Showing papers in "Stochastic Analysis and Applications in 2010"
TL;DR: In this paper, the authors extend the notion of g-evaluation, in particular g-expectation, of Peng [8, 9] to the case where the generator g is allowed to have a quadratic growth (in the variable “z”).
Abstract: In this article we extend the notion of g-evaluation, in particular g-expectation, of Peng [8, 9] to the case where the generator g is allowed to have a quadratic growth (in the variable “z”). We show that some important properties of the g-expectations, including a representation theorem between the generator and the corresponding g-expectation—and consequently the reverse comparison theorem of quadratic BSDEs as well as the Jensen inequality—remain true in the quadratic case. Our main results also include a Doob–Meyer type decomposition, the optional sampling theorem, and the upcrossing inequality. The results of this article are important in the further development of the general quadratic nonlinear expectations (cf. [5]).
53 citations
TL;DR: In this article, the problem of parameter estimation for an ergodic diffusion with the symmetric scaled Student invariant distribution is considered, where the spectral representation of the transition density is given in terms of the finite number of polynomial eigenfunctions (Routh-Romanovski polynomials) and the absolutely continuous spectrum of the negative infinitesimal generator of observed diffusion.
Abstract: We consider the problem of parameter estimation for an ergodic diffusion with the symmetric scaled Student invariant distribution, where the spectral representation of the transition density is given in terms of the finite number of polynomial eigenfunctions (Routh–Romanovski polynomials) and absolutely continuous spectrum of the negative infinitesimal generator of observed diffusion. We prove the consistency and asymptotic normality of the proposed estimators and, based on the Stein equation for Student diffusion, consider the statistical test for the Student distributional assumptions.
40 citations
TL;DR: In this article, a broad class of functional dependence of the right-hand side on the current random state is investigated and the existence and uniqueness of solutions is obtained as a limiting process by freezing the coefficients over short time intervals.
Abstract: Stochastic ordinary differential equations are investigated for which the coefficients depend on nonlocal properties of the current random variable in the sample space such as the expected value or the second moment. The approach here covers a broad class of functional dependence of the right-hand side on the current random state and is not restricted to pathwise relations. Existence and uniqueness of solutions is obtained as a limiting process by freezing the coefficients over short time intervals and applying existence and uniqueness results and appropriate estimates for stochastic ordinary differential equations.
39 citations
TL;DR: In this paper, a sector-wise allocation in a portfolio consisting of a very large number of stocks is considered, and the dependence of the drift coefficient of each stock on an averaged effect of the sectors is captured.
Abstract: This article considers a sector-wise allocation in a portfolio consisting of a very large number of stocks. Their interdependence is captured by the dependence of the drift coefficient of each stock on an averaged effect of the sectors. This leads to a decoupled dynamics in the limit of large numbers, akin to the "mean field" limit leading to the McKean-Vlasov equation in statistical physics. This gives a more compact description using a time-varying drift characterized in terms of a measure-valued process that satisfies a nonlinear parabolic equation. The classical portfolio optimization problem is then addressed in this framework.
36 citations
TL;DR: In this paper, the authors consider a class of control systems governed by neutral stochastic functional differential equations with unbounded delay and study the approximate controllability of the system.
Abstract: In this article, we consider a class of control systems governed by the neutral stochastic functional differential equations with unbounded delay and study the approximate controllability of the system. An example is given to illustrate the result.
35 citations
TL;DR: In this paper, strong stability for weighted sums of negatively associated random variables is studied and strong convergence results for the weighted sums are obtained, which generalize the corresponding results for independent sequences without adding extra conditions.
Abstract: Let {X n , n ≥ 1} be a sequence of negatively associated random variables with identical distribution. Some properties for negatively associated sequences are discussed. Some strong convergence results for the weighted sums are obtained, which generalize the corresponding results for independent sequences without adding extra conditions. In addition, strong stability for weighted sums of negatively associated random variables is studied.
32 citations
TL;DR: In this paper, the authors investigated backward stochastic Volterra integral equations in Hilbert spaces and established the regularity of the adapted solutions to such equations by means of Malliavin calculus.
Abstract: This article investigates backward stochastic Volterra integral equations in Hilbert spaces. The existence and uniqueness of their adapted solutions is reviewed. We establish the regularity of the adapted solutions to such equations by means of Malliavin calculus. For an application, we study an optimal control problem for a stochastic Volterra integral equation driven by a Hilbert space-valued fractional Brownian motion. A Pontryagin-type maximum principle is formulated for the problem and an example is presented.
32 citations
TL;DR: In this article, it was shown that the expectation of the cube of the G-Brownian motion is positive, which is qualitatively different from the classical Brownian motion case.
