P. E. Kloeden and E. Platen. Numerical Solutions of Stochastic Differential Equations. SpringerVerlag, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is an excitable dynamical system which has been the paradigm for studying stochastic resonance [2 5] We consider the following periodic signal: r sin # with amplitude r and frequency # 0 . Because of the two independent noise terms, we find it convenient to use the Milshtein method [12] to numerically integrate the set of stochastic differential equations in (2) For concreteness, we choose r 0.34 (arbitrarily) Other parameters are: # 0.005, a 0.5, b 0.15, K 1.0and # 0 15.0. The jamming is assumed to be a Gaussianwhite noise with amplitude varying from 0.1 to ....

P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.

....and volatility a = 2.10 . dX = a(b X)ds aXdW (53) We start the process at x = 1.2.10 and simulate paths one year ahead with 250 sample points. Thus the time interval between the sample points will be roughly one business day. In the simulation we use a strong Taylor scheme of order 2. 0, see [38], p. 356. As the simulated paths do not come from the real mean reversion process, but an approximation to it, we use 10 internal steps in each step of the simulation in the hope of reducing the approximation error. The diffusion parameter estimation is performed using a quadratic variation of the ....

P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Springer Verlag, 1995.

....the reader instead to [24] for information on difference methods in general and Lax Friedrichs method in particular, and to [25, 12, 11] for a description of the front tracking method. We also need to integrate the stochastic differential equation, we have used the Euler Deltat order method, [20]. In figure 2.2 we show the solution where we have no phase transitions, i.e. both k 1 and k 2 are zero. Figure 2.2a shows the solution in the x Gamma t plane, the thin lines shown can be thought of as characteristics, although they are only approximations to these. For a complete discussion of ....

P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.

....Formulae, Adams methods, predictor corrector methods. 1 Introduction Many real world phenomena are (or appear to be) liable to random noisy perturbation. This is the case, for example, in investment finance, turbulent diffusion, chemical kinetics, VLSI circuit design, etc. see, for example, [7, 8, 10]) The mathematical modelling of such phenomena is, therefore, not well Work supported by CNR, contract n. 98.01037.CT01, and by the Universit a di Firenze. y Dipartimento di Matematica U. Dini , Universit a di Firenze, 50134 Firenze, Italy. z Department of Mathematics, University of ....

....without loss of generality, we have assumed to be autonomous, in order to simplify the notation. In the formulation (1) the W j (t) j = 1; d, are independent Wiener processes, modelling independent Brownian motions, which satisfy the initial condition W j (0) 0 with probability 1 [8]. The deterministic term f(y) is sometimes called the drift. The Wiener processes are known to be Gaussian processes satisfying E(W j (t) 0; E(W j (t)W j (s) minft; sg; whose increments Z s t dW j are, if not overlapping, independent and N(0; jt Gamma sj) distributed. The solution of ....

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P. E. Kloeden, E. Platen. Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.

.... bydW (t) y(0) 1: 9 We repeat M=20 batches and each batch contains N=500 different simulations of sample paths of the Ito process and their Euler approximationands corresponding to the same sample paths of the Wiener process. We use the mean and the confidence interval of the absolute error [8] as the criteria, but only give the mean of the absolute errors because the radius of the confidence intervals are of the same order of those of the corresponding mean of the absolute errors. Here we only consider the numerical results using q in the T(A) stability region. The maximum q which is ....

P.E.Kloeden, E.Platen and H.Schurz, The numerical solution of stochastic differential equations through computer experiments, Springer-Verlag, 1994.

.... drift coefficient, g j (t; y) is called the diffusion coefficient, W j (t) is the standard Wiener process whose increment DeltaW j (t) W j (t Deltat) Gamma W j (t) is a Gaussian random variable N(0; Deltat) Recently much work has been done in developing numerical schemes for SDEs [1] 2] [7], and these schemes can be divided into three categories. 1) Explicit numerical schemes in which both the drift coefficient and the diffusion coefficients are explicit. 2) Semi implicit numerical schemes in which the drift coefficient is implicit but the diffusion coefficients are explicit. 3) ....

....0 : 7) For the strong solution up to now, nearly all of the derived numerical methods are explicit or semi implicit methods in nature. The difficulty with implicit methods is their 2 stability properties. Applying the implicit Euler method to the linear Ito test equation (7) Kloeden and Platen [7] point out that the implicit Euler method is not suitable as an approximation because the update values given by y n 1 = 1 1 Gamma ah Gamma b DeltaW n y n may not exist or may be very large depending on W n . Furthermore, it can be shown that this numerical approximation converges to y(t) ....

