Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

Mohammed, Salah-Eldin A.; Nilssen, Torstein; Proske, Frank

doi:10.1214/14-aop909

Cited by 87 publications

(82 citation statements)

References 29 publications

(62 reference statements)

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“…We remark that using analogue techniques as in [30] one could even establish that the strong solution gives rise to a flow of Sobolev diffeomorphisms. This, however, is beyond the scope of this paper.…”

Section: Bsmentioning

confidence: 99%

Computing deltas without derivatives

Baños

Meyer‐Brandis²,

Proske

et al. 2017

Finance Stoch

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A well-known application of Malliavin calculus in Mathematical Finance is the probabilistic representation of option price sensitivities, the socalled Greeks, as expectation functionals that do not involve the derivative of the pay-off function. This allows for numerically tractable computation of the Greeks even for discontinuous pay-off functions. However, while the pay-off function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example excludes already simple regime switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to Itô diffusions with irregular drift coefficients, whereat we here focus on the computation of the Delta, which is the option price sensitivity with respect to the initial value of the underlying. To this purpose we first show existence, Malliavin differentiability, and (Sobolev) differentiability in the initial condition of strong solutions of Itô diffusions with drift coefficients that can be decomposed into the sum of a bounded but merely measurable and a Lipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin and Sobolev derivatives in terms of the local time of the diffusion, respectively. We then turn to the main objective of this article and analyse the existence and probabilistic representation of the corresponding Deltas for European and path-dependent options. We conclude with a small simulation study of several regime-switching examples.

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Section: Bsmentioning

confidence: 99%

Computing deltas without derivatives

Baños

Meyer‐Brandis²,

Proske

et al. 2017

Finance Stoch

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show abstract

“…Moreover, it has been proven (see, for example, Fedrizzi and Flandoli [24], Flandoli, Gubinelli and Priola [28] and Mohammed, Nilssen and Proske [44]), that solutions to (1.15) are more regular than their deterministic analogues. For example, [24] shows that, if u 0 ∈ λ≥1 W λ,1 and…”

Section: 4mentioning

confidence: 99%

Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws

Gess

Souganidis

2016

Comm Pure Appl Math

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Abstract. We study the long-time behavior and regularity of the pathwise entropy solutions to stochastic scalar conservation laws with random in time spatially homogeneous fluxes and periodic initial data. We prove that the solutions converge to their spatial average, which is the unique invariant measure of the associated random dynamical system, and provide a rate of convergence, the latter being new even in the deterministic case for dimensions higher than two. The main tool is a new regularization result in the spirit of averaging lemmata for scalar conservation laws, which, in particular, implies a regularization by noise-type result for pathwise quasi-solutions.

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“…Due to the additive decomposition assumption b(t, ω, x) = b 1 (t, x) + b 2 (t, ω), the analysis of the flow turns out to be much easier than that of the Malliavin derivative considered above. Most of the result of this section will follow as adaptations of some arguments of Mohammed et al [33]. Throughout this section, we denote b n := b 1,n + b 2 where b 1,n : [0, T ] × R → R, is the sequence of smooth functions with compact support approximating b 1 as introduced in the proof of Theorem 1.1, see Step 3.2.2.…”

Section: Stochastic Differentiable Flow For Sdes With Random Driftsmentioning

confidence: 99%

Strong Solutions of Some One-dimensional SDEs with Random and Unbounded Drifts

Pamen¹,

Tangpi²

2019

SIAM J. Math. Anal.

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In this paper, we are interested in the following one dimensional forward stochastic differential equation (SDE)where the driving noise Bt is a d-dimensional Brownian motion. The drift coefficient b : [0, T ] × Ω × R −→ R is Borel measurable and can be decomposed into a deterministic and a random part, i.e., b(t, x, ω) = b1(t, x) + b2 (t, x, ω). Assuming that b1 is of spacial linear growth and b2 satisfies some integrability conditions, we obtain the existence and uniqueness of a strong solution. The method we use is purely probabilitic and relies on Malliavin calculus. As byproducts, we obtain Malliavin differentiability of the solutions, provide an explicit representation for the Malliavin derivative and prove existence of weighted Sobolev differentiable flows. KEYWORDS:Malliavin calculus, random drift, measurable drift, compactness criterion, explicit representation, Sobolev differentiable flow. MSC 2010: 60G15, 60G60, 60H07, 60H10 * We thank an associate editor and the anonymous referees for their feedback and suggestions, one of which led to Theorem 1.3. We also thank Daniel Ocone for his comments. Probabilistic settingLet T ∈ (0, ∞) and d ∈ N be fixed and consider a probability space (Ω, F , P ) equipped with the completed filtration (F t ) t∈[0,T ] of a d-dimensional Brownian motion B. Throughout the paper, the product Ω × [0, T ] is endowed with the predictable σ-algebra. Subsets of R k , k ∈ N, are always endowed with the Borel σ-algebra induced by the Euclidean norm | · |. The interval [0, T ] is equipped with the Lebesgue measure. Unless otherwise stated, all equalities and inequalities

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Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation

Cited by 87 publications

References 29 publications

Computing deltas without derivatives

Computing deltas without derivatives

Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws

Strong Solutions of Some One-dimensional SDEs with Random and Unbounded Drifts

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