Home > Error Analysis > Error Analysis Of The Quantization Algorithm For Obstacle Problems# Error Analysis Of The Quantization Algorithm For Obstacle Problems

## Bally, G.

This phase admits several variants (optimal **or random quantization, see** Bally and Pagès, 2000) which all rely on some projection on the grid following some nearest neighbor rules. Pages / Stochastic Processes and their Applications 106 (2003) 1 – 40As a consequence of (50), we already have an evaluation of the error |u(tk;x)−ˆuk(Projk(x))|in a probabilistic sense. A discretization scheme for RBSDEs2.1. The pricing of multi-asset American style vanilla options is a typical example of such problems. http://axishost.net/error-analysis/error-analysis-of-the-quantization-algorithm.php

Download Info If you experience problems downloading a file, check if you have the proper application to view it first. Close Fields of science Bibliography Bally and Pagès 2000, Bally, V., Pagès, G., 2000. ElsevierAbout ScienceDirectRemote accessShopping cartContact and supportTerms and conditionsPrivacy policyCookies are used by this site. The book is geared toward quantitative analysts, probabilists, and applied mathematicians interested in financial applications. http://www.sciencedirect.com/science/article/pii/S0304414903000267

Of course, such an approach is limited to low-dimensionalproblems (say d63) because of the use of the safety grid.As a second step, the approximation procedure of the (lowest) optimal stopping time∗is Touzi: Maturity randomization for stochastic control **problems, in preparation. [Carr]** Carr P.: Randomization and the American Put, The Review of Financial Studies, 11, 597-626, 1997. [C] Carriere E.: Valuation of the Let u(t; x) be the function thatwe want to approximate, say the solution of the variational inequality which modelsthe problem. Louis You can import bibliographic info in various formats into you bibliographic tool, or just into your word processor.

At thisstage, one simply combines the optimized grid kof the kth layer with a uniform gridU(1=√n;R):= {i=√n; i ∈Z;|i|6R√n}⊗dof step 1=√n. They have been originally introduced by Cox–Ross–Rubinstein (see Cox et al., 1979 or Lamberton and Lapeyre, 1996) for pricing 1-dimensional American style vanilla options but they no longer work in higher In case of further problems read the IDEAS help page. Lyons: **Cubature on Wiener space, Proc.**

Then, as a consequence of Theorem3(see (50) with k=0)|ˆun0(Projn0(x)) −u(tn0;x)|=|ˆut00(Proj0(x)) −ut0(0;x)|6CpeCpT(1 + |x|)n:3.3. Lapeyre et al., 1998). It also allows you to accept potential citations to this item that we are uncertain about. Pages / Stochastic Processes and their Applications 106 (2003) 1 – 40Notations•|:|will denote the canonical Euclidean norm on Rdand :|:the corresponding innerproduct.•For every matrix A(with drows and qcolumns), set A2:= Tr(AA∗)

Screen reader users, click the load entire article button to bypass dynamically loaded article content. et modèles aléatoires, Univ. Then a dynamic programming formula is naturally designed on it. If (Yt)t∈[0;T]solves RBSDE (8), thenYt= esssup∈TtEtf(s; Xs;Ys)ds+h(; X)=Ft;(16)where Ttis the family of (t; T ]-valued stopping times.(Note that when the functionfdoes not depend upon Yt,Eq. (16)can be taken as a denition

From amathematical point of view, this means that the kij’s are replaced by˜kij =M‘=1 1{ˆX‘k+1=xk+1j}1{ˆX‘k=xki}M‘=1 1{ˆX‘k=xki};where (X‘k)06k6n;16‘6Mare Mindependent “copies” either of the diusion(Xtk)06k6nor of its Euler scheme ( Xtk)06k6n. (From now https://www.infona.pl/resource/bwmeta1.element.elsevier-19c8741e-acc4-3e15-a4fc-b84bd08f42a2 Historically, it motivated the use of the celebrated Monte Carlo (MC) method (see e.g. Under the hypothesis of the above proposition−f; 6∗f6+f; and +‘; 6∗‘6−‘;:(56)4. For some appropriate grids, we can produce a function un(t,x) such thatwhere or 1 according to some regularity and simulability properties.

We introduce the natural ltra-tion (Gk)06k6nof the “original” Markov chain Xand its M“copies” X‘;16‘6Mi.e.Gk:= (X‘p;Xp;16p6k; 16‘6M);06k6n:In this section, we evaluate the error obtained by replacing the weights kij’s by the˜kij’s in this page The key fact about theseoptimal grids is that they can be recursively computed by a simulation of the underlyingMarkov chain (either (Xtk)kor ( Xtk)k). An algorithm using projections of random trajectories on grids has already beendevised and successfully implemented by Chevance (1997) to produce discretizationschemes for 1-dimensional BSDEs (without reection). Optimal stopping timeIn the PDE terminology the complementary of the set={(t; x)∈[0;T]×Rq=u(t; x)6h(t; x)}is called the continuation set.

