The closed-form solution

by applying the Fourier inversion technique we derive the well known formula for the price of an option on a discount bond. Therefore, we first compute the exponential affine solution of the transform 

together with our (log) bond price dynamics under the 7b-forward measure (5.9) we obtain the dynamics of the new variable X(t) as follows 

Note that z G C and we therefore have to apply the complex Ito calculus (see. e.g. Protter [66] and Duffle, Pan and Singleton [28]) to compute the derivative 

plugging the (log) bond price d5mamics (5.9) under the To-forward measure in (5.15) and collecting the deterministic and stochastic terms leads to 

The stochastic process d% (z) is driftless and thereupon a local martingale, if the deterministic function A t,z) solves the following ODE 

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Concerning the numerical accuracy, the closed form solutions of normal surface deformation have been compared to the numerical results calculated through the three methods of DS, DC-FFT, and MLMI. The influence coefficients used in the numerical analyses were obtained from three different schemes Green function, piecewise constant function, and bilinear interpolation. The relative errors, as defined in Eq (39), are given in Table 2 while Fig. 4 provides an illustration of the data. 

The closed-form solutions are more difficult to obtain than those previously obtained by means of the survival functions. Numerical integration or quadrature can be used to solve the differential equation or the integral. For instance ... 

In a different field of study, namely physics, Macdonald [41 ] successfully applied the Nonlinear Least Squares (NLS) method to invert a ID pure diffusion problem. He applied the method to recover a number of Dirac-delta sources with large measurement errors in the data. Alapati and Kabala [2] employed the NLS method without regularization to recover the release history of a ground-water contaminant plume from its current measured spatial distribution. The closed form solution of Eq. (2) subject to Eqs. (3-5) is [34] ... 

Another approach, namely a Minimum Relative Entropy (MRE) inversion was used by Woodbury et al. [70]. The MRE inversion is a method of statistical inference. Woodbury and co-workers also utilized the closed form solution, Eq. (46) and discretized it in the form... 

Chapters 7 to 9 review the closed-form solutions of the systems of Eq. 2.36 for a single component (Chapter 7) and a multicomponent mixture in elution (Chapter 8) and in the displacement mode (Chapter 9). These solutions have great importance because they describe clearly what it is that thermodynamics tries to accomplish in a chromatographic column. The imderstanding of these solutions gives precious clues to the behavior of high-concentration chromatographic bands in actual columns. 

In Table 5.1, the maximum error of formulas (5.3.8) and (5.3.9) are shown in the entire range of the parameter ky for six different kinds of spherical particles, drops, or bubbles. All these estimates were found by comparison with the closed-form solution of problem (5.3.1), (5.3.2) obtained in the diffusion boundary layer approximation [363]. 

We proceed with illustrative examples for application of the proposed up-scaling scheme to seven soil types with properties listed in Table 1-2. The closed-form solution for degree of saturation (Eq. [23]) was fitted to measured data by optimizing parameters p, go, X, and the chemical potential pd at air entry point (that defines Lmax). Note that the Hamaker constant was estimated beforehand, as described in Estimation of the Effective Hamaker Constant for Solid-Vapor Interactions for Different Soils above. The estimated parameters were then used to calculate the liquid-vapor interfacial area for each soil (Eq. [28]). We used square shaped central pores for all soil types except the artificial sand mixture, where triangular pores were applied to emphasize capillaiy processes over adsorption in sand. I lowcver, the closed-form solutions for retention and interfacial area were derived lo accommodate any regular polygon-shaped central pore. Constants for various shapes are described in Table I-1. The values of primary physical constants employed in (he calculations and (heir units are shown in Table 1-3. 

True Isothermal Contact Area. Veziroglu and Chandra [119] used a conformal mapping technique to obtain the closed-form solution for the true isothermal strip ... 

Tables 9.2 and 9.3 list some of the closed-form solutions and correlations for the longitudinal and transverse dispersion coefficients for ordered and disordered media.

We can now evaluate this power series result and compare it to the result we obtained from the closed form solution. That is to say, if we had not recognized, as Mathematica did, that the power series solution could be recast as a log, then we might have simply used the solution we had. Let s compare this new solution with the previous one by making a fimction of it, evaluating t at each y and then plotting it against the previous results ... 

These equations appear to be very similar to those we have just seen, and hence they seem to be simple. In fact they are not simple because the pressure of the gas is a function of time as is the concentration on the surface. The previous experiment has the advantage of being designed around an analysis that was simple to carry out and solve for an analytical expression. We can solve these two equations using Mathematica, but the closed-form solutions are anything but straightforward. To see this run the DSolve code ... 

