application of firstorder differential equations to real world systems

Ngày tải lên: 22/06/2014, 02:20

... 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...

Ngày tải lên: 27/04/2013, 11:17

... ODE integrator that implemented this idea was developed by Bulirsch and Stoer, and so extrapolation methods are often called Bulirsch-Stoer methods. 3. Predictor-corrector methods store the solution ... auxiliary variables. The generic problem in ordinary differential equations is thus reduced to the study of a set of N coupled first-order differential equations for the functions y i ,i=1,2, ,N, having ... be reduced to the study of sets of first-order differential equations. For example the second-order equation d 2 y dx 2 + q(x) dy dx = r(x)(16.0.1) can be rewritten as two first-order equations dy dx =...

Ngày tải lên: 15/12/2013, 04:15

... Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice-Hall). Acton, F.S. 1970, Numerical Methods That Work ; 1990, corrected edition (Washington: Mathe- matical Association of America), ... to the midpoint 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 the idea. In equations, k 1 = ... Keep in mind, however, that the old workhorse’s last trip may well be to take you to the poorhouse: Bulirsch-Stoer or predictor-corrector methods can be very much more efficient for problems where very...

Ngày tải lên: 15/12/2013, 04:15

... a convenient indicator of truncation error ∆ ≡ y 2 − y 1 (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 stepper routine 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);...

Ngày tải lên: 24/12/2013, 12:16

... 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 turns out 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 n substeps each of size h, h = H/n (16.3.1) In principle, one could ... nvar] and its deriva- tive vector dydx[1 nvar] . Also input is htot , the total step to be made, and nstep ,the number of substeps to be used. The output is returned as yout[1 nvar] , which need...

Ngày tải lên: 24/12/2013, 12:16

... solution of a set of equations. A second warning is that the techniques in this section are not particularly good for differential equations that have singular points inside the interval of integration. ... grand leap consisting of many (e.g., dozens to hundreds) substeps of modified midpoint method, which are 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 this equal to 8; the 8th value of the sequence (16.4.2) is 16, so this is the maximum number of subdivisions of...

Ngày tải lên: 24/12/2013, 12:16

... Second-Order Conservative Equations Usually when you have a system of high-order differential equations to 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 a system of n second-order equations. The values of y arestoredinthefirstnelements of y, while the first derivatives are stored in the second n elements. ... class of equations that occurs quite frequently in practice where you can gain about a factor of two in efficiency by differencing the equations directly. The equations are second-order systems...

Ngày tải lên: 24/12/2013, 12:16

... *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 they are no longer the method of choice ... Generalizations of the Bulirsch-Stoermethod, in particular a semi-implicit extrapolation method due to Bader and Deuflhard. • Predictor-corrector methods, most of which are descendants of Gear’s backward...

Ngày tải lên: 24/12/2013, 12:16

... name predictor-corrector is often loosely used to denote all these methods. We suspect that predictor-corrector integrators have had their day, and that they are no longer the method of choice ... predictor-corrector approach using 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 a lot of experience with predictor-corrector routines, and they see no reason to make a precipitous change of habit. It is not a bad idea for you to be familiar with the principles...

Ngày tải lên: 24/12/2013, 12:16

... context of derivative pricing. The book does not claim to be a complete account of the state of the art of the subject. Rather it attempts to provide a systematic framework for an understanding of ... collection of events, sigma-algebra A filtration E(X) expectation of X E(X |A) conditional expectation of X under A P (A) probability of A P (A |B) probability of A conditioned on B ∈ element of ∈ ... also A X t -measurable. Conditional Expectations The notion of conditional expectation is central to many of the concepts that arise in applications of stochastic processes and also in finance. The mean...

Ngày tải lên: 19/02/2014, 22:20

... 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....

Ngày tải lên: 05/03/2014, 14:20

... 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, H st 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...

Ngày tải lên: 07/03/2014, 15:20

Ngày tải lên: 14/03/2014, 13:20

... J. Kato, The asymptotic equivalence of functional differenttial equations, J. Differenttial Equat. 1 (1996) 306. [3] E.V. Voskoresenski, Asymptotic equivalence of systems of differential equations, ... conditions of stable and asymptotic equivalence (see [1-5]) of linear delay differential equations under nonlinear perturbation in Banach space. The obtained results thank to use of the theories of general ... Journal of Science, Mathematics - Physics 23 (2007) 63-69 2. Main results 2.1. The uniformly stable of null solution of delay differential equations Let us consider the delay differential equations dx(t) dt =...

Ngày tải lên: 22/03/2014, 11:20

... translates into Z L =(r a,b )G L . A proof of Lemma 1.29 finishes the 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 we have to verify ... O-submodules N 1 ,N 2 of N with N = N 1 ⊕N 2 and N i = F i for i =1, 2. Proof. The proof is similar to the proof of Proposition 3.17. Let S 1 and S 2 be the set of eigenvalues of E acting on F 1 and ... analytic sections of the book. Finally, Appendix D discusses systems of linear partial differential equations and the ex- tent to which the results of this book are known to generalize to this situation. Conspicuously...

Ngày tải lên: 31/03/2014, 15:56

Ngày tải lên: 23/04/2014, 16:11

Ngày tải lên: 24/04/2014, 16:48

Ngày tải lên: 20/06/2014, 04:20

Ngày tải lên: 20/06/2014, 20:20