Peter E. Kloeden and Eckhard Platen. Numerical Solution of Stochastic Di#erential Equations. Springer-Verlag,  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is to small step using an Euler scheme which is the commonly used approach by practitioners, albeit that this involves an obvious time penalty. Despite the availability of various textbooks on the numerical solution of stochastic di#erential equations such as the classic work by Kloeden and Platen [KP99], very little has been published on improvements over the simple Euler method in the context of the BGM J market model. An exception to this is a research paper by Kurbanmuradov et al. KSS99] in which the authors, nonetheless, only discuss approximations that either result in terminal ....

....C is an integral over time, and A is by dimension a square root of C and thus of dimension # t. There are a number of ways to improve on the Euler method for the numerical integration of stochastic di#erential equations, many of which may be found in the canonical reference by Kloeden and Platen [KP99]. Instead of using any of the well known explicit, implicit, or standard predictor corrector methods, we employ a hybrid technique whereby we integrate the terms # k dW k directly as if the drift coe#cient is constant over any one time step. This is essentially consistent with the standard Euler ....

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Peter E. Kloeden and Eckhard Platen. Numerical Solution of Stochastic Di#erential Equations. Springer-Verlag,

....numerical approximations of stochastic cocycles. Although there are pathwise convergence results available for the Euler method (Gyongy [15] in general convergence tends to be proved on average (known as strong convergence ) Many schemes are shown to converge in such a sense in Kloeden Platen [21], usually with E # sup 0#t#T x(n#t) x(t) 2 # # C T (#t) # for some # 0 (see also Higham et al. 18] The result of the previous section is therefore not immediately applicable. Nevertheless it is still possible to prove a result along similar lines as theorem 2.2 assuming only ....

....2p ] ds. 5.37) We saw above that ER #. The discrete analogue of (5.34) also guarantees that ER#t # M # 26 for all #t su#ciently small. That the backwards Euler scheme converges according to E # sup 0#t#T x #t (t) x(t) 2 # # 0 as #t # 0 is shown in Kloeden Platen [21]. The equicontinuity of the flows for fixed # is essentially straightforward: for the continuous time flow the di#erence between two solutions x(t) and y(t) of dx = f(x) dt # dW t dy = f(y) dt # dW t satisfies d dt (x y) f(x) f(y) From the one sided Lipschitz condition (5.29) d ....

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P.E. Kloeden and E. Platen, Numerical solution of stochastic di#erential equations (Springer, Berlin, 1992).

....The above paragraph argues that it is not a restriction to require the approximating filter process to be piecewise constant. This logic also holds for algorithms based on random sampling. It holds even if some higher order interpolation method (say of the Milstein or other types used in [16]) are used, since even then we would use the interpolation to get a better approximation to the signal process at discrete time instants n# h . Thus, in the approximate filters that are considered here, we approximate the conditional distribution at the instants n# h , and suppose that the filter ....

P.E. Kloeden and E. Platen. Numerical Solution of Stochastic Di#erential Equations. Springer-Verlag, Berlin and New York, 1992.

....scheme, in terms of which the forward rates are generated as: f(t #t,T , #) f(t,T , #)exp # #(t, T ) # # #t #(t, T ) v # (t, T)#t 1 2 #(t, T ) 2 #t # (13) where # is a deviate drawn from a standard normal distribution. Further schema are reviewed in [KP95] in particular most of these are refered to as higher order schema. Simulation schema for LIBOR Market models has been studied by [BR98] who compares the two Euler schema with the Order 2.0 weak scheme . The log Euler is found to have less bias than the direct Euler, whilst the Order 2.0 weak ....

P E Kloeden and E Platen, Numerical Solution of Stochastic Di#erential Equations, Applications of Mathematics, Springer-Verlag, 1995.

....sound methods proposed during the last years #Chang 1987, Wagner 1988, Newton 1994, Tanaka 1998#. The by far most elegant and versatile variance reduction technique for randomly excited dynamic systems, however, is presumably importance sampling utilizing the measuretransformation method #Kloeden and Platen 1992, Milstein 1995#. Measure Transformation Method The structural response at time t #s # t # T#isgiven byap dimensional Ito process X#t#=#X 1 #t#; X p #t## in terms of the stochastic di#erential equations dX#t#=a#t;X#dt b#t; X#dW #t#; X#s#=x #1# with the q dimensional unit Wiener ....

Kloeden, P. E. and Platen, E. #1992#. Numerical Solution of Stochastic Di#erential Equations. Springer, Berlin.

....of response of Du#ng systems. An alternate method through which the evolution of the TPDF can be evaluated is direct Monte Carlo simulation. Here, the sample functions can be generated directly from the stochastic di#erential equations using, for example, the weakly convergentnumerical schemes of Kloeden and Platen #1992#. Estimation of the density function as it evolves through time, then, involves the simultaneous processing of a large number of these sample functions into histograms, the accuracy of which, in the tail regions of the distribution, is inversely proportional to the number of samples. Recent ....

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Di#erential Equations, SpringerVerlag, Berlin, 1992.

....small. In [9] we studied the mean square approximation of stochastic di#erential equations with small noise. Mean square numerical methods are themselves significant. Moreover, they are the basis for the construction of weak schemes. In many cases numerical methods in the weak sense (cf. [5] [8] 12, 14] are appropriate for solving physical problems by a Monte Carlo technique. They are easier to implement than mean square methods and the required random variables can be e#ciently simulated. However, for general systems, weak methods of more than second order tend to require ....

