Some inequalities for martingales and stochastic convolutions

Ichikawa, Akira

doi:10.1080/07362998608809094

Cited by 58 publications

(41 citation statements)

References 9 publications

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“…In [26] the càd-làg property of the solution has been proven using Aldous criteria for compactness in the Skorohod space. We used instead Ichikawa's inequality in Lemma 5 of [28], for the stochastic integrals (6), which is more straightforward. Reference [26] considers the case where the state space is an M-type-p Banach space.…”

Section: Properties Of Stochastic Integrals Wrt Cprmsmentioning

confidence: 99%

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

Albeverio

Mandrekar

Rüdiger

2009

Stochastic Processes and their Applications

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Existence and uniqueness of the mild solutions for stochastic differential equations for Hilbert valued stochastic processes are discussed, with the multiplicative noise term given by an integral with respect to a general compensated Poisson random measure. Parts of the results allow for coefficients which can depend on the entire past path of the solution process. In the Markov case Yosida approximations are also discussed, as well as continuous dependence on initial data, and coefficients. The case of coefficients that besides the dependence on the solution process have also an additional random dependence is also included in our treatment. All results are proven for processes with values in separable Hilbert spaces. Differentiable dependence on the initial condition is proven by adapting a method of S. Cerrai.

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Section: Properties Of Stochastic Integrals Wrt Cprmsmentioning

confidence: 99%

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

Albeverio

Mandrekar

Rüdiger

2009

Stochastic Processes and their Applications

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“…Proof. The case of a pseudo-contraction semigroup is proved in [8]. For the general case we include the following argument for completeness.…”

Section: Equations With Lipschitz Coefficientsmentioning

confidence: 99%

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Gawarecki

Mandrekar²,

Richard

1999

Random Operators and Stochastic Equations

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Several results concerning the existence and uniqueness of solutions of Ito SDE's in a real separable Hubert space have recently been reported. In this work we first obtain the existence and uniqueness of strong solutions to (not necessarily) Ito SDE's in a Hubert space under Lipschitz-type conditions on the coefficients. We assume usual continuity and linear growth conditions. For non-Lipschitz coefficients an approximation technique of Gikhman and Skorokhod is then used to prove the existence of weak solutions taking values in a larger Hubert space H-\. This result depends on an assumption that H can be compactly embedded in H-\, such that the coefficients satisfy regularity conditions with respect to H-\. This assumption is not a limitation to our method as it is necessary even in the deterministic case. Additionally, we prove the existence of martingale solutions to infinite dimensional semilinear SDE's. In both cases, coefficients F(£, ·) and J9(£, ·) may depend on the entire past of X £ C([0, T], H) and not on the value of x(t) alone.Brought to you by | provisional account Unauthenticated Download Date | 7/2/15 1:26 AM

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“…Taking the square on both sides of this inequality, we get Using the Burkhölder-Davis-Gundy's inequality [7,11,12] …”

Section: −Au(s) − B(u(s) U(s)) U(s) Dsmentioning

confidence: 99%

Approximation of the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise

Deugoué

2017

Stoch. Dyn.

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Some inequalities for martingales and stochastic convolutions

Cited by 58 publications

References 9 publications

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Approximation of the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise

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