一类用于远期曲线的局部状态依赖模型
A class of locally state-dependent models for forward curves
AI总结：
本文在 Heath-Jarrow-Morton 框架下提出一类 Hilbert 空间值、局部状态依赖的远期曲线模型，证明其温和解存在唯一，并揭示其一维投影的马尔可夫性，从而与经典固定交割时间远期模型建立联系。
摘要：
我们在 Heath-Jarrow-Morton 框架下、采用 Musiela 参数化，提出了一个远期曲线的动态模型。远期曲线取值于函数空间 H，其动态服从一个系数具有状态依赖性的随机偏微分方程。特别地，这些系数通过 H 上的逐点运算映射来定义，从而形成局部状态依赖结构。我们首先探讨这些逐点算子在 H 上良定义的条件。接着，我们确定保证所得系数函数满足局部增长性和 Lipschitz 性质的条件，从而确保温和解的存在唯一性。所提出的模型通过单一方程刻画整条远期曲线的行为，同时保持了显著的简洁性。值得注意的是，我们证明了该模型的某些一维投影具有马尔可夫性，并满足一个一维随机微分方程。这将我们的 Hilbert 空间方法与针对固定交割时间远期合约的成熟模型联系起来，从而可以应用已有的公式和数值技术。这一联系还使我们能够考察保持解为正性的条件。作为具体例子，我们分析了指数模型和常弹性方差模型的 Hilbert 空间值变体。
英文摘要：
We present a dynamic model for forward curves within the Heath-Jarrow-Morton framework under the Musiela parametrization. The forward curves take values in a function space H, and their dynamics follows a stochastic partial differential equation with state-dependent coefficients. In particular, the coefficients are defined through point-wise operating maps on H, resulting in a locally state-dependent structure. We first explore conditions under which these point-wise operators are well defined on H. Next, we determine conditions to ensure that the resulting coefficient functions satisfy local growth and Lipschitz properties, so to guarantee the existence and uniqueness of mild solutions. The proposed model captures the behavior of the entire forward curve through a single equation, yet retains remarkable simplicity. Notably, we demonstrate that certain one-dimensional projections of the model are Markovian and satisfy a one-dimensional stochastic differential equation. This connects our Hilbert-space approach to well established models for forward contracts with fixed delivery times, for which existing formulas and numerical techniques can be applied. This link allows us to examine also conditions for maintaining positivity of the solutions. As concrete examples, we analyze Hilbert-space valued variants of an exponential model and of a constant elasticity of variance model.
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