Having removed the flow term, the analytical solution is found however, we also see that along the way the solver found indeterminance in addition to the closed-form solutions. If we look back at Chapter 5, we find that we already solved this problem, but there we made a substitution for C2[t] in terms of Cl[t], which thereby made the solution process easier and avoided an encoimter with the infinite expression. Nonetheless, we see that including the constant flow term makes the analytical solution difficult to obtain. On the other hand, the numerical solution is trivial to implement, just as long as we have proper parameter values to apply. 

In section (5.2.1), we have derived the closed-form solution for the price of a zero-coupon bond option. Then, later on in section (5.2.2) we introduced the FRFT-technique and showed that this method works excellent for a wide range of strike prices by solving the Fourier inversion numerically. Now, we show that also the IFF is an efficient and accurate method to compute the single exercise probabilities (see section (2.2)) for a G 0,1 via... 

As in section (5.2.2), we introduce the lEE technique by starting from a simpler model such that we obtain a closed-form solution for the option price. Then, the numerical approximations are directly comparable with the equivalent findings for the closed-form solution. Therefore, we derive the option pricing formula of a receiver swaption with only one (u = l) payment date in T. ... 

Given a particular parameter vector of n, the closed-form solution for the conditional optimal values for b and can be obtained. For example, with the prior PDF used in Section 2.4.1.1, the conditional optimal value for b can be computed in a similar fashion as Equation (2.103) ... 

Once the conditional optimal values for b and are obtained, the value of the goodness-of-flt function can be computed by Equation (2.121) whereas the normalizing constant in the posterior PDF does not affect the parametric identification results. By maximizing the goodness-of-fit function with respect to n, the updated model parameters can be obtained. Therefore, the closed-form solution of the conditional optimal parameters reduces the dimension of the original optimization problem from - - - - 1 to N only. 

These properties can be used to obtain the closed-form solution of the evidence integral for the linear regression model classes. By using Equation (6.51), the evidence integral in Equation (6.10) can be rewritten as ... 

By solving the following simultaneous linear algebraic equations 97g(b V, Cj)/dbi = 0, I = 1,2,..., Nb, the closed-form solution of the most probable coefficient vector b can be... 

A CCD-sensored robot operated on the basis of an analytical solution of the 3d re-/intersection problem of dimension four is reduced to dimension one by using prior information. Numerical results from an implementation of the closed-form solution of the 3d re-/intersection algorithm on a robot of the Institut fur Informatik, Technische Universitat Munchen are presented. 

The solution depends on the form of the velocity distribution. Unfortunately, the closed form solutions are only possible for the following three forms... 

The blister test was analyzed by the finite element method (16). To evaluate G, Andrews and Stevenson (17) have tentatively added the elastic energy computed by plate theory (far field) to that computed for an internal crack in an infinite medium (near field) for the same radius and pressure. But the closed form solution (analog to the JKR solution (3) for spheres, and the Kanninen solution (12) for DCB) is not yet known. 

The closed form solutions used in the design program for the pad distortions resulting from pressure and thermal loads are similar to those developed in (1). For this specific construction (Fig. 1) and with pad aspect ratios of approximately 1.0, this method of calculating pad distortions (and ultimately film shape) appears reasonable. 

Substituting hy Ct) of Eq. (A12) into Eq.(B4) and carrying out the integration with respect to t gives the closed form solution. 

Remark on Muskat s solution. We have given the closed form solution, but several subtleties deserve further discussion. The first concerns volumetric calculations for Q using Equation 2-32. We observe that the fracture half-length c and the farfield reservoir pressure Pr do not explicitly appear in that formula,... 

Here, v(x,y,t) = Vg(t) is the prescribed filtration rate, for example, as determined from the Vt law or its radial extensions. While we have defined IJq as the annular velocity parallel to the hole axis in a nonrotating flow, we can more generally take it as the maximum velocity (when both axial and circumferential speeds are present) in flows with rotation, providing increased modeling flexibility. The closed-form solution to this problem is... 

Using the average stress across the adhesive thickness makes the solution somewhat dependent on the mesh that is used to represent the adhesive. In order to calculate the average stresses, the values of stress were determined at the integration points within the finite element mesh as it was at these points that the solution was most accurate. In addition, it was shown that the closed form solution predicts very similar stresses to the finite element method and so it was decided just to report results from the finite element study to avoid repetition (Table 5). 

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