Abstract: In continuing his study of the intrinsically nonlinear expectation and conditional expectation under the so-called G-framework, Peng introduced a nonlinear Ito calculus; here, the G refers to the generator of a nonlinear heat equation. There, he derived the corresponding Ito formula for C 2-functions with bounded Lipschiz derivatives. This restrictive class of functions limits its applicatory value to stochastic finances and cannot be applied to study the powers of the G-Brownian motion. We extend the Ito formula to a slightly more general class of functions (C 2-functions with uniformly continuous derivatives). This enables us to compute the G-expectations of the even powers of the G-Brownian motion. The G-expectation of odd powers behave differently; in particular, we show that the G-expectation of the cube of the G-Brownian motion is positive, which is qualitatively different from the classical Brownian motion case. We remark that we are not able to get a formula for the G-expectation of the general od...
30 citations
TL;DR: In this paper, the importance sampling technique was generalized from simulating expectations to computing the initial value of backward stochastic differential equations (SDEs) with Lipschitz continuous driver.
Abstract: In this article, we explain how the importance sampling technique can be generalized from simulating expectations to computing the initial value of backward stochastic differential equations (SDEs) with Lipschitz continuous driver. By means of a measure transformation we introduce a variance reduced version of the forward approximation scheme by Bender and Denk [4] for simulating backward SDEs. A fully implementable algorithm using the least-squares Monte Carlo approach is developed and its convergence is proved. The success of the generalized importance sampling is illustrated by numerical examples in the context of Asian option pricing under different interest rates for borrowing and lending.
28 citations
TL;DR: In this paper, it was shown that if the generator f is uniformly continuous in (y, z), uniformly with respect to (t, ω), and if the terminal value ξ ∈ L p (Ω, ℱ T, P) with 1 < p ≤ 2, then the backward stochastic differential equation has a unique L p solution.
Abstract: In this article, we study one-dimensional backward stochastic differential equations with continuous coefficients. We show that if the generator f is uniformly continuous in (y, z), uniformly with respect to (t, ω), and if the terminal value ξ ∈L p (Ω, ℱ T , P) with 1 < p ≤ 2, the backward stochastic differential equation has a unique L p solution.
24 citations
TL;DR: The asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity.
Abstract: In this article, the asymptotic behavior of multitype Markov branching processes with discrete or continuous time is investigated in the positive regular and nonsingular case when both the initial number of ancestors and the time tend to infinity. Some limiting distributions are obtained as well as multivariate asymptotic normality is proved. The article also considers the relative frequencies of distinct types of individuals motivated by applications in the field of cell biology. We obtained non-random limits for the frequencies and multivariate asymptotic normality when the initial number of ancestors is large and the time of observation increases to infinity. In fact this paper continues the investigations of Yakovlev and Yanev [32] where the time was fixed. The new obtained limiting results are of special interest for cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement.
TL;DR: In this article, the authors considered the optimal reinsurance and dividend strategy for an insurer by modeling the surplus process of the insurer by the classical compound Poisson risk model modulated by an observable continuous-time Markov chain.
Abstract: In this article, we consider the optimal reinsurance and dividend strategy for an insurer. We model the surplus process of the insurer by the classical compound Poisson risk model modulated by an observable continuous-time Markov chain. The object of the insurer is to select the reinsurance and dividend strategy that maximizes the expected total discounted dividend payments until ruin. We give the definition of viscosity solution in the presence of regime switching. The optimal value function is characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation and a verification theorem is also obtained.
Abstract: The mean density of a random closed set Θ in ℝ d with Hausdorff dimension n is the Radon–Nikodym derivative of the expected measure [ℋ n (Θ ∩·)] induced by Θ with respect to the usual d-dimensional Lebesgue measure. Starting from an open problem posed by Matheron in [24, pp. 50–51], we consider here inhomogeneous Boolean models Ξ in ℝ d with integer Hausdorff dimension n ∊ {0,…, d}, and we study the mean density of their boundary (which is their mean density if n < d) and the differentiability of their spherical contact distribution function H Ξ, under general regularity assumptions on the typical grain, related to the existence of its (outer) Minkowski content. In particular, we provide an explicit formula for ∂2 H Ξ(r, x)/(∂r 2) at r = 0 for a class of Boolean models, whose typical grain has positive reach; known results for stationary Boolean models with convex grains follows then as a particular case. Examples and statistical applications are also discussed.
TL;DR: In this article, the authors considered the SDE with α, β ∊ ℝ, α≠ β, β ≥ β and showed that the probability measures induced by the processes X (α) and X (β) are singular on (C[0, T), ℬ(C[ 0, T))).
Abstract: Let us consider the process given by the SDE , t ∊ [0, T), where α ∊ ℝ, T ∊ (0, ∞), and (B t ) t≥0 is a standard Wiener process. In case of α > 0, the process X (α) is known as an α-Wiener bridge, in case of α = 1 as the usual Wiener bridge. We prove that for all α, β ∊ ℝ, α ≠ β, the probability measures induced by the processes X (α) and X (β) are singular on (C[0, T), ℬ(C[0, T))). Further, we investigate regularity properties of as t ↑ T.
TL;DR: In this paper, continuous time Markov chain (CTMC) and It stochastic differential equation (SDE) models are derived for a population with births, immigration and deaths (BID model).
