Stochastic evolution equations in Banach spaces and applications to the Heath–Jarrow–Morton–Musiela equations

Abstract

In this paper we study the stochastic evolution equation (1.1) in martingale-type 2 Banach spaces (with the linear part of the drift being only a generator of a C 0 -semigroup). We prove the existence and the uniqueness of solutions to this equation. We apply the abstract results to the Heath-Jarrow-Morton-Musiela (HJMM) equation (6.3). In particular, we prove the existence and the uniqueness of solutions to the latter equation in the weighted Lebesgue and Sobolev spaces L p ν and W 1,p ν respectively. We also find a sufficient condition for the existence and the uniqueness of an invariant measure for the Markov semigroup associated to equation (6.3) in the weighted spaces L p ν . Mathematical preliminariesWe begin with introducing some notations. Key words and phrases. (SEE)s and (HJMM) Equation and mild and strong solutions and the existence and the uniqueness of solutions and Markov semigroup and the existence and the uniqueness of an invariant measure. 1 arXiv:1608.05814v1 [q-fin.MF] 20 Aug 2016 2 Z. BRZEŹNIAK AND T. KOKDefinition 2.1. A Banach space X with the norm · X is called a martingale-type 2 Banach space if there exists a constant C > 0 depending only on X such that for any X-valued martingale {M n } n∈N , the following inequality holdsThe Lebesgue function spaces L p , p ≥ 2, are examples of martingale-type 2 Banach spaces.Proposition 2.2.[32] If X is a Banach space satisfying the H p condition, then X is an martingale-type 2 Banach space.Definition 2.3. Let X be a Banach space and L(X) be the space of all bounded linear operators from X to X. A C 0 -semigroup S = {S (t)} t≥0 on X is called contraction type iff there exists a constant β ∈ R such that S (t) L(X) ≤ e βt , t ≥ 0.(2.1)Definition 2.4. Let X be a martingale-type 2 Banach space and S be a contraction type C 0 -semigroup on X with the infinitesimal generator A. A process u is called an X-valued mild solution to SEE (1.1) if for each t ≥ 0, u(t) = S (t)u 0 + t 0 S (t − r)F(r, u(r))dr + t 0

Stochastic analysis with modelled distributions

“…Having shown that the Besov spaces of modelled distributions are UMD spaces and of M -type 2, gives us access to the solution theories of SPDEs in these Banach spaces, see e.g. [Brz95,Brz97,vNVW08,BK18], and, consequently, we obtain novel existence and uniqueness results for mild solutions of semilinear SPDEs in spaces of modelled distributions. In the following we briefly illustrate this for SPDEs with finite dimensional noise but we would like to emphasize that the theory of SPDEs in Banach spaces works, of course, also in the case of infinite dimensional noises, cf.…”

“…In the following we briefly illustrate this for SPDEs with finite dimensional noise but we would like to emphasize that the theory of SPDEs in Banach spaces works, of course, also in the case of infinite dimensional noises, cf. [vNVW08,BK18].…”

“…It depends on the purpose in mind, which property is actually needed, but for integrals with respect to Brownian motion martingale type 2 or UMD is favorable and both allow for treating stochastic partial differential equations like stochastic evolution equations in Banach spaces, see e.g. [Brz95], [BK18] or [vNVW08]. We shall prove here that the Besov space D γ p,q of modelled distributions (for p, q ≥ 2) has indeed the martingale type 2 and the UMD property, respectively, see Proposition 3.2.…”

Singular perturbations and asymptotic expansions for SPDEs with an application to term structure models

“…For details on the financial background we refer to, e.g., [5,9,11,22,23]. There is a large literature on the well-posedness of (5.3) in the mild sense, also in the (more interesting) case where (σ k ), hence α 0 , depend explicitly on the unknown u, with different choices of state space as well as with more general noise (see, e.g., [1,3,9,14,17,28], [25, §20.3]). Here we limit ourselves to the case where (σ k ) are possibly random, but do not depend explicitly on u, and use as state space H(R + ), which we define as the space of locally integrable functions on R + such that f ′ ∈ L 2 (R + , e wx dx), endowed with the inner product…”

Harnack Inequalities and Ergodicity of Stochastic Reaction-Diffusion Equation in $L^p$

“…The authors in [BLSa10] studied invariant measures for SPDEs in M-type 2 Banach spaces, under Lipschitz and dissipativity conditions, driven by regular noise. Recently, their method was extended in [BK18] to an SPDE, arisen in stochastic finance, in a weighted L p -space. We note that the authors in [BR16] showed the strong Feller property and irreducibility of (P t ) t≥0 , and thus the uniqueness of the invariant measure, if it exists, for the stochastic heat equation with white noise on L p (0, 1) with p > 4.…”