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Sensitivity analysis using ItôMalliavin calculus and application to stochastic optimal control
, 2002
"... We consider a multidimensional diffusion process (X t ) 0tT whose dynamics depends on parameters . Our rst purpose is to give representation formulae of the sensitivity rJ() for the expected cost J() = E(f(X T )) as an expectation: this issue is motivated by stochastic control problems (where th ..."
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Cited by 30 (10 self)
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We consider a multidimensional diffusion process (X t ) 0tT whose dynamics depends on parameters . Our rst purpose is to give representation formulae of the sensitivity rJ() for the expected cost J() = E(f(X T )) as an expectation: this issue is motivated by stochastic control problems (where the controller is parameterized and the optimization problem is then reduced to a parametric optimization one) or by model misspecifications in nance. Known results concerning the evaluation of rJ() by simulations concern the case of smooth cost functions f or of diusion coecients not depending on (see Kushner and Yang, SIAM J. Control Optim. 29 (5), pp. 12161249, 1991). Here, we handle the general case removing these two restrictions, deriving three new type formulae to evaluate rJ(): we call them Malliavin calculus approach, adjoint approach and martingale approach. For this, our basic tools are Itô calculus, Malliavin calculus and martingale arguments. In the second part of this work, we provide discretization procedures to simulate the relevant random variables and we analyze the associated weak error: the nature of the results are new in that context. We prove that the discretization error is essentially linear w.r.t. the time step. Finally, some numerical experiments deal with some examples in random mechanics and finance: we compare different methods in terms of variance, complexity, computational time and inuence of the time discretization step.
Hypoelliptic heat kernel inequalities on Lie groups
, 2005
"... This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associate ..."
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Cited by 16 (2 self)
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This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature ” takes on the value − ∞ at points of degeneracy of the semiRiemannian metric associated to the operator. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are invariant under left translation. In particular, “L ptype ” gradient estimates hold for p ∈ (1, ∞), and the p = 2 gradient estimate
A stochastic approach to a priori estimates and Liouville theorems for harmonic maps
 Bull. Sci. Math
"... Abstract Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain thr ..."
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Cited by 2 (1 self)
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Abstract Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, nonpositively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.
ScienceDirect A stochastic approach to the harmonic map heat flow on manifolds with timedependent Riemannian metric
"... Abstract We first prove stochastic representation formulae for spacetime harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Spacetime harmonic mappings which are defined glo ..."
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Abstract We first prove stochastic representation formulae for spacetime harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Spacetime harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations.
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"... Abstract. This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) are equivalent to a lower bound on the Ricci te ..."
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Abstract. This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature ” takes on the value − ∞ at points of degeneracy of the semiRiemannian metric. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are given by the sum of squares of left invariant vector fields. In particular, “Lptype ” gradient estimates hold for p ∈ (1,∞), and the
Riesz transform on manifolds and heat . . .
, 2004
"... One considers the class of complete noncompact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel s ..."
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One considers the class of complete noncompact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same