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Lingjiong Zhu

Publications and source records attributed to Lingjiong Zhu.

108 records · Page 6Linked to original sources
arXiv2016

Discrete Sums of Geometric Brownian Motions, Annuities and Asian Options

The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of geometric Brownian motions with geometric stopping time, and the finite sum of the geometric Brownian motions. These results are extended to the discrete sum of the exponential Lévy process. We derive tail asymptotics and compute numerically the asymptotic distribution function. We compare the results against the known results for the continuous time integral of the geometric Brownian motion up to an exponentially distributed time. The results are illustrated with numerical examples for life annuities with discrete payments, and Asian options.

q-fin.PR↗
arXiv2016

Short Maturity Asian Options in Local Volatility Models

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows a local volatility model. The asymptotics for out-of-the-money, in-the-money, and at-the-money cases are derived, considering both fixed strike and floating strike Asian options. The asymptotics for the out-of-the-money case involves a non-trivial variational problem which is solved completely. We present an analytical approximation for Asian options prices, and demonstrate good numerical agreement of the asymptotic results with the results of Monte Carlo simulations and benchmark test cases in the Black-Scholes model for option parameters relevant in practical applications.

q-fin.PR↗
arXiv2016

Asymptotics for Sparse Exponential Random Graph Models

We study the asymptotics for sparse exponential random graph models where the parameters may depend on the number of vertices of the graph. We obtain exact estimates for the mean and variance of the limiting probability distribution and the limiting log partition function of the edge-(single)-star model. They are in sharp contrast to the corresponding asymptotics in dense exponential random graph models. Similar analysis is done for directed sparse exponential random graph models parametrized by edges and multiple outward stars.

math.PR↗
arXiv2015

Reciprocity in directed networks

Reciprocity is an important characteristic of directed networks and has been widely used in the modeling of World Wide Web, email, social, and other complex networks. In this paper, we take a statistical physics point of view and study the limiting entropy and free energy densities from the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble whose sufficient statistics are given by edge and reciprocal densities. The sparse case is also studied for the grand canonical ensemble. Extensions to more general reciprocal models including reciprocal triangle and star densities will likewise be discussed.

cond-mat.stat-mech↗
arXiv2015

Moderate and Large Deviations for the Erdős-Kac Theorem

The Erdős-Kac theorem is a celebrated result in number theory which says that the number of distinct prime factors of a uniformly chosen random integer satisfies a central limit theorem. In this paper, we establish the large deviations and moderate deviations for this problem in a very general setting for a wide class of additive functions.

math.PR↗
arXiv2015

Dynamics of Order Positions and Related Queues in a Limit Order Book

Order positions are key variables in algorithmic trading. This paper studies the limiting behavior of order positions and related queues in a limit order book. In addition to the fluid and diffusion limits for the processes, fluctuations of order positions and related queues around their fluid limits are analyzed. As a corollary, explicit analytical expressions for various quantities of interests in a limit order book are derived.

q-fin.TR↗
arXiv2015

On the growth rate of a linear stochastic recursion with Markovian dependence

We consider the linear stochastic recursion $x_{i+1} = a_{i}x_{i}+b_{i}$ where the multipliers $a_i$ are random and have Markovian dependence given by the exponential of a standard Brownian motion and $b_{i}$ are i.i.d. positive random noise independent of $a_{i}$. Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments $λ_q = \lim_{n\to \infty} \frac{1}{n} \log\mathbb{E}[(x_n)^q]$ with $q\in \mathbb{Z}_+$. We show that the Lyapunov exponents $λ_q$ exist, under appropriate scaling of the model parameters, and have non-analytic behavior manifested as a phase transition. We study the properties of the phase transition and the critical exponents using both analytic and numerical methods.

math.PR↗
arXiv2015

Asymptotic structure and singularities in constrained directed graphs

We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward $p$-stars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph densities are controlled by parameters. The idea of directly constraining edge and other subgraph densities comes from Radin and Sadun. Such modeling circumvents a phenomenon first made precise by Chatterjee and Diaconis: that in ERGMs it is often impossible to independently constrain edge and other subgraph densities. In all our models, we find that large graphs have either uniform or bipodal structure. When edge density (resp. $p$-star density) is fixed and $p$-star density (resp. edge density) is controlled by a parameter, we find phase transitions corresponding to a change from uniform to bipodal structure. When both edge and $p$-star density are fixed, we find only bipodal structures and no phase transition.

