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2011 | Buch

Stochastic Analysis with Financial Applications

Hong Kong 2009

herausgegeben von: Arturo Kohatsu-Higa, Nicolas Privault, Shuenn-Jyi Sheu

Verlag: Springer Basel

Buchreihe : Progress in Probability

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Über dieses Buch

Stochastic analysis has a variety of applications to biological systems as well as physical and engineering problems, and its applications to finance and insurance have bloomed exponentially in recent times. The goal of this book is to present a broad overview of the range of applications of stochastic analysis and some of its recent theoretical developments. This includes numerical simulation, error analysis, parameter estimation, as well as control and robustness properties for stochastic equations. The book also covers the areas of backward stochastic differential equations via the (non-linear) G-Brownian motion and the case of jump processes. Concerning the applications to finance, many of the articles deal with the valuation and hedging of credit risk in various forms, and include recent results on markets with transaction costs.

Inhaltsverzeichnis

Frontmatter

Stochastic Analysis

Frontmatter
Dirichlet Forms for Poisson Measures and Lévy Processes: The Lent Particle Method
Abstract
We present a new approach to absolute continuity of laws of Poisson functionals. The theoretical framework is that of local Dirichlet forms as a tool for studying probability spaces. The argument gives rise to a new explicit calculus that we present first on some simple examples: it consists in adding a particle and taking it back after computing the gradient. Then we apply the method to SDE’s driven by Poisson measure.
Nicolas Bouleau, Laurent Denis
Stability of a Nonlinear Equation Related to a Spatially-inhomogeneous Branching Process
Abstract
Consider the nonlinear equation
$$\frac{\partial}{\partial t}u(x,t)=\Delta_\alpha u (x, t) + a(x) \sum\limits_{k=2}^{\infty} pk^{{u^k}} (x, t)+(p0 + p1 u(x, t))\phi(x), x\in \mathbb{R}^d,$$
where α ∈ (0, 2], u (x, 0) is nonnegative, {pk, k = 0, 1,...} is a probability distribution on ℤ+, and a and φ are positive functions satisfying certain growth conditions. We prove existence of non-trivial positive global solutions when p0, p1 and u(x, 0) are small.
S. Chakraborty, E. T. Kolkovska, J. A. López-Mimbela
Backward Stochastic Difference Equations with Finite States
Abstract
We define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. Solutions exist and are unique under weaker assumptions than are needed in the continuous time setting. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are explored, including a representation result.
Samuel N. Cohen, Robert J. Elliott
On a Forward-backward Stochastic System Associated to the Burgers Equation
Abstract
We describe a probabilistic construction of H-regular solutions for the spatially periodic forced Burgers equation by using a characterization of this solution through a forward-backward stochastic system.
Ana Bela Cruzeiro, Evelina Shamarova
On the Estimate for Commutators in DiPerna–Lions Theory
Abstract
In this note, we will exhibit the estimates on the commutator of semi-groups, motivated by commutator estimates in Di Perna-Lions theory.
Shizan Fang, Huaiqian Lee
Approximation Theorem for Stochastic Differential Equations Driven by G-Brownian Motion
Abstract
We present an approximation theorem for stochastic differential equations driven by G-Brownian motion, i.e., solutions of stochastic differential equations driven by G-Brownian motion can be approximated by solutions of ordinary differential equations.
Fuqing Gao, Hui Jiang
Stochastic Flows for Nonlinear SPDEs Driven by Linear Multiplicative Space-time White Noises
Abstract
For a nonlinear stochastic partial differential equation driven by linear multiplicative space-time white noises, we prove that there exists a bicontinuous version of the solution with respect to the initial value and thetime variable.
Benjamin Goldys, Xicheng Zhang
Optimal Stopping Problem Associated with Jump-diffusion Processes
Abstract
In this paper we study an optimal stopping problem associated with jump-diffusion processes. We use a viscosity solution approach for the solution to HJB equality, which the value function should obey. Using the penalty method we obtain the existence of the value function as a viscosity solution to the HJB equation, and the uniqueness.
Yasushi Ishikawa
A Review of Recent Results on Approximation of Solutions of Stochastic Differential Equations
Abstract
