Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds

We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for β = 1, namely ∂τ u = σ 2 1 2 (∂ 2The terms u j are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of u that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" τ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of ν, similar to Hagan's formula, but including also the mean reverting term. We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility. Contents32 5.2. The SABR model (with general β) 34 References 35

show abstract

“…Let also h(x) = |e x − K| + = (e x − K) + be the usual pay-off of a European call option, Equation (4). Recall that a function φ :…”

Section: In View Of Equationsmentioning

confidence: 99%

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

2014

View full text Add to dashboard Cite

show abstract

“…We fix δ ∈ C ∂M, {0, 1} and set Proof. This is a special case of the much more general Theorem 1.2.3(i) of [7] (also see [9]).…”

Section: Parabolic Problems On Uniformly Regular Riemannian Manifoldsmentioning

confidence: 87%

“…All this will be exposed in detail in [9]. The reader may also consult our earlier papers [6] and [7].…”

Section: Parabolic Problems On Uniformly Regular Riemannian Manifoldsmentioning

confidence: 99%

Linear parabolic equations with strong boundary degeneration

Amann

2020

J Elliptic Parabol Equ

Self Cite

View full text Add to dashboard Cite

As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in Euclidean spaces. The new feature of our result is the fact that-besides of obtaining an optimal solution theory-we consider the 'natural' case where the degeneration occurs only in the normal direction.2010 Mathematics Subject Classification. 35K65 35K45 53C44 Key words and phrases: Degenerate parabolic boundary value problems, Riemannian manifolds with bounded geometry.

show abstract

“…For manifolds with bounded geometry (no boundary), this result was proved in [49]. The result in [49] was generalized to higher order equations in [7]. For the particular case mixed (Dirichlet/Neumann) boundary conditions and scalar equations, this result was proved in [6].…”

Section: A Uniform Agmon Conditionmentioning

confidence: 93%

“…However, if C 11 is not invertible, the behavior of the problem becomes completely different and, to the best of our knowledge, it is not fully understood at this time (see, however, [54] and the references therein). (8) and the related form B from Equation (7). Let us assume that M = U ⊂ R m is a submanifold with boundary of dimension m. (Here ∂U denotes the boundary of U as a manifold with boundary, not as a subset of R m !)…”

Section: 21mentioning

confidence: 99%

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

Große

Nistor

2019

Potential Anal

View full text Add to dashboard Cite

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P, C) is equivalent to the uniform regularity of the natural family (Px, Cx) of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincaré inequality of (Ammann-Große-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, wellknown theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition. 12 4. Variational boundary conditions and regularity 13 4.

show abstract

Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds

Cited by 13 publications

References 39 publications

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

A Volatility-of-Volatility Expansion of the Option Prices in the SABR Stochastic Volatility Model

Linear parabolic equations with strong boundary degeneration

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

Contact Info

Product

Resources

About