Well-posedness and invariant measures for HJM models with deterministic volatility and Lévy noise

Marinelli, Carlo

doi:10.1080/14697680802595692

Cited by 5 publications

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“…In particular, denoting by u ( t , x ) the forward rate prevailing at time t with maturity , we shall consider the mild formulation of a stochastic PDE of the form where and M is a locally square integrable martingale with independent increments taking values in a Hilbert space with inner product . Moreover, no‐arbitrage conditions uniquely determine the functional form of f in terms of σ (see, e.g., Björk et al 1997; Jakubowski and Zabczyk 2007; Marinelli 2007). The properties of this SPDE in the case of M being a Wiener process are rather well studied: let us just mention Filipović (2001) for a self‐contained treatment of existence of solutions, no‐arbitrage conditions, and finite dimensional realizations, Goldys and Musiela (2001) for connections with Kolmogorov equations and option pricing, and Tehranchi (2005) for the ergodic properties.…”

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confidence: 99%

Local Well-Posedness of Musiela’s Spde With Lévy Noise

Marinelli

2010

Mathematical Finance

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We determine sufficient conditions on the volatility coefficient of Musiela's stochastic partial differential equation driven by an infinite dimensional Lévy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight.KEY WORDS: HJM model, Musiela's stochastic PDE, stochastic PDEs with jumps, maximal inequalities.where u x (t) := {u(t, y) : y ∈ [0, x]} and M is a locally square integrable martingale with independent increments taking values in a Hilbert space with inner product ·, · . Moreover, no-arbitrage conditions uniquely determine the functional form of f in terms of σ (see, e.g., Björk et al. 1997;Jakubowski and Zabczyk 2007;Marinelli 2007). The properties of this SPDE in the case of M being a Wiener process are rather well studied: let us just mention Filipović (2001) for a self-contained treatment of existence of solutions, no-arbitrage conditions, and finite dimensional realizations, Goldys and Musiela (2001) for connections with Kolmogorov equations and option pricing, and Tehranchi (2005) for the ergodic properties. A basic question to ask is clearly whether a solution exists to (1.1), and under what conditions on σ and f . If M is a Wiener process, one can draw on a large body of results (see, e.g., Da Prato and Zabczyk 1992) to establish existence of solutions. In the general case of discontinuous M, the situation turns out to be more involved, not only because the no-arbitrage "constraint" on the drift term f is relatively more complicated, but also (perhaps mainly) because the theory of stochastic PDEs driven by jump noise is not so well developed (see, however, for instance, Métivier

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Section: Introductionmentioning

confidence: 99%

Local Well-Posedness of Musiela’s Spde With Lévy Noise

Marinelli

2010

Mathematical Finance

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

Brzeźniak

Kok

2018

Finance Stoch

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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

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On a stochastic heat equation with first order fractional noises and applications to finance

Jiang

Wang

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tIn this paper, we propose a class of stochastic heat equations with first order fractional noises. We define a first order noise through the adjoint operator of the first order operator, where the operation of the stochastic integral can be avoided. In this framework, the existence and uniqueness of the solution of the equation will be established. Further, we give the regularity of the solution. Finally, we model the term structure of forward rate with the solutions and give the conditions under which the market is arbitrary-free.

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Well-posedness and invariant measures for HJM models with deterministic volatility and Lévy noise

Cited by 5 publications

References 24 publications

Local Well-Posedness of Musiela’s Spde With Lévy Noise

Local Well-Posedness of Musiela’s Spde With Lévy Noise

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

On a stochastic heat equation with first order fractional noises and applications to finance

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