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Existence and uniqueness results for solutions of stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coeffcient is nondegenerate. Here, existence and uniqueness of strong solutions is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solutions of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coeffcient is exactly half that of the state. In addition to this degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential. The first type consists of discontinuities that may occur on a possibly highdimensional manifold. The second consists of singularities that may occur on a smoothly parameterized curve.

A new optimal control based representation for stationary action trajectories is constructed by exploiting connections between semiconvexity, semiconcavity, and stationarity. This new representation is used to verify a known twopoint boundary value problem characterization of stationary action.

Existence and uniqueness results for stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coefficient is nondegenerate. Here, existence and uniqueness of a strong solution is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solution of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coefficient is exactly half that of the state. In addition to the degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential and the resulting solutions to the dequantized Schrödinger equation. The first type consists of discontinuities that may occur on a possibly highdimensional manifold (up to codimension one). The second consists of singularities that may occur on a lowerdimensional manifold (up to codimension two).

A class of nonlinear, stochastic staticization control problems (including minimization problems with smooth, convex, coercive payoffs) driven by diffusion dynamics with constant diffusion coefficient is considered. A fundamental solution form is obtained where the same solution can be used for a limited variety of terminal costs without resolution of the problem. One may convert this fundamental solution form from a stochastic control problem form to a deterministic control problem form. This yields an equivalence between certain secondorder (in space) HamiltonJacobi partial differential equations (HJ PDEs) and associated firstorder HJ PDEs. This reformulation has substantial numerical implications.

A class of nonlinear, stochastic staticization control problems (including minimization problems with smooth, convex, coercive payoffs) driven by diffusion dynamics and constant diffusion coefficient is considered. Using dynamic programming and tools from static duality, a fundamental solution form is obtained where the same solution can be used for a variety of terminal costs without resolution of the problem. Further, this fundamental solution takes the form of a deterministic control problem rather than a stochastic control problem.

By exploiting minplus linearity, semiconcavity, and semigroup properties of dynamic programming, a fundamental solution semigroup for a class of approximate finite horizon linear infinite dimensional optimal control problems is constructed. Elements of this fundamental solution semigroup are parameterized by the time horizon, and can be used to approximate the solution of the corresponding finite horizon optimal control problem for any terminal cost. They can also be composed to compute approximations on longer horizons. The value function approximation provided takes the form of a minplus convolution of a kernel with the terminal cost. A general construction for this kernel is provided, along with a spectral representation for a restricted class of subproblems.

A new optimal control based representation for stationary action trajectories is constructed by exploiting connections between semiconvexity, semiconcavity, and stationarity. This new representation is used to verify a known twopoint boundary value problem characterization of stationary action.