math.PR↗
arXiv2015

Large deviations for Markovian nonlinear Hawkes processes

Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience and many other fields. In this paper, we study the large deviations for nonlinear Hawkes processes. The large deviations for linear Hawkes processes has been studied by Bordenave and Torrisi. In this paper, we prove first a large deviation principle for a special class of nonlinear Hawkes processes, that is, a Markovian Hawkes process with nonlinear rate and exponential exciting function, and then generalize it to get the result for sum of exponentials exciting functions. We then provide an alternative proof for the large deviation principle for a linear Hawkes process. Finally, we use an approximation approach to prove the large deviation principle for a special class of nonlinear Hawkes processes with general exciting functions.

math.PR↗
arXiv2015

The Speed of a Biased Walk on a Galton-Watson Tree without Leaves is Monotonic with Respect to Progeny Distributions for High Values of Bias

Consider biased random walks on two Galton-Watson trees without leaves having progeny distributions $P_1$ and $P_2$ (GW$(P_1)$ and GW$(P_2)$) where $P_1$ and $P_2$ are supported on positive integers and $P_1$ dominates $P_2$ stochastically. We prove that the speed of the walk on GW$(P_1)$ is bigger than the same on GW$(P_2)$ when the bias is larger than a threshold depending on $P_1$ and $P_2$. This partially answers a question raised by Ben Arous, Fribergh and Sidoravicius.

math.PR↗
arXiv2014

Asymptotics for a Class of Self-Exciting Point Processes

In this paper, we study a class of self-exciting point processes. The intensity of the point process has a nonlinear dependence on the past history and time. When a new jump occurs, the intensity increases and we expect more jumps to come. Otherwise, the intensity decays. The model is a marriage between stochasticity and dynamical system. In the short-term, stochasticity plays a major role and in the long-term, dynamical system governs the limiting behavior of the system. We study the law of large numbers, central limit theorem, large deviations and asymptotics for the tail probabilities.

math.PR↗
arXiv2014

Central Limit Theorem for Nonlinear Hawkes Processes

Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance and many other fields. In this paper, we obtain a functional central limit theorem for nonlinear Hawkes process. Under the same assumptions, we also obtain a Strassen's invariance principle, i.e. a functional law of the iterated logarithm.

math.PR↗
arXiv2014

Ruin Probabilities for Risk Processes with Non-Stationary Arrivals and Subexponential Claims

In this paper, we obtain the finite-horizon and infinite-horizon ruin probability asymptotics for risk processes with claims of subexponential tails for non-stationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, non-stationary and non-renewal. We give three examples of non-stationary and non-renewal point processes: Hawkes process, Cox process with shot noise intensity and self-correcting point process. We also show some aggregate claims results for these three examples.

q-fin.RM↗
arXiv2014

Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

In this paper, we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. Laplace transforms and limit theorems have been obtained, including law of large numbers, central limit theorems and large deviations.

math.PR↗
arXiv2014

Optimal Strategies for a Long-Term Static Investor

The optimal strategies for a long-term static investor are studied. Given a portfolio of a stock and a bond, we derive the optimal allocation of the capitols to maximize the expected long-term growth rate of a utility function of the wealth. When the bond has constant interest rate, three models for the underlying stock price processes are studied: Heston model, 3/2 model and jump diffusion model. We also study the optimal strategies for a portfolio in which the stock price process follows a Black-Scholes model and the bond process has a Vasicek interest rate that is correlated to the stock price.

q-fin.PM↗
arXiv2013

Nonlinear Hawkes Processes

The Hawkes process is a simple point process that has long memory, clustering effect, self-exciting property and is in general non-Markovian. The future evolution of a self-exciting point process is influenced by the timing of the past events. There are applications in finance, neuroscience, genome analysis, seismology, sociology, criminology and many other fields. We first survey the known results about the theory and applications of both linear and nonlinear Hawkes processes. Then, we obtain the central limit theorem and process-level, i.e. level-3 large deviations for nonlinear Hawkes processes. The level-1 large deviation principle holds as a result of the contraction principle. We also provide an alternative variational formula for the rate function of the level-1 large deviations in the Markovian case. Next, we drop the usual assumptions on the nonlinear Hawkes process and categorize it into different regimes, i.e. sublinear, sub-critical, critical, super-critical and explosive regimes. We show the different time asymptotics in different regimes and obtain other properties as well. Finally, we study the limit theorems of linear Hawkes processes with random marks.

math.PR↗