Subordinator

About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.

Two recursive decompositions of Brownian bridge

Abstract: Aldous and Pitman (1994) studied asymptotic distributions, as n tends to infinity, of various functionals of a uniform random mapping of a set of n elements, by constructing a mapping-walk and showing these mapping-walks converge weakly to a reflecting Brownian bridge Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian bridge, each defined by cutting the path of the bridge at an increasing sequence of recursively defined random times in the zero set of the bridge The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the bridge fragments defined by these decompositions We derive various extensions of these identities for Brownian and Bessel bridges, and characterize the distributions of various path fragments involved, using the theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index between 0 and 1

Spectral projections correlation structure for short-to-long range dependent processes

Abstract: Let X = (Xt)t≥0 be a stochastic process issued from that admits a marginal stationary measure v, i.e. vPt f = vf for all t ≥ 0, where . In this paper, we introduce the (resp. biorthogonal) spectral projections correlation functions which are expressed in terms of projections.” Also, update first published online date, if available. into the eigenspaces of Pt (resp. and of its adjoint in the weighted Hilbert space L2 (v)). We obtain closed-form expressions involving eigenvalues, the condition number and/or the angle between the projections in the following different situations: when X = X with X = (Xt)t ≥ 0 being a Markov process, X is the subordination of X in the sense of Bochner, and X is a non-Markovian process which is obtained by time-changing X with an inverse of a subordinator. It turns out that these spectral projections correlation functions have different expressions with respect to these classes of processes which enables to identify substantial and deep properties about their dynamics. This interesting fact can be used to design original statistical tests to make inferences, for example, about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to-long-range dependence. To reveal the usefulness of our results, we apply them to a class of non-self-adjoint Markov semigroups studied in Patie and Savov (to appear, Mem. Amer. Math. Soc., 179p), and then time-change by subordinators and their inverses.

Indifference pricing and the minimal entropy martingale measure in a stochastic volatility model with jumps

Abstract: We use the dynamic programming approach to derive an equation for the utility indierence price of Markovian claims in a stochastic volatility model proposed by Barndor-Nielsen and Shephard (3). The pricing equation is a Black & Scholes equation with a nonlinear integral term involving the risk preferences of the investor. Passing to the zero risk aversion limit, we present a Feynman-Kac representation of the minimal entropy price. The density of the minimal entropy martingale measure is found via the Girsanov transform of the Brownian motion and a subordinator process controlling the jumps in the volatility model. The density is represented by the logarithm of the value function for an investor with exponential utility and no claim issued, and a Feynman-Kac representation of this function is provided. We calculate the function explicitly in a special case, and show some properties in the general case.

Logarithmic L\'{e}vy process directed by Poisson subordinator.

Abstract: Let $\{L(t),t\geq 0\}$ be a Levy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Levy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic Levy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the Levy measure of the subordinated process $Z(t)$ is a shifted Levy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding Levy measures are equal.