In this article, we give a brief review of some recent results concerning the study of the Euler-Maruyama scheme and its high-order extensions. These numerical schemes are used to approximate solutions of stochastic differential equations, which enables to approximate various important quantities including solutions of partial differential equations. Some have been implemented in Premia [56]. In this article we mainly consider results about weak approximation, the most important for financial applications.
Benjamin Jourdain, Arturo Kohatsu-Higa
Strong Consistency of Bayesian Estimator Under Discrete Observations and Unknown Transition Density
Abstract
We consider the asymptotic behavior of a Bayesian parameter estimation method under discrete stationary observations. We suppose that the transition density of the data is unknown, and therefore we approximate it using a kernel density estimation method applied to the Monte Carlo simulations of approximations of the theoretical random variables generating the observations. In this article, we estimate the error between the theoretical estimator, which assumes the knowledge of the transition density and its approximation which uses the simulation. We prove the strong consistency of the approximated estimator and find the order of the error. Most importantly, we give a parameter tuning result which relates the number of data, the number of time-steps used in the approximation process, the number of the Monte Carlo simulations and the bandwidth size of the kernel density estimation.
Arturo Kohatsu-Higa, Nicolas Vayatis, Kazuhiro Yasuda
Exponentially Stable Stationary Solutions for Delay Stochastic Evolution Equations
Abstract
We establish some sufficient conditions ensuring existence, uniqueness and exponential stability of non-trivial stationary mild solutions for a class of delay stochastic partial differential equations. Some known results are generalized and improved.
Jiaowan Luo
Robust Stochastic Control and Equivalent Martingale Measures
Abstract
We study a class of robust, or worst case scenario, optimal control problems for jump diffusions. The scenario is represented by a probability measure equivalent to the initial probability law. We show that if there exists a control that annihilates the noise coefficients in the state equation and a scenario which is an equivalent martingale measure for a specific process which is related to the control-derivative of the state process, then this control and this probability measure are optimal. We apply the result to the problem of consumption and portfolio optimization under model uncertainty in a financial market, where the price process S(t) of the risky asset is modeled as a geometric Ito-L00E9vy process. In this case the optimal scenario is an equivalent local martingale measure of S(t). We solve this problem explicitly in the case of logarithmic utility functions.
Bernt Øksendal, Agnès Sulem
Multi-valued Stochastic Differential Equations Driven by Poisson Point Processes
Abstract
We prove the existence and uniqueness of solutions of multi-valued stochastic differential equations driven by Poisson point processes when the domain of the multi-valued maximal monotone operator is the whole space Rd.
Jiagang Ren, Jing Wu
Sensitivity Analysis for Jump Processes
Abstract
Consider stochastic differential equations with jumps. The goal in this paper is to study the sensitivity of the solution with respect to the initial point, under the conditions on the Lévy measure and the uniformly elliptic condition on the coefficients. The key tool is the martingale property based upon the Kolmogorov backward equation for the infinitesimal generator associated with the equation.
Atsushi Takeuchi
Quantifying Model Uncertainties in Complex Systems
Abstract
Uncertainties are abundant in complex systems. Appropriate mathematical models for these systems thus contain random effects or noises. The models are often in the form of stochastic differential equations, with some parameters to be determined by observations. The stochastic differential equations may be driven by Brownian motion, fractional Brownian motion, or Lévy motion. After a brief overview of recent advances in estimating parameters in stochastic differential equations, various numerical algorithms for computing parameters are implemented. The numerical simulation results are shown to be consistent with theoretical analysis. Moreover, for fractional Brownian motion and a-stable Lévy motion, several algorithms are reviewed and implemented to numerically estimate the Hurst parameter H and characteristic exponent a.
Jiarui Yang, Jinqiao Duan

