By Nicolas Bouleau
Many fresh advances in modelling in the utilized sciences and engineering have desirous about the expanding significance of sensitivity analyses. For a given actual, monetary or environmental version, elevated emphasis is now put on assessing the results of adjustments in version outputs that end result from small alterations or mistakes in either the hypotheses and parameters. The technique proposed during this booklet is completely new and lines major features. even if tremendous small, blunders own biases and variances. The equipment provided listed here are capable, because of a particular differential calculus, to supply information regarding the correlation among mistakes in various parameters of the version, in addition to information regarding the biases brought by way of non-linearity. The process uses very robust mathematical instruments (Dirichlet forms), which permit one to deal with blunders in limitless dimensional areas, corresponding to areas of services or stochastic procedures. the tactic is consequently acceptable to non-elementary versions alongside the traces of these encountered in glossy physics and finance. this article has been drawn from displays of analysis performed over the last ten years and that's nonetheless ongoing. The paintings used to be provided along side a path taught together on the Universities of Paris 1 and Paris 6. The booklet is meant for college kids, researchers and engineers with strong wisdom in chance thought.
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Additional info for Error calculus for finance and physics: The language of Dirichlet forms
There exists u, f ∈ L2 such that un → u in L2 un → f in L2 . ∞ ]0, 1[ , we have Let ϕ ∈ CK 1 1 ϕ(x)f (x) dx = lim n↑∞ 0 0 ϕ(x)un (x) dx 1 = lim − n↑∞ 1 ϕ (x)un (x) = − ϕ u dx. e. u ∈ D and un → u in D. 4. In this example, the convergence in D preserves continuity (to be more formal, the existence of a continuous version is actually preserved). This stems from both equicontinuity and the Ascoli theorem. Indeed let un be a Cauchy sequence in D and u˜ n be continuous versions of the un ’s. We then have u˜ n (y) − u˜ n (x) 2 2 y = x un (t) dt 1 ≤ |y − x| 0 2 u n (t) dt, but the un L2 are bounded (a Cauchy sequence is bounded): the u˜ n ’s are equi-uniformly-continuous on [0, 1].
2 The Ornstein–Uhlenbeck semigroup on R and the associated Dirichlet form 27 In other words, f ∈ DA and 1 1 f (x) − xf (x). 2 2 Af (x) = (10) A fortiori f ∈ D, and E [f ] = −Af, f L2 (m) 1 x f (x) − f (x) f (x) dm(x). 2 2 =− Integration by parts yields (11) E [f ] = 1 2 2 f (x) dm(x). If we consider the bilinear operator , defined by (12) [f, g] = f g , and the associated quadratic operator 2 [f ] = f , we have E [f ] = 1 2 [f ] dm and for Cb2 functions (13) [f ] = Af 2 − 2f Af. Recapitulation Let us emphasize the important properties obtained for the one-dimensional Ornstein– Uhlenbeck semigroup.
E. Proof . In this instance dx is the Lebesgue measure on R. The lemma easily follows from the fact that continuous functions with compact support are dense in L1 (dx+µ). 42 III Error structures Let us define x Gn (x) = gn (t) dt. 0 From the functional calculus of class C 1 ∩ Lip, we have Gn ◦ u = gn2 ◦ u [u] and E G n ◦ u − Gm ◦ u = 1 2 gn − gm 2 ◦ u · [u] dP. This expression, coupled with the fact that Gn ◦u tends to G◦u in L2 (P) by dominated convergence, implies that the sequence Gn ◦u is Cauchy in D, and, therefore, converges to G ◦ u in D.
Error calculus for finance and physics: The language of Dirichlet forms by Nicolas Bouleau