$\varepsilon$-Strong simulation for multidimensional stochastic differential equations via rough path analysis

Abstract

Consider a multidimensional diffusion process X = {X (t) : t ∈ [0, 1]}. Let ε > 0 be a deterministic, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of X, we construct a probability space, supporting both X and an explicit, piecewise constant, fully simulatable process Xε such that sup 0≤t≤1with probability one. Moreover, the user can adaptively choose ε ∈ (0, ε) so that X ε (also piecewise constant and fully simulatable) can be constructed conditional on Xε to ensure an error smaller than ε with probability one. Our construction requires a detailed study of continuity estimates of the Itô map using Lyons' theory of rough paths. We approximate the underlying Brownian motion, jointly with the Lévy areas with a deterministic ε error in the underlying rough path metric.MSC 2010 subject classifications: Primary 34K50, 65C05, 82B80; secondary 97K60

A dichotomy for sampling barrier-crossing events of random walks with regularly varying tails

“…(S2) We have P(X + Z > t) ≤ P(Z > t) for all sufficiently large t, where Z has the residual life distribution (8) and is independent of X.…”

“…Suppose that Assumptions (A1)-(A3) hold. Let Z be a random variable independent of X with the residual life distribution (8). Then the following statements hold:…”

“…sampling, since naive Monte Carlo sampling devotes much computational time to paths that never cross the barrier and must therefore be ultimately discarded.The ability to sample up to the first barrier-crossing time plays a central role in several related problems, such as for sampling paths up to their maximum [12] or for sampling only the maximum itself [15]. In turn these have applications to perfect sampling from stationary distributions [7,13] and to approximately solving stochastic differential equations [8].Main contributions. The central question in this paper is: Can the change of measure proposed in Blanchet and Glynn [9] be used for exact, i.e.…”

Controlled rough paths on manifolds I

“…Recall that Taylor's formula with integral remainder states for any smooth function g on [0,1] , that…”

“…Lyons' theory has found numerous applications to stochastic calculus and stochastic differential equations, for example see [4], [5], [6], [8], and the references therein. For some more recent applications, see [1], [19], [18], [9] , and [2].…”