P.E.Kloeden and E.Platen, The numerical solution of stochastic differential equations, Springer-Verlag, 1992.

....1 2 b 2 )t bW (t) We repeat M=20 batches and each batch contains N=500 different simulations of sample paths of the Ito process and their Euler approximations corresponding to the same sample paths of the Wiener process. We use mean and confidence interval of absolute errors as criteria [10], but only give means of absolute errors because the radius of confidence intervals are nearly the same order as the corresponding means of the absolute errors. The first test equation is solved with a fixed step size h = 0:05, so p = Gamma5 0:05 = Gamma0:25. Table 1 lists some T values, ....

....Euler method. This is in accord with the definition of T (A) stability. We can obtain stable results by the semi implicit Euler method at t 1 = 15 with the same p and q. In this case , the number of time step is 300. Next we consider the following nonlinear SDE as the second test equation [10] dy t = Gamma(ff fi 2 y t ) 1 Gamma y 2 t )dt fi(1 Gamma y 2 t )dW t ; t 2 [0; 3] 13) The exact solution of this equation is y t = 1 y 0 )exp( Gamma2fft 2fiW t ) y 0 Gamma 1 (1 y 0 )exp( Gamma2fft 2fiW t ) 1 Gamma y 0 ; 9 Table 1: The T values of the methods ....

P.E.Kloeden and E.Platen, Numerical solution of stochastic differential equations, Springer-Verlag, Berlin, 1992.

....complicated transformations, a sample from a normal distribution can at least theoretically be converted into a sample from any other distribution. However, we have in mind simulations of discretized diffusion processes using, for example, an Euler scheme or higher order discretization (see Kloeden and Platen 1992), or an exact solution to a stochastic differential equation if available. We do not address the issue of discretization bias. Rather, we assume that an acceptable discretization has already been determined and thus we focus attention on obtaining precise estimates at that level of discretization. ....

....of simply increasing the probability of an event, the change of measure provided by this analysis balances the magnitude and probability of payoffs and puts the mean on the trajectory that effectively maximizes the product of the two. Importance sampling for diffusion processes is treated in Kloeden and Platen (1992) and Newton (1994) in a general setting; further developments in the application of the method to option pricing appear in Andersen (1995) Fournie, Lasry, and Touzi (1997) Newton (1997) and Schoenmakers and Heemink (1997) For problems that can be formulated as path independent options in ....

KLOEDEN, P., and E. PLATEN (1992): Numerical Solution of Stochastic Differential Equations, Berlin: Springer-Verlag.

....paths on can proceed as follows. Consider a onedimensional process dx(t) x; t)dt oe(x; t)dB(t) Suppose that at time t the particle is at location x(t) x. Now for the location at time t dt take x(t dt) x Sigma oe(x; t) p dt with prob 1 2 Sigma (x; t) 2oe(x; t) p dt See [17, 22]. In the bivariate case one generates x and y processes. Figure 1 presents examples of such simulations in the case of the process dr(t) Gamma rH(r)dt dB(t) and two particular potential functions. In the first example H(r; t) r r, i.e. the process is Ornstein Uhlenbeck reverting to the ....

Kloeden, P. E. and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations. Springer, New York.

....observed. The ideal framework to deal with estimation of partially observed dynamical system is the extended Kalman filter (see, for example, Harvey (1989) Wells (1996) In order to apply the Kalman filter it is necessary to discretise system (14) this we do using the EulerMaruyama scheme (Kloeden and Platen (1992)) and the details are available in the appendix. 4. Data and Results The estimation methodology is applied to daily data from the Australian market. We use the market index (All Ordinaries Index) index futures, and call options on the index futures for all the four delivery months (March, June, ....

Kloeden, P. E. and Platen, E. (1992), The Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin.

....of each individual particle is determined numerically. An Euler scheme (O( Deltat) is used to compute the advective step. The diffusive step is determined by a random vector with mean 0 and covariance I . Higher order schemes for stochastic differential equations are described in Kloeden et al. [10]. The O( Deltat) integration scheme for the tidal model is X l = X l Gamma1 (u l ( X l Gamma1 ) u cl ( X l Gamma1 ) Deltat (2 D Deltat) 1 2 l (27) 11 with X l the numerical approximation of X(t 0 l Deltat) The numerical scheme for the long term averaged model is exactly the ....

Kloeden, P.E., Platen, E., and Schurz, H., Numerical Solution of Stochastic Differential Equations through Computer Experiments, Springer, Berlin, 1994.

