Affine Volterra processes with jumps

Abstract

The theory of affine processes has been recently extended to the framework of stochastic Volterra equations with continuous trajectories. These so-called affine Volterra processes overcome modeling shortcomings of affine processes because they can have trajectories whose regularity is different from the regularity of the paths of Brownian motion. More specifically, singular kernels yield rough affine processes. This paper extends the theory by considering affine stochastic Volterra equations with jumps. This extension is not straightforward because the jump structure together with possible singularities of the kernel may induce explosions of the trajectories.

The Rough Hawkes Heston Stochastic Volatility Model

“…The same arguments as in the proof of [18,Theorem 5], which essentially rely on the stochastic Fubini's theorem (see, e.g., [66, Theorem 65, Chapter IV]), allow us to prove that…”

“…If we assume that K and the shifted kernels K(• + 1/n), n ∈ N, satisfy Hypothesis 1, the (weak) existence of the spot variance process σ 2 , satisfying (1), is ensured by [1,Theorem 2.13] and [18,Lemma 9]. Assuming weak existence, weak uniqueness is established in [18,Corollary 12] under Hypothesis 1. We refer to [2] and [18] for more information about stochastic Volterra equations and stochastic convolution for processes with jumps.…”

“…Assuming weak existence, weak uniqueness is established in [18,Corollary 12] under Hypothesis 1. We refer to [2] and [18] for more information about stochastic Volterra equations and stochastic convolution for processes with jumps.…”

“…In order to argue the martingale property of H(w, T ), first we observe that by Lemma 2 the realvalued process H(Rw, T ) = (H t (Rw, T )) t∈[0,T ] = (exp{V t (Rw, T )}) t∈[0,T ] is a true martingale. Secondly, we invoke [18,Corollary 8] to obtain the following alternative expression for V (w, T ) (an analogous one holds for V (Rw, T )) (14) where for every…”

“…To keep mathematical and numerical tractability, we choose an affine specification of the model. As such, our model belongs to the class of affine Volterra processes [5], which has been recently extended to jump processes in [18,29,30]. In particular, the Fourier-Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati-Volterra equations, see Theorems 3 and 10.…”