....(3.1) 3.3) must be such that h k (#h 1 2 ) s k = O(h p 1 P l#S h l 1 # J(l) for k = 0, 1, s. To construct a one step approximation, one frequently uses a truncated expansion of the exact solution in terms of Ito integrals (for the stochastic Taylor type expansion, see [5], 8] 17] On the basis of such an expansion, a one step approximation X(t h) of weak order 3 can be derived for a general system (# = 1) which in the case of the system (1.1) has the form X(t h) x # q X r=1 # r I r h(a # 2 b) # 2 q X i,r=1 # i # r I ir # q X ....

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P. E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Di#erential Equations, Springer-Verlag, Berlin, 1992.

....we consider only a single delay of #xed length r # 0 the SDDE #1.1# generalises already a range of practically important classes of DDEs and SDEs. The extension of our results to a #nite number of di#erent time delays is straightforward. The numerical analysis of SDEs is well studied, see, e.g. Kloeden Platen #1999# and Platen #1999# for a recent survey. As we will show, many of the established numerical methods for SDEs can be extended to SDDEs. 2 Within the following we shall concentrate on strong discrete time approximations of solutions of SDDEs of the type #1.1#. These will include Euler and ....

....#0;T#. To be able to judge the basic asymptotic properties of a given discrete time approximation and to classify schemes accordingly we can introduce a concept of convergence. In this paper we are interested in pathwise approximation and use the concept of strong order convergence as de#ned in Kloeden Platen #1999# to distinguish between di#erent discrete time approximations. We shall say that a discrete time approximation Y # converges strongly with order # # 0 at time T if there exists a positive constant C, which does not depend on # and an L 2f2; 3; g such that ## #=E ,# # X T , Y # ....

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Kloeden, P. E. & E. Platen #1999#. Numerical Solution of Stochastic Di#erential Equations,Volume 23 of Appl. Math. Springer. Third corrected printing.

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Springer. Kloeden, P. E. & E. Platen #1999#. Numerical Solution of Stochastic Di#erential Equations,Volume 23 of Appl. Math. Springer. Third corrected printing.

....has a Gamma distribution as stationary distribution. Let us assume that the di#usion process Z is observed at discrete times t i = i#;i=1; 2; with a time step size # # 0. Here we suppose that # is small or, more precisely, will tend to zero asymptotically. Under rather weak assumptions, see Kloeden Platen #1999#, on the functions m and # 2 ,itcan be shown that the Euler approximation Z # #t#=Z # #0# Z t 0 m # Z # #t i s # # ds Z t 0 # # Z # #t i s # # dW #s# #2.22# with t i s = maxft i ;t i # sg, converges in a mean square sense to Z as # 0, i.e. lim # 0 E # sup ....

Kloeden, P. E. & Platen, E. #1999#, Numerical Solution of Stochastic Di#erential Equations,Vol. 23 of Applications of Mathematics, Springer Verlag Berlin Heidelberg.

....with m driving independent standard Wiener processes W 1 ; W m . Extensions of our results to nonautonomous coe#cients and a #nite number of di#erent time delays are straightforward and therefore omitted. Weak numerical methods for ordinary SDEs are well studied, see, for example, 2 Kloeden Platen #1999# and Platen #1999# for a recent survey. In the present paper we will demonstrate that well known weak approximation methods for SDEs can be extended to cover SDDE of the type described by #1.1#. The paper is organized as follows. In Section 2 we establish the existence and uniqueness of a ....

....in basic properties of such approximations. Since we only deal with the approximation of expectations of functions of solutions of SDDEs it is appropriate to introduce some kind of weak convergence. In this paper we shall use the concept of weak order convergence as de#ned in Section 9. 7 in Kloeden Platen #1999#. This allows us to classify given discrete time approximations. We denote by C p the set of all polynomials g : # d #. A discrete time approximation Y # converges weakly with order # # 0 towards X at time T if for each g 2 C p there exists a constant C, which does not depend on # ....

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Kloeden, P. E. & E. Platen #1999#. Numerical Solution of Stochastic Di#erential Equations,Volume 23 of Appl. Math. Springer. Third corrected printing.

....equations with respect to strong and weak convergence criterions. We refer here to the papers of Talay ( 9] 1982) Klauder and Petersen ( 3] 1985) Milstein ( 6] 1988) Hernandez and Spigler ( 2] 1990) Saito and Mitsui ( 8] 1993) Drummond and Mortimer ( 1] 1991) Kloeden and Platen ([4], 5] 1992) just to mention a few of them. As in the deterministic case implicit methods are necessary to integrate sti# systems. However, the introduction of implicitness is restricted in the above mentioned papers to the deterministic terms, e.g. the drift term in the Euler scheme. Let us call ....

....time discretization points # n = n#, n = 0, 1, N with step size # = T N,N = 1, 2, Here we shall use the abbreviation Y n to denote the value of the approximation at time n#. To classify di#erent methods with respect to the rate of strong convergence as in Kloeden and Platen ([4], 1992) we say that a discrete time approximation Y # converges with strong order # 0 if there exist constants # 0 # (0, #) and K #, not depending on #, such that we have a mean global error, Eps(T ) E X T Y # N # K# # (2.1) for all # # (0, # 0 ) The simplest useful ....

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P.E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Di#erential Equations, Appl. Math. 23, Springer-Verlag, Berlin, 1992.

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Kloeden, P. E., and E. Platen (1991): The Numerical Solution of Stochastic Di#erential Equations. New York: Springer.

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