Multidimensional Diffusion Processes

Stroock, Daniel W.; Varadhan, S. R. S.

doi:10.1007/3-540-28999-2

Cited by 1,312 publications

(1,715 citation statements)

References 10 publications

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“…Then for any X 0 ∈ C tem there is a solution X to (1.1) on (Ω, F W ∞ , F W Proof. The Borel measurability of the law is proved as in Exercise 6.7.4 in (SV79). We now apply Theorem 3.14 of (Kur07), with the Polish state spaces S 1 and S 2 for the driving process (W ) and solution (X) in that work both equal to C(R + , C tem ).…”

Section: (14)mentioning

confidence: 99%

Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case

Mytnik

Perkins

2009

Probab. Theory Relat. Fields

215

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Section: (14)mentioning

confidence: 99%

Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case

Mytnik

Perkins

2009

Probab. Theory Relat. Fields

215

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“…A solution to the martingale problem for (L, ν) always exists. This follows from Theorem 6.1.7 of Stroock and Varadhan (1979) because Condition 3 implies that both coefficients in L (those of f (x) and f (x)) are bounded in t and x and continuous in x for each t.…”

Section: Condition 3 the Functions A Bmentioning

confidence: 90%

A Diffusion Limit for Generalized Correlated Random Walks

Gruber¹,

Schweizer

2006

J. Appl. Probab.

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A generalized correlated random walk is a process of partial sums X k = k j =1 Y j such that (X, Y ) forms a Markov chain. For a sequence (X n ) of such processes in which each Y n j takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Y n . Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

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“…Towards this end, we set F + t = s>t F s and F + = (F + t ) t∈[0,∞) . Next, we observe that the F-martingale X is also an F + -martingale due to Exercise 1.5.8 in Stroock and Varadhan (2006). Now, Lemma 1.1 in Föllmer (1972) yields the existence of a right-continuous version of X, which we call again X.…”

Section: Introductionmentioning

confidence: 89%

The uniform integrability of martingales. On a question by Alexander Cherny

Ruf

2015

Stochastic Processes and their Applications

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Let X be a progressively measurable, almost surely right-continuous stochastic process such that X τ ∈ L 1 and E[X τ ] = E[X 0 ] for each finite stopping time τ . In 2006, Cherny showed that X is then a uniformly integrable martingale provided that X is additionally nonnegative. Cherny then posed the question whether this implication also holds even if X is not necessarily nonnegative. We provide an example that illustrates that this implication is wrong, in general. If, however, an additional integrability assumption is made on the limit inferior of |X| then the implication holds. Finally, we argue that this integrability assumption holds if the stopping times are allowed to be randomized in a suitable sense.

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Multidimensional Diffusion Processes

Cited by 1,312 publications

References 10 publications

Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case

Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case

A Diffusion Limit for Generalized Correlated Random Walks

The uniform integrability of martingales. On a question by Alexander Cherny

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