The Parametrix Method for Skew Diffusions

Kohatsu‐Higa, Arturo; Taguchi, Dai; Zhong, Jie

doi:10.1007/s11118-016-9547-0

Cited by 13 publications

(17 citation statements)

References 16 publications

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“…a.e., y ∈ R d . Moreover, the Gaussian lower bound (8) for q(0, x; t, y), and estimations ( 27), (31) with r = −1, we obtain…”

Section: Gaussian Two-sided Bound and Continuity Of Pdfmentioning

confidence: 68%

“…The following lemma plays a crucial role in our argument. Lemma 3.25 (Lemma 7.3 in [31]). Let R = (R t ) t≥0 be a counting process with jumps times π := (τ n ) n∈N and ζ.…”

Section: Application To Unbiased Simulation Schemementioning

confidence: 99%

“…For the Euler-Maruyama scheme X (x,n) , many authors studies the strong error E[sup 0≤t<T |X x t −X )]| for some measurable function f : R d → R, (see, e.g. [5,8,30,33,40,44,46,47,49,50,64,66], see also [1,4,12,23,26,31,34] as an unbiased simulation scheme based on the Euler-Maruyama scheme with random grid and parametrix expansions/Malliavin weight). On the other hand, recently, the convergence rate for a pdf of the Euler-Maruyama scheme studied by using the parametrix method and Malliavin calculus (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

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Probability density function of SDEs with unbounded and path-dependent drift coefficient

Taguchi

Tanaka

2020

Stochastic Processes and their Applications

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In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path-dependent, and diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama-Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super-linear growth condition), Gaussian two-sided bound and Hölder continuity (under sub-linear growth condition) of a probability density function of a solution of SDEs with path-dependent drift coefficient. As an application of explicit representation, we provide the rate of convergence for an Euler-Maruyama (type) approximation, and an unbiased simulation scheme. = σ(t, Y 0,x t )dW t , Y 0,x 0 = x and "local" Novikov condition, in order to use Maruyama-Girsanov transformation. We will also show that the existence of a pdf

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“…a.e., y ∈ R d . Moreover, the Gaussian lower bound (8) for q(0, x; t, y), and estimations ( 27), (31) with r = −1, we obtain…”

Section: Gaussian Two-sided Bound and Continuity Of Pdfmentioning

confidence: 68%

“…The following lemma plays a crucial role in our argument. Lemma 3.25 (Lemma 7.3 in [31]). Let R = (R t ) t≥0 be a counting process with jumps times π := (τ n ) n∈N and ζ.…”

Section: Application To Unbiased Simulation Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Probability density function of SDEs with unbounded and path-dependent drift coefficient

Taguchi

Tanaka

2020

Stochastic Processes and their Applications

Self Cite

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“…Moreover, under assumptions (HR) and (HE), separating the four cases: x, y ≥ 0, x ≥ 0 > y, y ≥ 0 > x and x, y < 0, after some cumbersome but simple computations that we do not detail here (see Lemma 4.5 in [KHTZ16] for more details), one gets…”

Section: Proofs Of Theorems 21 and 22mentioning

confidence: 99%

“…Once again, for more details, we refer the reader to Section 5 in [KHTZ16] and notably to Corollary 5.5 for the extension to the case of bounded measurable drift b.…”

Section: Proofs Of Theorems 21 and 22mentioning

confidence: 99%

On the weak approximation of a skew diffusion by an Euler-type scheme

Frikha¹

2018

Bernoulli

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We study the weak approximation error of a skew diffusion with bounded measurable drift and Hölder diffusion coefficient by an Euler-type scheme, which consists of iteratively simulating skew Brownian motions with constant drift. We first establish two sided Gaussian bounds for the density of this approximation scheme. Then, a bound for the difference between the densities of the skew diffusion and its Euler approximation is obtained. Notably, the weak approximation error is shown to be of order h η/2 , where h is the time step of the scheme, η being the Hölder exponent of the diffusion coefficient.

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