Jacobi equations with unbounded data
We provide a stochastic representation for a general class of viscous
Hamilton-Jacobi (HJ) equations, which has convexity and superlinear
nonlinearity in its gradient term, via a type of backward stochastic
differential equation (BSDE) with constraint in the martingale part. We compare
our result with the classical representation in terms of (super)quadratic BSDE,
and show in particular that existence of a solution to the viscous HJ equation
can be obtained under more general growth assumptions on the coefficients,
including both unbounded diffusion coefficient and terminal data.Comment: to appear in Annales de l'Institut Henri Poincar{\'e} (B),
Probabilit{\'e}s et statistiqu
Robust feedback switching control: dynamic programming and viscosity solutions
We consider a robust switching control problem. The controller only observes
the evolution of the state process, and thus uses feedback (closed-loop)
switching strategies, a non standard class of switching controls introduced in
this paper. The adverse player (nature) chooses open-loop controls that
represent the so-called Knightian uncertainty, i.e., misspecifications of the
model. The (half) game switcher versus nature is then formulated as a two-step
(robust) optimization problem. We develop the stochastic Perron method in this
framework, and prove that it produces a viscosity sub and supersolution to a
system of Hamilton-Jacobi-Bellman (HJB) variational inequalities, which
envelope the value function. Together with a comparison principle, this
characterizes the value function of the game as the unique viscosity solution
to the HJB equation, and shows as a byproduct the dynamic programming principle
for robust feedback switching control problem.Comment: to appear on SIAM Journal on Control and Optimizatio
Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics
We analyze a stochastic optimal control problem, where the state process
follows a McKean-Vlasov dynamics and the diffusion coefficient can be
degenerate. We prove that its value function V admits a nonlinear Feynman-Kac
representation in terms of a class of forward-backward stochastic differential
equations, with an autonomous forward process. We exploit this probabilistic
representation to rigorously prove the dynamic programming principle (DPP) for
V. The Feynman-Kac representation we obtain has an important role beyond its
intermediary role in obtaining our main result: in fact it would be useful in
developing probabilistic numerical schemes for V. The DPP is important in
obtaining a characterization of the value function as a solution of a
non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman
equation), in this case on the Wasserstein space of measures. We should note
that the usual way of solving these equations is through the Pontryagin maximum
principle, which requires some convexity assumptions. There were attempts in
using the dynamic programming approach before, but these works assumed a priori
that the controls were of Markovian feedback type, which helps write the
problem only in terms of the distribution of the state process (and the control
problem becomes a deterministic problem). In this paper, we will consider
open-loop controls and derive the dynamic programming principle in this most
general case. In order to obtain the Feynman-Kac representation and the
randomized dynamic programming principle, we implement the so-called
randomization method, which consists in formulating a new McKean-Vlasov control
problem, expressed in weak form taking the supremum over a family of equivalent
probability measures. One of the main results of the paper is the proof that
this latter control problem has the same value function V of the original
control problem.Comment: 41 pages, to appear in Transactions of the American Mathematical
Societ
Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations
The paper reminds the basic ideas of stochastic calculus via regularizations
in Banach spaces and its applications to the study of strict solutions of
Kolmogorov path dependent equations associated with "windows" of diffusion
processes. One makes the link between the Banach space approach and the so
called functional stochastic calculus. When no strict solutions are available
one describes the notion of strong-viscosity solution which alternative (in
infinite dimension) to the classical notion of viscosity solution.Comment: arXiv admin note: text overlap with arXiv:1401.503
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