Additive process

An additive process, in probability theory, is a cadlag, continuous in probability stochastic process with independent increments. An additive process is the generalization of a Lévy process (a Lévy process is an additive process with stationary increments). An example of an additive process that is not a Lévy process is a Brownian motion with a time-dependent drift. The additive process was introduced by Paul Lévy in 1937. There are applications of the additive process in quantitative finance (this family of processes can capture important features of the implied volatility) and in digital image processing.

An additive process, in probability theory, is a cadlag, continuous in probability stochastic process with independent increments. An additive process is the generalization of a Lévy process (a Lévy process is an additive process with stationary increments). An example of an additive process that is not a Lévy process is a Brownian motion with a time-dependent drift. The additive process was introduced by Paul Lévy in 1937. There are applications of the additive process in quantitative finance (this family of processes can capture important features of the implied volatility) and in digital image processing. A stochastic process \{X_t\}_{t \geq 0} on \mathbb R^d such that X_0=0 almost surely is an additive process if it satisfy the following hypothesis: It has independent increments. It is continuous in probability. \lim_{s \to t^-} \Pr \left(\big| X_s- X_t \big| \geq \varepsilon \right) = 0. \nu_t(\{0\})=0 and \int_{\mathbb R^d}(1\wedge x^2)\nu_t(dx) . \gamma_t is called drift term, A_t covariance matrix and \nu_t Lévy measure. It is possible to write explicitly the additive process characteristic function using the Lévy–Khintchine formula: \varphi_X(u)(t) := \operatorname E \left[e^{iu'X_t}\right] = \exp \left(u' \gamma_t i - \frac{1}{2}u' A_t u + \int_{\mathbb R^d} \left(e^{i u' x}-1 -iu'x\mathbf{I}_{|x| where u is a vector in \mathbb R^d and \mathbf{I_C} is the indicator function of the set C . A Lèvy process characteristic function has the same structure but with \gamma_t =t\gamma, \nu_t = t\nu and A_t = At with \gamma a vector in \mathbb R^d , A a positive definite matrix in \mathbb R^{d \times d} and \nu is a measure on \mathbb R^d . == Existence and uniqueness in law of additive process== The following result together with the Lévy–Khintchine formula characterizes the additive process. Let \{X_t\}_{t \geq 0} be an additive process on \mathbb R^d . Then, its infinitely divisible distribution is such that: For all t , A_t is a positive definite matrix. \gamma_0=0, A_0=0, \nu_0=0 and for all s, t is such that s , A_t-A_s is a positive definite matrix and \nu_t(B)\geq \nu_s(B) for every B in \mathbf{B}(\mathbb R^d) . If s\to t \gamma_s\to \gamma_t, A_s \to A_t and \nu_s(B)\to \nu_t(B) every B in \mathbf{B}(\mathbb R^d) , 0\not\in B . Conversely for family of infinitely divisible distributions characterized by a generating triplet (\gamma_t, A_t, \nu_t) that satisfies 1, 2 and 3, it exists an additive process \{X_t\}_{t \geq 0} with this distribution. ===Additive Logistic Process=== Family of additive processes with generalized logistic distribution. Their 5 parameters characteristic function is \operatorname E \left[e^{iuX_t}\right] = \left(\frac{B(\alpha_t+i \sigma_t u, \beta_t- i \sigma u )}{B(\alpha_t, \beta_t)} \right)^{\delta_t} e^{ i \mu_t u}\;\;. Two subcases of additive logistic process are the symmetric logistic additive process with standard logistic distribution ( \alpha_t=1 , \beta_t=1 , \delta_t=1 ) and the conjugate-power Dagum additive process with Dagum distribution ( \alpha_t=1 , \beta_t=1-\sigma(t) , \alpha_t=1 ). The function \mu_t can always be chosen s.t. the additive process is a martingale. ===Additive Normal Tempered Stable Process=== Extension of the Lévy normal tempered stable processes; some well-known Lévy normal tempered stable processes have normal-inverse Gaussian distribution and the variance-gamma distribution. Additive normal tempered stable processes have the same characteristic function of Lévy normal tempered stable processes but with time dependent parameters \sigma_t (the level of the volatility), k_t (the variance of jumps) and \eta_t (linked to the skew): \operatorname E \left[e^{iuX_t}\right] = {\cal L}_t \left(iu \left(\frac{1}{2}+\eta_t \right)\sigma_t^2+\frac{u^2\sigma^2_t}{2};\;k_t,\;\alpha \right)e^{iu\varphi_tt}, where \ln {\cal L}_t \left(u;\;k_t,\;\alpha\right) := \begin{cases} \displaystyle \frac{t}{k_t} \displaystyle \frac{1-\alpha}{\alpha} \left \{1- \left(1+\frac{u \; k_t}{1-\alpha}\right)^\alpha \right \} & \mbox{if } \; 0 The function \varphi_t can always be chosen s.t. the additive process is a martingale. === Additive Subordinator=== A positive non decreasing additive process \{S_t\}_{t \geq 0} with values in \mathbb R is an additive subordinator. An additive subordinator is a semimartingale (thanks to the fact that it is not decreasing) and it is always possible to rewrite its Laplace transform as \operatorname E\left[ e^{-u S_t} \right] = \exp\left(u b_t + \int_{\mathbb R^d} (e^{i u x}-1) \nu_t(dx)\right). It is possible to use additive subordinator to time-change a Lévy process obtaining a new class of additive processes. The characteristic function of the Variance gamma at time t=1 is \operatorname E \left[e^{iuX_1}\right] = \left(\frac{1}{1-iu\theta\nu+0.5\sigma^2\nu u^2}\right)^{1/\nu}, where \theta, \nu and \sigma are positive constant. The characteristic function of the variance gamma SSD is \operatorname E \left[e^{iuZ_t}\right] = \left(\frac{1}{1-iut^h\theta\nu+0.5\sigma^2\nu u^2t^2h}\right)^{1/\nu} ===Jump simulation=== Jump simulation is a generalization to the class of additive processes of the jump simulation technique developed for Lévy processes. The method is based on truncating small jumps below a certain threshold and simulating the finite number of independent jumps. Moreover, Gaussian approximation can be applied to replace small jumps with a diffusive term. It is also possible to use the Ziggurat algorithm to speed up the simulation of jumps. ===Characteristic function inversion=== Simulation of Lévy process via characteristic function inversion is a well established technique in the literature. This technique can be extended to additive processes. The key idea is obtaining an approximation of the cumulative distribution function (CDF) by inverting the characteristic function. The inversion speed is enhanced by the use of the Fast Fourier transform. Once the approximation of the CDF is available is it possible to simulate an additive process increment just by simulating a uniform random variable. The method has similar computational cost as simulating a standard geometric Brownian motion. ==Applications== Moreover, an interesting power scaling characteristic arises in calibrated parameters k_t=\bar{k}t^\beta and \eta_t=\bar{\eta}t^\delta . There is statistical evidence that \beta=1 and \delta=-1/2 . Lévy subordination is used to construct new Lévy processes (for example variance gamma process and normal inverse Gaussian process). There is a large number of financial applications of processes constructed by Lévy subordination. An additive process built via additive subordination maintains the analytical tractability of a process built via Lévy subordination but it reflects better the time-inhomogeneus structure of market data. Additive subordination is applied to the commodity market and to VIX options. ===Digital image processing=== An estimator based on the minimum of an additive process can be applied to image processing. Such estimator aims to distinguish between real signal and noise in the picture pixels. ==References== ==Sources==

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