Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions

Karatzas, Ioannis; Ruf, Johannes

doi:10.1214/14-aihp660

Cited by 7 publications

(6 citation statements)

References 78 publications

(105 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…having used [19, Thm 6.13, (c)]. Using again the same result, the term (S2), together with (16), gives (with the constant C changing from line to line)…”

Section: The Kolmogorov Pdementioning

confidence: 86%

“…Coming back to the one-dimensional case, the main idea of [11] was the so called Zvonkin transform which allows to transform the candidate solution process X into a solution of a stochastic differential equation with continuous non-degenerate coefficients without drift. Recently [16] has considered other types of transforms to study similar equations. Indeed the transformation introduced by Zvonkin in [27], when the drift is a function, is also stated in the multidimensional case.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multidimensional stochastic differential equations with distributional drift

Flandoli¹,

Issoglio²,

Russo³

2016

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…having used [19, Thm 6.13, (c)]. Using again the same result, the term (S2), together with (16), gives (with the constant C changing from line to line)…”

Section: The Kolmogorov Pdementioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Multidimensional stochastic differential equations with distributional drift

Flandoli¹,

Issoglio²,

Russo³

2016

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…This approach has been generalized in an effort to study the pathwise solvability of stochastic differential equations by [12,32,37,52].…”

Section: Proposition 71 (Conditions For Explosions In Special Cases)mentioning

confidence: 99%

Distribution of the time to explosion for one-dimensional diffusions

Karatzas

Ruf

2015

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to the computation of expectations of suitable nonnegative local martingales. Moreover, we characterize the distribution function of the time to explosion as the minimal solution to a certain Cauchy problem for an appropriate parabolic differential equation; this leads to alternative characterizations of Feller's criterion for explosions. We discuss in detail several examples for which To the Memory of Marc Yor. for their insightful remarks that improved the manuscript. We thank Alex Mijatović for raising the issues of strict positivity and full support and for prompting us to think about them (Sects. 4.2 and 4.3, respectively). We are grateful to the referees for their meticulous reading of the paper and their many and incisive suggestions. J.R. acknowledges generous support from the Oxford-Man Institute of Quantitative Finance, University of Oxford, where a major part of this work was completed. The problem discussed here was suggested to us by Marc Yor, who also gave us generous expert advice on several of its aspects. We dedicate the paper to his memory with deep sorrow at his passing. I.

show abstract

“…One-dimensional stochastic differential equations with distributional drift were examined by several authors, see [14,15,2,23] and references therein, with a recent contribution by [19]. Such an equation appears formally as…”

Section: Introductionmentioning

confidence: 99%

Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time

Russo

Wurzer

2017

Stoch. Dyn.

View full text Add to dashboard Cite

We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator L has a generalized drift. We investigate existence and uniqueness of generalized solutions of class C 1 . The generator L is associated with a Markov process X which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is X. Since X is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of solutions of a BSDE with random terminal time when the driving process is a general càdlàg martingale.KEY WORDS AND PHRASES: Backward stochastic differential equations; random terminal time; martingale problem; distributional drift; elliptic partial differential equations.MSC 2010: 60H10; 60H30; 35H99.

show abstract

Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions

Cited by 7 publications

References 78 publications

Multidimensional stochastic differential equations with distributional drift

Multidimensional stochastic differential equations with distributional drift

Distribution of the time to explosion for one-dimensional diffusions

Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time

Contact Info

Product

Resources

About