... extended to two risk factors, while the portfolio consists of vanilla options on stocks. 3.1 One Risk Factor To facilitate a better understanding of the behaviour of generalized Pareto distribution ... the analysis of the DAX stocks. Given the lack of extreme events in the time period considered (October 4, 1988 to October 1, 2003), when fitting the tail to the generalized Pareto distribution ... main drivers of risk in the portfolio. This tool is employed to measure the sensitivity of the overall portfolio to individual risk factors, thus allowing us to have a clear view of the risk...
... ODE integrator that implemented this idea wasdeveloped by Bulirsch and Stoer, and so extrapolation methods are often calledBulirsch-Stoer methods.3. Predictor-corrector methods store the solution ... auxiliary variables.The generic problem in ordinary differentialequations is thus reduced to thestudy of a set of N coupled first-order differentialequations for the functionsyi,i=1,2, ,N, having ... bereduced to the study of sets of first-order differential equations. For example thesecond-order equationd2ydx2+ q(x)dydx= r(x)(16.0.1)can be rewritten as two first-order equations dydx=...
... Ordinary Differential Equations (EnglewoodCliffs, NJ: Prentice-Hall).Acton, F.S. 1970,Numerical Methods That Work; 1990, corrected edition (Washington: Mathe-matical Association of America), ... to themidpoint of the interval. Then use the value of both x and y at that midpoint to compute the real step across the whole interval. Figure 16.1.2 illustrates theidea. In equations, k1= ... Keepin mind, however, that the old workhorse’s last trip may well be to take you to thepoorhouse: Bulirsch-Stoer or predictor-corrector methods can be very much moreefficient for problems where very...
... a convenient indicator of truncation error∆ ≡ y2− y1(16.2.2)It is this difference that we shall endeavor to keep to a desired degree of accuracy,neither too large nor too small. We do this ... these considerations into a generally useful stepperroutine is this: One of the arguments of the routine will of course be the vector of dependent variables at the beginning of a proposed step. ... dc1=c1-2825.0/27648.0,dc3=c3-18575.0/48384.0,dc4=c4-13525.0/55296.0,dc6=c6-0.25;float *ak2,*ak3,*ak4,*ak5,*ak6,*ytemp;ak2=vector(1,n);ak3=vector(1,n);ak4=vector(1,n);ak5=vector(1,n);ak6=vector(1,n);ytemp=vector(1,n);for (i=1;i<=n;i++) First step.ytemp[i]=y[i]+b21*h*dydx[i];(*derivs)(x+a2*h,ytemp,ak2);...
... section.The usefulness ofthe modified midpointmethod tothe Bulirsch-Stoertechnique(§16.4) derives from a “deep” result about equations (16.3.2), due to Gragg. It turnsout that the error of (16.3.2), expressed ... the modified midpoint method, which advances a vector of dependent variables y(x) from a point x to a point x + H by a sequence of nsubsteps each of size h,h = H/n (16.3.1)In principle, one could ... nvar]and its deriva-tive vectordydx[1 nvar]. Also input ishtot, the total step to be made, andnstep,thenumber of substeps to be used. The output is returned asyout[1 nvar], which need...
... solution of a set of equations. A second warning is that the techniques in this section arenot particularly good for differentialequations that have singular points inside theinterval of integration. ... grand leap consisting of many (e.g., dozens to hundreds) substeps of modified midpoint method, whichare then extrapolated to zero stepsize.The sequence of separate attempts to cross the interval ... number of n’s to be tried is called KMAXX. For reasons described below we usually take thisequal to 8; the 8th value of the sequence (16.4.2) is 16, so this is the maximumnumber of subdivisions of...
... Second-Order Conservative Equations Usually when you have a system of high-order differentialequationsto solve it is best to reformulate them as a system of first-order equations, as discussed ... bsstep the arrays y and d2y are of length 2n for asystem of n second-order equations. The values of y arestoredinthefirstnelements of y,while the first derivatives are stored in the second n elements. ... class ofequations that occurs quite frequently in practice where you can gainabout a factor of two in efficiency by differencing the equations directly. The equations aresecond-order systems...
