We show how a minimal deformation of the geometry of the classical Hamilton-Jacobi equation provides a probabilistic theory whose cornerstone is the Hamilton-Jacobi-Bellman equation. This is the basis for a novel dynamical system approach to Stochastic Analysis. 1. Stochastic deformation of classical dynamical systems. The geometrical study of the Hamilton-Jacobi theory lies at the heart of Ana...

Abstract We develop a semiclassical approximation scheme for the constraint equations of supersymmetric canonical quantum gravity. This is achieved by a Born–Oppenheimer type of expansion, in analogy to the case of the usual Wheeler–DeWitt equation. The formalism is only consistent if the states at each order depend on the gravitino field. We recover at consecutive orders the Hamilton–Jacobi eq...

We show that the recently formulated Equivalence Principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one–dimension is sufficient to fix the Schwarzian equation [6], implies a fundamental higher dimensional Möbius invariance which in turn univocally fixes the quantum version of the Hamilton–Jacobi equation...

In this paper we study the convergence of the Galerkin approximation method applied to the Generalized Hamilton-Jacobi-Bellman (GHJB) equation over a compact set containing the origin. The GHJB equation gives the cost of an arbitrary control law and can be used to improve the performance of this control. The GHJB equation can also be used to successively approximate the Hamilton-Jacobi-Bellman ...

The paper concerns with the infinite dimensional Hamilton-JacobiBellman equation related to optimal control problem regulated by a linear transport equation with boundary control. A suitable viscosity solution approach is needed in view of the presence of the unbounded controlrelated term in the state equation in Hilbert setting. An existence-anduniqueness result is obtained.

In this paper we propose a relaxation scheme for solving discrete HJB equations based on scheme II [1] of Lions and Mercier. The convergence of the new scheme has been established. Numerical example shows that the scheme is efficient.

In this paper, optimal consumption and investment decisions are studied for an investor who can invest in a fixed interest rate bank account and a stock whose price is a log normal diffusion. We present the method of the HJB equation in order to explicitly solve problems of this type with modifications such as a fixed percentage transaction cost and a mandatory bequest function. It is shown tha...

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We analyze the associated forward performance SPDE and provide explicit formulae...

An ordinary unambiguous integral representation for the finite propagator of a quantum system is obtained by path integrating in phase space. The skeletonization of the canonical action by means of pieces described by a complete solution of the Hamilton-Jacobi equation leads to a simple composition law that reduces the multiple integration to a sole one. Thus the finite quantum propagator can b...

We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutati...