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Register a new userLimit theorems for Hawkes processes including inhibition
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In this paper we consider some non linear Hawkes processes with signed reproduction function (or memory kernel) thus exhibiting both self-excitation and inhibition. We provide a Law of Large Numbers, a Central Limit Theorem and large deviation results, as time growths to infinity. The proofs lie on a renewal structure for these processes introduced in Costa et al. (2020) which leads to a comparison with cumulative processes. Explicit computations are made on some examples. Similar results have been obtained in the literature for self-exciting Hawkes processes only.
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This paper investigates Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of this point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows to apply results known for Galton-Watson trees. In the present paper, we establish limit theorems for Hawkes process with signed reproduction functions by using renewal techniques. We notably prove exponential concentration inequalities, and thus extend results of Reynaud-Bouret and Roy (2007) which were proved for nonnegative reproduction functions using this cluster representation which is no longer valid in our case. An important step for this is to establish the existence of exponential moments for renewal times of M/G/infinity queues that appear naturally in our problem. These results have their own interest, independently of the original problem for the Hawkes processes.
In this paper, we study the asymptotic behavior of a supercritical $(xi,psi)$-superprocess $(X_t)_{tgeq 0}$ whose underlying spatial motion $xi$ is an Ornstein-Uhlenbeck process on $mathbb R^d$ with generator $L = frac{1}{2}sigma^2Delta - b x cdot abla$ where $sigma, b >0$; and whose branching mechanism $psi$ satisfies Greys condition and some perturbation condition which guarantees that, when $zto 0$, $psi(z)=-alpha z + eta z^{1+beta} (1+o(1))$ with $alpha > 0$, $eta>0$ and $betain (0, 1)$. Some law of large numbers and $(1+beta)$-stable central limit theorems are established for $(X_t(f) )_{tgeq 0}$, where the function $f$ is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding the branching rate being relatively small, large or critical at a balanced value.
We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $log Z_n$.
This paper is a continuation of our recent paper (Elect. J. Probab. 24 (2019), no. 141) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes $(X_t)_{tgeq 0}$ with branching mechanisms of infinite second moment. In the aforementioned paper, we proved stable central limit theorems for $X_t(f) $ for some functions $f$ of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for $X_t(f) $ for all functions $f$ of polynomial growth. In this note, we show that the limit stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for $X_t(f) $ for all functions $f$ of polynomial growth.
We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinators Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.
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