The euler scheme for anticipating stochastic differential equations

Ahn, Hyungsok; Kohatsu‐Higa, Arturo

doi:10.1080/17442509508834008

Cited by 11 publications

(13 citation statements)

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“…The method of proof is different. Theorem 4.2 in [2] is strongly based in some generator of a highly complex process which in spirit resembles the classical proofs that one can find in e.g. [16], Chapter 14.…”

Section: The Condition Supmentioning

confidence: 75%

“…This method is not well defined in the whole sample space; therefore we will need a localization procedure. Second, the approximation to X 0 does not satisfy the requirements of the weak approximation results in [2]. Third, we are interested in approximating the density of a process with the possible complication that the approximating process may not have a density in itself, although the limit may have one.…”

Section: E(f (φ T (X))) − E(f (φ T (X)))| ≤ Ke(|φ T (X) − φ T (X)|) mentioning

confidence: 99%

“…In Ahn and Kohatsu-Higa [2], we defined and analyzed the weak and strong rate of convergence for an Euler type scheme in the case of Φ t (X 0 ), where X 0 is a smooth random variable (in the Malliavin sense) but not necessarily adapted to F 0 . We assumed that the joint distribution of the vector (X 0 , W t1 , .…”

Section: E(f (φ T (X))) − E(f (φ T (X)))| ≤ Ke(|φ T (X) − φ T (X)|) mentioning

confidence: 99%

“…This type of result measures the path-by-path difference between the solution of (1) and its Euler-Maruyama approximation (2). For this reason this type of result is usually called a strong approximation theorem.…”

Section: Introductionmentioning

confidence: 99%

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Weak Approximations. A Malliavin Calculus Approach

Kohatsu‐Higa

1999

SSRN Journal

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Abstract. We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore, if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable, then approximations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions.

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Section: The Condition Supmentioning

confidence: 75%

Section: E(f (φ T (X))) − E(f (φ T (X)))| ≤ Ke(|φ T (X) − φ T (X)|) mentioning

confidence: 99%

Section: E(f (φ T (X))) − E(f (φ T (X)))| ≤ Ke(|φ T (X) − φ T (X)|) mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weak Approximations. A Malliavin Calculus Approach

Kohatsu‐Higa

1999

SSRN Journal

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“…The procedure of the evaluation of an approximate solution is clear for pathwise integrals (such as the forward stochastic integral or Stratonovich integral); namely, it is sufficient to approximate the flow generated by the nonanticipating equation and use the approximate initial value. Results concerning approximations for the Stratonovich integral can be found in [1,2]. The forward integral can be defined with the help of an enlargement of filtration that reduces it to an integral with respect to a semimartingale.…”

Section: Introductionmentioning

confidence: 99%

Euler approximations of anticipating quasilinear stochastic differential equations

Shevchenko

2006

Theor. Probability and Math. Statist.

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Abstract. We obtain the rate of convergence of Euler type approximations for an anticipating quasilinear stochastic differential equation involving the white noise integral. IntroductionThere are numerous applications of anticipating stochastic differential equations. A typical application is presented by models of a market of securities where there exists an "insider" possessing partial information about the future fluctuations of prices in the market. Thus there is a real need to develop a method of solving such stochastic differential equations (or, at least, of finding approximations of their solutions).There are several known generalizations of the Itô integral for the case of anticipating integrands. The procedure of the evaluation of an approximate solution is clear for pathwise integrals (such as the forward stochastic integral or Stratonovich integral); namely, it is sufficient to approximate the flow generated by the nonanticipating equation and use the approximate initial value. Results concerning approximations for the Stratonovich integral can be found in [1,2]. The forward integral can be defined with the help of an enlargement of filtration that reduces it to an integral with respect to a semimartingale. Approximations of solutions of stochastic differential equations with semimartingales are discussed in [5]. The case of the Skorokhod integral (or, which is the same, of the white noise integral) is not so easy, since it is not defined pathwise and is not a continuous operator in any space L p . This results not only in the discretization in time but also in a shift in ω at every step of the discretization. Approximations of solutions obtained by the method of variation of parametersLet (Ω, F, P) = (S (R), B(S (R)), µ) be a standard white noise space, · ♦ · be the Wick product, B(t) = ω, [0,t] be the Wiener process (Brownian motion), W (t) =Ḃ(t) be a white noise, and More details concerning the white noise space can be found in [3, Chapter 1]. Below we denote by C all constants whose values are not important for the purposes of this paper. Consider the quasilinear stochastic differential equation, where X 0 ∈ R, σ is a certain real bounded nonrandom function, and b(s, x, ω) is, generally speaking, an anticipating random function. Our goal is to construct discrete-time Euler type approximations of a given equation. At a first glance, a natural way is to take δ = T/N for a positive integer N , to consider a uniform partition τ n = nδ, n = 0, . . . , N, of the interval [0, T ], and to construct approximations as follows:However, there are some problems when following this idea. First, it is not easy to evaluate the Wick product on the right-hand side. Second, even if one were able to approximate the solution by using the equalitythen one would face a problem when proving that approximations (1.2) converge, namely, when estimating the stochastic integral, since it is unbounded in every space L p (thus the image of a function under this operator may not belong to L p ). Therefore this natural way le...

