The Martingale Problem in Hilbert Spaces

Giuseppe Da Prato; Luciano Tubaro

Bollettino dell'Unione Matematica Italiana (2008)

  • Volume: 1, Issue: 3, page 839-855
  • ISSN: 0392-4041

Abstract

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We consider an SPDE in a Hilbert space H of the form d ⁢ X ⁢ ( t ) = ( A ⁢ X ⁢ ( t ) + b ⁢ ( X ⁢ ( t ) ) ) ⁢ d ⁢ t + σ ⁢ ( X ⁢ ( t ) ) ⁢ d ⁢ W ⁢ ( t ) , X ⁢ ( 0 ) = x ∈ H and the corresponding transition semigroup P t ⁢ f ⁢ ( x ) = 𝔼 ⁢ [ f ⁢ ( X ⁢ ( t , x ) ) ] . We define the infinitesimal generator L ¯ of P t through the Laplace transform of P t as in [1]. Then we consider the differential operator L ⁢ φ = 1 2 ⁢ Tr ⁡ [ σ ⁢ ( x ) ⁢ σ * ⁢ ( x ) ⁢ D 2 ⁢ φ ] + ⟨ b ⁢ ( x ) , D ⁢ φ ⟩ defined on a suitable set V of regular functions. Our main result is that if V is a core for L ¯ , then there exists a unique solution of the martingale problem defined in terms of L . Application to the Ornstein-Uhlenbeck equation and to some regular perturbation of it are given.

How to cite

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Da Prato, Giuseppe, and Tubaro, Luciano. "The Martingale Problem in Hilbert Spaces." Bollettino dell'Unione Matematica Italiana 1.3 (2008): 839-855. <http://eudml.org/doc/290491>.

@article{DaPrato2008,
abstract = {We consider an SPDE in a Hilbert space $H$ of the form $dX(t) = ( AX(t) + b(X(t)) ) \, dt + \sigma(X(t)) \, dW(t)$, $X(0) = x \in H$ and the corresponding transition semigroup $P_t f (x)= \mathbb\{E\}[ f(X(t, x)) ]$. We define the infinitesimal generator $\bar L$ of $P_t$ through the Laplace transform of $P_t$ as in [1]. Then we consider the differential operator $L\varphi = \frac\{1\}\{2\} \operatorname\{Tr\}[\sigma(x)\sigma^*(x)D^2\varphi] + \langle b(x), D\varphi \rangle$ defined on a suitable set $V$ of regular functions. Our main result is that if $V$ is a core for $\bar L$, then there exists a unique solution of the martingale problem defined in terms of $L$. Application to the Ornstein-Uhlenbeck equation and to some regular perturbation of it are given.},
author = {Da Prato, Giuseppe, Tubaro, Luciano},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {839-855},
publisher = {Unione Matematica Italiana},
title = {The Martingale Problem in Hilbert Spaces},
url = {http://eudml.org/doc/290491},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Da Prato, Giuseppe
AU - Tubaro, Luciano
TI - The Martingale Problem in Hilbert Spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/10//
PB - Unione Matematica Italiana
VL - 1
IS - 3
SP - 839
EP - 855
AB - We consider an SPDE in a Hilbert space $H$ of the form $dX(t) = ( AX(t) + b(X(t)) ) \, dt + \sigma(X(t)) \, dW(t)$, $X(0) = x \in H$ and the corresponding transition semigroup $P_t f (x)= \mathbb{E}[ f(X(t, x)) ]$. We define the infinitesimal generator $\bar L$ of $P_t$ through the Laplace transform of $P_t$ as in [1]. Then we consider the differential operator $L\varphi = \frac{1}{2} \operatorname{Tr}[\sigma(x)\sigma^*(x)D^2\varphi] + \langle b(x), D\varphi \rangle$ defined on a suitable set $V$ of regular functions. Our main result is that if $V$ is a core for $\bar L$, then there exists a unique solution of the martingale problem defined in terms of $L$. Application to the Ornstein-Uhlenbeck equation and to some regular perturbation of it are given.
LA - eng
UR - http://eudml.org/doc/290491
ER -

References

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