File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(03)00026-7Download Restriction: Full text for ScienceDirect subscribers only As the access to this document is restricted, you may want to look for a different version under "Related research" (further The key result about optimalquantization that we will need further on is the so-called Zador Theorem: if PX(du)=f(u)d(du)+(du);⊥d(dfor Lebesgue measure), thenmin; ||6NX−ˆXpp∼Jp;dfd=(d+p)Np=d as N→+∞:The real constant Jp;d corresponds to the case SP/I/1/77065/10 by the strategic scientific research and experimental development program: SYNAT - “Interdisciplinary System for Interactive Scientific and Scientific-Technical Information”. http://axishost.net/error-analysis/error-analysis-problems-and-procedures.php Furthermore, uturns out to be theminimal solution of the associated variational inequality (see Bensoussan and Lions,1982). 26 V.

Namely, under the other assumptions of the above theorem,(a) For every p¿2;max06k6n|Ytk−Uk|p6max4;p(x)√nwithmax4;p (x):=0Te0T((1 + T)DpC0(x)+2T(Cmax1(x)+Cmax2(x)));where Cmax1(x) and Cmax2(x) are specied in Lemmas 5and 2below, respec-tively.(b) For every p¿2;max06k6n|Ytk−Uk|p6˜max4;p (x)nwith˜max4;p (x):=(1+0Te0T)( ˜Cmax1(x)+ The pricing ofmulti-asset American style vanilla options is a typical example of such problems. For some appropriate grids, we can produce a function un(t; x) such thatsup|x|6R|u(t; x)−un(t; x)|6CR1n+n(N=n)1=d :where =12or 1 according to some regularity and simulability properties.

Pages / Stochastic Processes and their Applications 106 (2003) 1 – 40discretization scheme for the corresponding reected backward stochastic dierentialequation (RBSDE). and N. JavaScript is disabled on your browser. One simply processes a standardMC simulation of the diusion at times tk(or the Euler scheme) and uses the originaldenition i.e.safe;kij := P'Xtk+1 ∈C(xsafe;k+1j)=Xtk∈C(xsafe;ki)(: V.

Pages / Stochastic Processes and their Applications 106 (2003) 1 – 40 29vanilla options like American Puts on a basket of traded assets. Here, we essentiallyinvestigate the problem corresponding to the pricing of (bounded) American style V. and J. useful reference If the function f(only)satises assumption (H2), then,for every p¿1,max06k6nRk−Lkp6C1(x)√nwith C1(x):=2T0(1 + 0Te0T)+2c’0˜C1(x)+ ˜C1(x):(41)For every p¿1,max06k6n|Rk−Lk|p6Cmax1(x)√n(42)withCmax1(x):=2T0(1 + 0Te0T)+2c’0˜Cmax1(x)+ ˜Cmax1(x):Proof.

Pages / Stochastic Processes and their Applications 106 (2003) 1 – 40Proof of Lemma 3.First, one has for every k∈{0;:::;n},|Rk−Uk|6Tnni=k+1Etk|f(ti;Xti;Ui)−f(ti;Xti;Yti)|60Tnni=k+1Etk|Ui−Yti|60Tnni=k+1(Etk|Ui−Ri|+Etk|Ri−Yti|):Let k0∈{0;:::;k}and take Etk0. RachevIngen förhandsgranskning - 2012Handbook of Computational and Numerical Methods in FinanceGeorge A Anastassiou,Svetlozar T RachevIngen förhandsgranskning - 2004Vanliga ord och frasera-stable algorithm applications approach approximation Asian options asset returns assume binary In particular, if x0is the starting point of thediusion process, then ˆU0=ˆu0(x0). It follows from (43) and (44) thatˆUk:= ˆuk(ˆXk) is given byˆUn=h(T; ˆXn);ˆUk= maxh(tk;ˆXk);EˆUk+1 +Tnf(tk+1;ˆXk+1;ˆUk+1)=ˆXk V.

Close Close Accessibility options High contrast On Off Change font size You can adjust the font size by pressing a combination of keys: CONTROL + + increase font size CONTROL + The pricing of multi-asset American style vanilla options is a typical example of such problems. By extension, it also provides an original treatment of Monte Carlo methods for the recursive computation of conditional expectations and solutions of BSDEs and generalized multiple optimal stopping problems and their We come backon uand its approximation in Section 3.The important point in this paper is that the solution does exist and that, furthermore,it may be represented by means of a “reduite”.Proposition

Note that in the above error bound the quantization needs not being optimal(i.e. This happens for instance when Xt=’(t; Bt) for an explicit function’like in the multi-dimensional Black and Scholes model. Let R¿1; assume that ˆuis constructedusing the safety grids sk;06k6ngiven by (53). The first part of this paper is devoted to the analysis of the Lp-error induced by the quantization procedure.

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