Financial Applications

Frontmatter
Convertible Bonds in a Defaultable Diffusion Model
Abstract
In this paper, we study convertible securities (CS) in a primary market model consisting of: a savings account, a stock underlying a CS, and an associated CDS contract (or, alternatively to the latter, a rolling CDS more realistically used as an hedging instrument). We model the dynamics of these three securities in terms of Markovian diffusion set-up with default. In this model, we show that a doubly reflected Backward Stochastic Differential Equation associated with a CS has a solution, meaning that super-hedging of the arbitrage value of a convertible security is feasible in the present setup for both issuer and holder at the same initial cost, and we provide the related (super-)hedging strategies. Moreover, we characterize the price of a CS in terms of viscosity solutions of associated variational inequalities and we prove the convergence of suitable approximation schemes. We finally specify these results to convertible bonds and their straight bond and game exchange option components, and provide numerical results.
Tomasz R. Bielecki, Stéphane Crépey, Monique Jeanblanc, Marek Rutkowski
A Convexity Approach to Option Pricing with Transaction Costs in Discrete Models
Abstract
We study the option pricing problems under transaction costs in a discrete financial model with a riskless bond and one risky asset. For general models with a transaction fee, a perfectly replicating portfolio (if it exists) may not be optimal, i.e., a perfectly replicating portfolio may cost more than certain portfolios which super replicate the contingent claim.
Tzuu-Shuh Chiang, Shuenn-Jyi Sheu
Completeness and Hedging in a Lévy Bond Market
Abstract
In this paper we analyze the completeness problem in a bond market where the short rate is driven by a non-homogeneous Lé process. Even though it is known that under certain conditions we have a kind of uniqueness of the risk neutral measure, little is known about how to hedge in this market. We elucidate that perfect replication formulas are not, in general, possible to obtain and an approximate hedging, in an L2 sense, is then the appropriate approach.
José M. Corcuera
Asymptotically Efficient Discrete Hedging
Abstract
The notion of asymptotic efficiency for discrete hedging is introduced and a discretizing strategy which is asymptotically efficient is given explicitly. A lower bound for asymptotic risk of discrete hedging is given, which is attained by a simple discretization scheme. Numerical results for delta hedging in the Black-Scholes model are also presented.
Masaaki Fukasawa
Efficient Importance Sampling Estimation for Joint Default Probability:The First Passage Time Problem
Abstract
Motivated from credit risk modeling, this paper extends the twodimensional first passage time problem studied by Zhou (2001) to any finite dimension by means of Monte Carlo simulation. We provide an importance sampling method to estimate the joint default probability, and apply the large deviation principle to prove that the proposed importance sampling is asymptotically optimal. Our result is an alternative to the interacting particle systems proposed by Carmona, Fouque, and Vestal (2009).
Chuan-Hsiang Han
Market Models of Forward CDS Spreads
Abstract
The paper re-examines and generalizes the construction of several variants of market models for forward CDS spreads, as first presented by Brigo [10]. We compute explicitly the joint dynamics for some families of forward CDS spreads under a common probability measure. We first examine this problem for single-period CDS spreads under certain simplifying assumptions. Subsequently, we derive, without any restrictions, the joint dynamics under a common probability measure for the family of one- and two-period forward CDS spreads, as well as for the family of one-period and co-terminal forward CDS spreads. For the sake of generality, we work throughout within a general semimartingale framework.
Libo Li, Marek Rutkowski
Optimal Threshold Dividend Strategies under the Compound Poisson Model with Regime Switching
Abstract
In this paper, we consider the optimal dividend strategy for an insurer whose surplus process is modeled by the classical compound Poisson risk model modulated by an observable continuous-time Markov chain. The object of the insurer is to select the dividend strategy that maximizes the expected total discounted dividend payments until ruin. We assume that the company only allows to pay dividend at a small rate. Given some conditions, similar to the results of Sotomayor and Cadenillas (2008) and Jiang and Pistorius (2008), the optimal strategy of our model is also a modulated threshold strategy which depends on the environment state. For the case of two regimes and exponential claim sizes, we obtain an analytical solution.
Jiaqin Wei, Hailiang Yang, Rongming Wang
Metadaten
Titel
Stochastic Analysis with Financial Applications
herausgegeben von
Arturo Kohatsu-Higa
Nicolas Privault
Shuenn-Jyi Sheu
Copyright-Jahr
2011
Verlag
Springer Basel
Electronic ISBN
978-3-0348-0097-6
Print ISBN
978-3-0348-0096-9
DOI
https://doi.org/10.1007/978-3-0348-0097-6