....the population size is large, this will also result in small steps. One can then reformulate the problem as a continuous diffusion and solve the stochastic differential equations (Fox and Lu, 1994) Special Runge Kutta methods have been developed to solve these with second or higher order accuracy (Kloeden et al. 1993). Integrate and fire networks raise similar issues, with the additional complication that each element s firing can influence the other elements. Fixed step size integration is again only first order accurate, regardless of the accuracy between events, and can also artifactually synchronize the ....

Kloeden, P., Platen, E., and Schurz, H. (1993) Numerical Solution of Stochastic Differential Equations Through Computer Experiments, Springer-Verlag, Heidelberg.

....volatility have tractable analytical expressions. 4 This latter class is quite general, including the affine class of stochastic differential equations popularized by Duffie and Kan (1996) and Dai and Singleton (2000) as well as the quadratic stochastic volatility class of models (see, e.g. Kloeden and Platen, 1992). 2.2 Conditional Moments of Integrated Volatility The square root volatility model, or scalar affine diffusion, is succinctly defined by, dp t = t dt p V t dB t ; dV t = Gamma V t )dt oe p V t dW t ; 3) where V t is a scalar latent volatility process. While this first order ....

Kloeden, Perter E. and Eckhard Platen (1992), Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Springer-Verlag.

....given by eqs. 12) up to order oe 2 (solid line) and (13) up to order a 2 (dashed line) are shown with the numerical solution of eqs. 3) for N = 5 000, a = 1, 1, 1:1 and = 0 (squares) or = 10 (circles) The numerical calculation was done with an explicit 2. 0 order weak scheme [10]. to the values of and a. We want to stress that, except for a small interval 0:5 oe 1:2, our analytical predictions (12) and (13) describe the numerical solution for any noise intensity to a high degree of accuracy, providing a global picture of the main features of the linearly coupled ....

Kloeden P. E. and Platen E., Numerical Solution of Stochastic Differential Equations (Springer, Berlin) 1992.

....coefficients satisfy, kb(Y ) Gamma b( Y )k joe(Y ) Gamma oe( Y )j K kY Gamma Y k ; 4.2) where again kbk 2 = P d j=1 kb j k 2 and joej 2 = P d j;k=1 joe jk j 2 . It is well known that the stochastic differential equation (4. 1) determines a strong Markov process [12]. Define the transition probabilities of this d dimensional process by, P(r; x; s; B) P(Y s 2 BjY r = x) s 0 r s ; B 2 L : 4.3) 15 The probability density functions corresponding to these transition probabilities are denoted p(r; x; s; Delta) and for any Borel set B, P(r; x; s; B) ....

....d dimensional process by, P(r; x; s; B) P(Y s 2 BjY r = x) s 0 r s ; B 2 L : 4.3) 15 The probability density functions corresponding to these transition probabilities are denoted p(r; x; s; Delta) and for any Borel set B, P(r; x; s; B) Z BaeR d p(r; x; s; y) d(y) 4. 4) From [12], for fixed r and x, p(s; y) p(r; x; s; y) satisfy the Fokker Planck equation, s p d X i=1 y i Gamma b i (y) p Delta Gamma 1 2 d X i;j=1 2 y i y j Gamma (A(y) ij p Delta = 0 ; 4.5) with A(y) oe(y) oe t (y) and the initial condition, lim s#r ....

Kloeden, P. E. & Platen, E., Numerical solution of stochastic differential equations, Applications of Mathematics 23, Springer-Verlag (1992). 24

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P. E. Kloeden and E. Platen. Numerical Solutions of Stochastic Differential Equations. SpringerVerlag, 1995.

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P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations. Number 23 in Applications of Mathematics. Springer, 1999. Corrected Third Printing.

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P.E. Kloeden and E. Platen. Numerical solution of stochastic differential equations. Springer-Verlag, Berlin, 1992.

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P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Vol. 23 of Applications of Mathematics: Stochastic Modelling and Applied Probability (Springer-Verlag, New York, 1992.

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Peter E. Kloeden and Eckhard Platen. Numerical Solution of Stochastic Differential Equations, volume 23 of Applications of Mathematics: Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 1992.

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P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations, volume 23 of Applications of Mathematics: Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 1992.

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P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer, 1994.

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P.E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Springer, 1995.

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Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, Berlin Heidelberg New York: Springer 1992

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Kloeden, P. E., and E. Platen, 1992, Numerical Solution of Stochastic Differential Equation, Springer-Verlag, New York.

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