We study the existence theory for parabolic variational inequalities in weighted $L^2$ spaces wit... more We study the existence theory for parabolic variational inequalities in weighted $L^2$ spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted $L^2$ setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.
The optimal control problem governed by the stochastic reflection problem associated with a close... more The optimal control problem governed by the stochastic reflection problem associated with a closed convex set $K$ in $\mathbb{R}^d$ is reduced via the corresponding Kolmogorov equation to a determi...
Semigroups of Operators: Theory and Applications, 2000
We will introduce a new quantization method for the vorticity equation which converts the sum of ... more We will introduce a new quantization method for the vorticity equation which converts the sum of the linear and nonlinear operators in this equation to be α—m-accretive and thus allows us to use nonlinear semigroup theory to establish solvability. This method also sets the stage for the study of vorticity dynamics with state constraints and control forcing, which can be formulated as a control problem for parabolic variational inequalities. Applications of this mathematical work include atmospheric weather prediction, flow and combustion control and control of information flow in massive command-control networks.
We study the existence theory for parabolic variational inequalities in weighted $L^2$ spaces wit... more We study the existence theory for parabolic variational inequalities in weighted $L^2$ spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted $L^2$ setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.
The optimal control problem governed by the stochastic reflection problem associated with a close... more The optimal control problem governed by the stochastic reflection problem associated with a closed convex set $K$ in $\mathbb{R}^d$ is reduced via the corresponding Kolmogorov equation to a determi...
Semigroups of Operators: Theory and Applications, 2000
We will introduce a new quantization method for the vorticity equation which converts the sum of ... more We will introduce a new quantization method for the vorticity equation which converts the sum of the linear and nonlinear operators in this equation to be α—m-accretive and thus allows us to use nonlinear semigroup theory to establish solvability. This method also sets the stage for the study of vorticity dynamics with state constraints and control forcing, which can be formulated as a control problem for parabolic variational inequalities. Applications of this mathematical work include atmospheric weather prediction, flow and combustion control and control of information flow in massive command-control networks.
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