... *g1,*g2,*g3,*g4,*ysav;indx=ivector(1,n);a=matrix(1,n,1,n);dfdx=vector(1,n);dfdy=matrix(1,n,1,n);dysav=vector(1,n);err=vector(1,n);g1=vector(1,n);g2=vector(1,n);g3=vector(1,n);g4=vector(1,n);ysav=vector(1,n);xsav=(*x); ... name predictor-corrector is often loosely used to denote all these methods.We suspect that predictor-corrector integrators have had their day, and that theyare no longer the method of choice ... Generalizations of the Bulirsch-Stoermethod, in particular a semi-implicitextrapolation method due to Bader and Deuflhard.• Predictor-corrector methods, most of which are descendants of Gear’sbackward...
... name predictor-corrector is often loosely used to denote all these methods.We suspect that predictor-corrector integrators have had their day, and that theyare no longer the method of choice ... predictor-corrector approachusing an explicit method for the predictor, or by Newton iteration for stiff systems. Why go to all the trouble of introducing a whole new method that turns out to be ... capable researchers have alot of experience with predictor-corrector routines, and they see no reason to makea precipitous change of habit. It is not a bad idea for you to be familiar with theprinciples...
... context of derivative pricing.The book does not claim to be a complete account of the state of theart of the subject. Rather it attempts to provide a systematic framework foran understanding of ... collection of events, sigma-algebraAfiltrationE(X) expectation of XE(X |A) conditional expectation of X under AP (A) probability of AP (A |B) probability of A conditioned on B∈ element of ∈ ... alsoAXt-measurable.Conditional ExpectationsThe notion of conditional expectation is central to many of the concepts thatarise in applications of stochastic processes and also in finance. The mean...
... Generator): to issue controlling pulses according with sample pulses to fire thyristor valves. - Pulse Amplifier: to amplify both sample pusles and controlling pulses to fire desired thyristor ... applied to determine the time of issueing pulse in the positive half-cycles of voltage signal applied to thyristor valves. According to this principle [4,5], there are two voltage signals fed to ... always change in spite of using Automatic Reactive Power Regulator. To regulate voltage and cosφ in an expected range, it is necessary to change the parameters of regulators as the loads change....
... languages of the top ten, N° foreign languages in the top ten, HHI distribution of films produced by language, HHI distribution of top ten film admissions by languages, Hst on distribution of top ... concentration of demand for a narrow segment of the market and is a strong indicator of the propensity of demand to be driven by a “star system” logic. Thus, the market share of the top ten films ... represent only 15% of the total number of cinemas, the size of the theatres is one of the factors that explains the relatively high number of admissions per cinema. Table 8. Number of admissions...
... J. Kato, The asymptotic equivalence of functional differenttial equations, J. Differenttial Equat. 1 (1996) 306.[3] E.V. Voskoresenski, Asymptotic equivalence ofsystemsofdifferential equations, ... conditions of stable and asymptoticequivalence (see [1-5]) of linear delay differentialequations under nonlinear perturbation in Banachspace. The obtained results thank to use of the theories of general ... Journal of Science, Mathematics - Physics 23 (2007) 63-692. Main results2.1. The uniformly stable of null solution of delay differential equations Let us consider the delay differential equations dx(t)dt=...
... translates into ZL=(ra,b)GL. A proof of Lemma 1.29 finishesthe proof of the theorem. ✷Proof of lemma 1.29.The proof is rather similar to the one of lemma 1.23. The only thing that wehave to verify ... O-submodules N1,N2 of N with N = N1⊕N2and Ni= Fifor i =1, 2.Proof. The proof is similar to the proof of Proposition 3.17. Let S1and S2be the set of eigenvalues of E acting on F1and ... analytic sections of the book. Finally,Appendix D discusses systemsof linear partial differential equations and the ex-tent to which the results of this book are known to generalize to this situation.Conspicuously...