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Numerical Analysis of Stochastic Differential Systems and its Applications in Finance

Zheng

2004

Handbook of Computational and Numerical Methods in Finance

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In this note, we provide a survey of recentresults on numerical analysis of stochastic differential systems and its applications in Finance. S. T. Rachev (ed.), Handbook of Computational and Numerical Methods in Finance © Springer Science+Business Media New York 2004 404 Z. Zheng • Numerical methods for stochastic differential equations (SDEs), stochastic optimal control problems and stochastic differential games.! The recognized procedures for stochastic system simulation include time discretization schemes (Euler scheme, Milshtein scheme, Runge-Kutta scheme, Romberg extrapolations) & Monte Carlo/quasi-Monte Carlo method (cf. Asmussen et al. [6], Boyle [27], Boyle et al. [28], Clark [36], Duffie and Glynn [48], Jacod and Protter [72], Kloeden and Platen [77], Kohatsu-Higa and Ogawa [79], Kohatsu-Higa and Protter [80], Kurtz and Protter [84], Milshtein [103], [104], Newton [110], [111], [112], Niederreiter [113], [114], [115], Talay [126], [127], Talay and Tubaro [129]), the Markov chain approximation method (cf. Kushner and Dupuis [87]), the lattice binomial & multinomial approximation (cf. Cox-Ross-Rubinstein [37]), the forward shooting grid method (cf. Barraquand and Pudet [19]), etc. On one hand, these methods provide probabilistic algorithms for deterministic nonlinear PDEs (cf. Kushner [86], [85], Talay and Tubaro [128], Talay and Zheng [131]), on the other hand, they provide direct approaches to simulate the sophisticated stochastic Finance models (cf. Rogers and Talay [123], Gobet and Temam [68]) and to access some quantities which are impossible or difficult to achieve via deterministic algorithms, e.g., the numerical computation of Valueat-Risk (cf. Talay and Zheng [130], [133]). An important advantage of such probabilistic algorithms is that they themselves can be interpreted as natural discrete models of the same financial situation, and thus have an inherent value independent of their use as a numerical method. In addition, as a well-known merit of the Monte Carlo method, probabilistic algorithms are less dimension-sensitive than their deterministic counterparts, which is especially important for large scale Financial Engineering. Finally we would like to point out the recent successful applications of Malliavin calculus in the numerical methods for stochastic systems (cf. Bally and Talay [13], [14], Protter and Talay [121], Fournie et al [58], [59], Gobet [66], Kohatsu-Higa [78]). • Statistical procedures and Filtering to identify model structures (cf. Bamdorff-Nielsen and Shephard [17], Del Moral et al. [105], Florens-Zmirou [56], Fournie and Talay [60], Kutoyants [91], [92], Viens [138]). • Numerical methods for backward stochastic differential equations (BSDEs). The notion of backward stochastic differential equations (BSDEs) was introduced by Pardoux and Peng [118]. Independently, Duffie and Epstein [46], [47] introduced stochastic differential utilities in economics models, as solutions to certain BSDEs . From then on, BSDEs found numerous applications in mathematical economics and finance (cf....

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The euler scheme for anticipating stochastic differential equations

Cited by 11 publications

References 12 publications

Weak Approximations. A Malliavin Calculus Approach

Weak Approximations. A Malliavin Calculus Approach

Euler approximations of anticipating quasilinear stochastic differential equations

Numerical Analysis of Stochastic Differential Systems and its Applications in Finance

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