71 research outputs found
Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables
We prove weak laws of large numbers and central limit theorems
of Lindeberg type for empirical centres of mass (empirical Fréchet means)
of independent nonidentically distributed random variables taking values in
Riemannian manifolds. In order to prove these theorems we describe and
prove a simple kind of LindebergâFeller central approximation theorem for
vector-valued random variables, which may be of independent interest and
is therefore the subject of a self-contained section. This vector-valued result
allows us to clarify the number of conditions required for the central limit
theorem for empirical Fréchet means, while extending its scope
A diffusion process associated with Fréchet means
This paper studies rescaled images, under expâ1ÎŒ, of the sample FrĂ©chet means of i.i.d. random variables {Xk|kâ„1} with FrĂ©chet mean ÎŒ on a Rie-mannian manifold. We show that, with appropriate scaling, these images converge weakly to a diffusion process. Similar to the Euclidean case, this limiting diffusion is a Brownian motion up to a linear transformation. However, in addition to the covariance structure of expâ1ÎŒ(X1), this linear transformation also depends on the global Riemannian structure of the manifol
The logarithm map, its limits and Fréchet means in orthant spaces
The first part of the paper studies the expression for, and the properties of, the logarithm map on an orthant space, which is a simple stratified space, with the aim of analysing Fréchet means of probability measures on such a space. In the second part, we use these results to characterise Fréchet means and to derive various of their properties, including the limiting distribution of sample Fréchet means
Markov decision process algorithms for wealth allocation problems with defaultable bonds
This paper is concerned with analysing optimal wealth allocation techniques within a defaultable financial market similar to Bielecki and Jang (2007). It studies a portfolio optimization problem combining a continuous-time jump market and a defaultable security; and presents numerical solutions through the conversion into a Markov decision process and characterization of its value function as a unique fixed point to a contracting operator. This work analyses allocation strategies under several families of utilities functions, and highlights significant portfolio selection differences with previously reported results
A diffusion approach to Stein's method on Riemannian manifolds
We detail an approach to develop Stein's method for bounding integral metrics
on probability measures defined on a Riemannian manifold . Our
approach exploits the relationship between the generator of a diffusion on
with target invariant measure and its characterising Stein
operator. We consider a pair of such diffusions with different starting points,
and investigate properties of solution to the Stein equation based on analysis
of the distance process between the pair. Several examples elucidating the role
of geometry of in these developments are presented
Conjugate duality in stochastic controls with delay
This paper uses the method of conjugate duality to investigate a class of stochastic optimal control problems where state systems are described by stochastic differential equations with delay. For this, we first analyse a stochastic convex problem with delay and derive the expression for the corresponding dual problem. This enables us to obtain the relationship between the optimalities for the two problems. Then, by linking stochastic optimal control problems with delay with a particular type of stochastic convex problem, the result for the latter leads to sufficient maximum principles for the former
Time-randomized stopping problems for a family of utility functions
This paper studies stopping problems of the form for strictly concave or convex utility functions U in a family of increasing functions satisfying certain conditions, where Z is a geometric Brownian motion and T is the time of the nth jump of a Poisson process independent of Z. We obtain some properties of and offer solutions for the optimal strategies to follow. This provides us with a technique to build numerical approximations of stopping boundaries for the fixed terminal time optimal stopping problem presented in [J. Du Toit and G. Peskir, Ann. Appl. Probab., 19 (2009), pp. 983--1014]
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