61 research outputs found
Singular control of SPDEs with space-mean dynamics
We consider the problem of optimal singular control of a stochastic partial
differential equation (SPDE) with space-mean dependence. Such systems are
proposed as models for population growth in a random environment. We obtain
sufficient and necessary maximum principles for such control problems. The
corresponding adjoint equation is a reflected backward stochastic partial
differential equation (BSPDE) with space-mean dependence. We prove existence
and uniqueness results for such equations. As an application we study optimal
harvesting from a population modelled as an SPDE with space-mean dependence.Comment: arXiv admin note: text overlap with arXiv:1807.0730
A stochastic HJB equation for optimal control of forward-backward SDEs
We study optimal stochastic control problems of general coupled systems of forward- backward stochastic di erential equations with jumps. By means of the It^o-Ventzell formula the system is transformed to a controlled backward stochastic partial di eren- tial equation (BSPDE) with jumps. Using a comparison principle for such BSPDEs we obtain a general stochastic Hamilton-Jacobi- Bellman (HJB) equation for such control problems. In the classical Markovian case with optimal control of jump di usions, the equation reduces to the classical HJB equation. The results are applied to study risk minimization in nancial markets
Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations
This paper is concerned with the stochastic Hamilton-Jacobi-Bellman equation
with controlled leading coefficients, which is a type of fully nonlinear
backward stochastic partial differential equation (BSPDE for short). In order
to formulate the weak solution for such kind of BSPDEs, the classical potential
theory is generalized in the backward stochastic framework. The existence and
uniqueness of the weak solution is proved, and for the partially non-Markovian
case, we obtain the associated gradient estimate. As a byproduct, the existence
and uniqueness of solution for a class of degenerate reflected BSPDEs is
discussed as well.Comment: 29 page
Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control
The present paper continues the study of infinite dimensional calculus via
regularization, started by C. Di Girolami and the second named author,
introducing the notion of "weak Dirichlet process" in this context. Such a
process \X, taking values in a Hilbert space , is the sum of a local
martingale and a suitable "orthogonal" process. The new concept is shown to be
useful in several contexts and directions. On one side, the mentioned
decomposition appears to be a substitute of an It\^o type formula applied to
f(t, \X(t)) where is a function
and, on the other side, the idea of weak Dirichlet process fits the widely used
notion of "mild solution" for stochastic PDE. As a specific application, we
provide a verification theorem for stochastic optimal control problems whose
state equation is an infinite dimensional stochastic evolution equation
Dynkin Game of Stochastic Differential Equations with Random Coefficients, and Associated Backward Stochastic Partial Differential Variational Inequality
A Dynkin game is considered for stochastic differential equations with random
coefficients. We first apply Qiu and Tang's maximum principle for backward
stochastic partial differential equations to generalize Krylov estimate for the
distribution of a Markov process to that of a non-Markov process, and establish
a generalized It\^o-Kunita-Wentzell's formula allowing the test function to be
a random field of It\^o's type which takes values in a suitable Sobolev space.
We then prove the verification theorem that the Nash equilibrium point and the
value of the Dynkin game are characterized by the strong solution of the
associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a
backward stochastic partial differential variational inequality (BSPDVI, for
short) with two obstacles. We obtain the existence and uniqueness result and a
comparison theorem for strong solution of the BSPDVI. Moreover, we study the
monotonicity on the strong solution of the BSPDVI by the comparison theorem for
BSPDVI and define the free boundaries. Finally, we identify the counterparts
for an optimal stopping time problem as a special Dynkin game.Comment: 40 page
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