分数阶核随机Volterra方程的马尔可夫逼近
Markovian approximations of stochastic Volterra equations with the fractional kernel
摘要：
我们考虑粗糙随机波动率模型，其中方差过程满足一个随机Volterra方程的分数核，在粗糙Bergomi和粗糙赫斯顿模型。特别地，方差过程因此不是马尔可夫过程或半鞅，并且具有相当低的Hölder正则性。在实践中，模拟这种粗糙的过程，因此往往导致高计算成本。为了弥补这一点，我们研究了随机Volterra方程的近似使用一个$N$维扩散过程定义为解决一个系统的普通随机微分方程。如果随机Volterra方程的系数是Lipschitz连续的，我们证明了这些逼近在$N$中以超多项式速度强收敛。最后，我们应用这个近似计算了粗糙Bergomi和粗糙Heston模型下欧式看涨期权的隐含波动率微笑。
英文摘要：
We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölder-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an $N$-dimensional diffusion process defined as solution to a system of ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz continuous, we show that these approximations converge strongly with superpolynomial rate in $N$. Finally, we apply this approximation to compute the implied volatility smile of a European call option under the rough Bergomi and the rough Heston model.
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