论文标题
非自主慢速随机微分方程的平均原理和对局部随机波动率模型的应用
The Averaging Principle for Non-autonomous Slow-fast Stochastic Differential Equations and an Application to a Local Stochastic Volatility Model
论文作者
论文摘要
在这项工作中，我们研究了随机微分方程的非自治慢速系统的平均原理。尤其是在第一部分中，我们证明了平均原理，假设系数的均衡性，Lipschitzianity和持有人的连续性，一个千古的假设和$ \ Mathcal {l}^2^2 $ bound。在这种情况下，我们证明了慢组分对平均方程解的弱收敛性。此外，我们提供了适当的散发条件，在该条件下，沿着千古化假设和快速组件的$ \ MATHCAL {l}^2 $ bounds满足了隐式条件。 在第二部分中，我们提出了此结果的财务应用：我们将开发的理论应用于缓慢的局部随机波动率模型。首先，我们证明该模型与局部波动率的收敛弱。然后，在风险中性度量下，我们表明，可能依赖路径的衍生物价格会融合使用限制模型计算的衍生物。
In this work we study the averaging principle for non-autonomous slow-fast systems of stochastic differential equations. In particular in the first part we prove the averaging principle assuming the sublinearity, the Lipschitzianity and the Holder's continuity in time of the coefficients, an ergodic hypothesis and an $\mathcal{L}^2$-bound of the fast component. In this setting we prove the weak convergence of the slow component to the solution of the averaged equation. Moreover we provide a suitable dissipativity condition under which the ergodic hypothesis and the $\mathcal{L}^2$-bound of the fast component, which are implicit conditions, are satisfied. In the second part we propose a financial application of this result: we apply the theory developed to a slow-fast local stochastic volatility model. First we prove the weak convergence of the model to a local volatility one. Then under a risk neutral measure we show that the prices of the derivatives, possibly path-dependent, converge to the ones calculated using the limit model.
