43 research outputs found
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
Continuous viscosity solutions to linear-quadratic stochastic control problems with singular terminal state constraint
This paper establishes the existence of a unique nonnegative continuous
viscosity solution to the HJB equation associated with a Markovian
linear-quadratic control problems with singular terminal state constraint and
possibly unbounded cost coefficients. The existence result is based on a novel
comparison principle for semi-continuous viscosity sub- and supersolutions for
PDEs with singular terminal value. Continuity of the viscosity solution is
enough to carry out the verification argument
Smooth Solutions to Portfolio Liquidation Problems under Price-Sensitive Market Impact
International audienceWe consider the stochastic control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price-dependent impact functions describing the trading costs in the primary market and price-dependent adverse selection costs associated with dark pool trading. We prove that the value function can be characterized in terms of the unique smooth solution to a PDE with singular terminal value, establish its explicit asymptotic behavior at the terminal time, and give the optimal trading strategy in feedback form
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