Abstract: Continuous time Markov chain (CTMC) and It stochastic differential equation (SDE) models are derived for a population with births, immigration and deaths (BID model). Differential equations are derived for the moments of the distribution for each stochastic model. Each moment differential equation depends on higher-order moments. Assumptions are made regarding higher-order moments to form a finite, solvable system. Conditions are given under which the CTMC and SDE BID models have the same moment solution or the same stationary solution. The close agreement between the CTMC and SDE models is illustrated in three numerical examples based on normal or log-normal moment closure assumptions.
TL;DR: In this article, a weaker sufficient condition than that of Theorem 5.2.1 in Fleming and Soner (2006) was determined for the continuity of the value functions of stochastic exit time control problems.
Abstract: We determine a weaker sufficient condition than that of Theorem 5.2.1 in Fleming and Soner (2006) for the continuity of the value functions of stochastic exit time control problems.
TL;DR: In this paper, the topological properties of the space of upper semicontinuous functions were discussed, and large deviation principles for random upper semiconvolutional functions under various topologies were obtained.
Abstract: We firstly discuss the topological properties of the space of upper semicontinuous functions, and then we obtain large deviation principles for random upper semicontinuous functions under various topologies. Finally, we prove moderate deviation principles for random sets and random upper semicontinuous functions.
Abstract: Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and L 1 convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire [19] and Bacry et al. [1] in the simpler framework of Mandelbrot cascades.
TL;DR: In this article, the generalized Clark-Ocone formula for change of measure was shown to be generalizable under Gaussian white noise analysis and Malliavin calculus, and it was shown that for any square integrable ℱ T -measurable random variable, where 𝔼 Q is the expectation under Q and D · F(ω) is the (Hida) Mallian derivative, the important point in this settlement is that F does not have to be in stochastic Sobolev space & #x 1d53
Abstract: We prove the white noise generalization of the Clark-Ocone formula under change of measure by using Gaussian white noise analysis and Malliavin calculus. Let W(t) be a Brownian motion on the filtered white noise probability space (Ω, ℬ, {ℱ t }0≤t≤T , P) and let be defined as , where u(t) is an ℱ t -measurable process satisfying certain conditions for all 0 ≤ t ≤ T. Let Q be the probability measure equivalent to P such that is a Brownian motion with respect to Q, in virtue of the Girsanov theorem. In this article, it is shown that for any square integrable ℱ T -measurable random variable, where 𝔼 Q is the expectation under Q and D · F(ω) is the (Hida) Malliavin derivative. The important point in this settlement is F does not have to be in stochastic Sobolev space 𝔻1, 2 ⊂ L 2(P). This makes the formula more useful in applications of finance. As an example, the replicating portfolio for a digital option with the payoff χ[K, ∞) W(T) ∉ 𝔻1, 2 is calculated by using this generalized Clark-Ocone formula under cha...
TL;DR: In this paper, the authors consider a typical portfolio of different insurance products and investigate the pricing process using the framework of a linear time invariant generalized stochastic discrete-time model, assuming that the resulting system is (regular) descriptor and calculate the solution using the tools of matrix pencil theory.
Abstract: We consider a typical portfolio of different insurance products and investigate the pricing process using the framework of a linear time invariant generalized stochastic discrete-time model. Moreover, we assume that, due to regulatory constraints, the resulting system is (regular) descriptor and calculate the solution using the tools of matrix pencil theory. Finally, we present a numerical application for two different portfolios.
TL;DR: In this paper, the authors considered a filtering problem for forward-backward stochastic systems that are driven by Brownian motions and Poisson processes, and the desired filtering equation was established.
Abstract: In this article, we consider a filtering problem for forward-backward stochastic systems that are driven by Brownian motions and Poisson processes. This kind of filtering problem arises from the study of partially observable stochastic linear-quadratic control problems. Combining forward-backward stochastic differential equation theory with certain classical filtering techniques, the desired filtering equation is established. To illustrate the filtering theory, the theoretical result is applied to solve a partially observable linear-quadratic control problem, where an explicit observable optimal control is determined by the optimal filtering estimation.
TL;DR: In this paper, an algorithm using rooted trees for expanding the weight functions occurring in this representation in terms of multiple integrals using multi-indices is presented, which results in an alternative approach to express relations between multiple integral components.
Abstract: The exact solution of stochastic differential equations can be expressed as stochastic B-series. In this article, we present an algorithm using rooted trees for expanding the weight functions occurring in this representation in terms of multiple integrals using multi-indices. This results in an alternative approach to express relations between multiple integrals.
TL;DR: In this article, several inequalities for the ℒ q (&#x 1d4ab;)-norm of the Wick product of random variables were derived via a positivity argument.
Abstract: We provide several inequalities for the ℒ q (𝒫)-norm of the Wick product of random variables. These estimates are based on a Jensen's type inequality for the Wick multiplication, which we derive via a positivity argument. As an application we study a certain type of anticipating stochastic differential equation whose solution is shown to be an element of ℒ q (𝒫) for some q ≥ 1.
TL;DR: In this article, the authors study nonlinear heat and wave equations on a Lie group and give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise.
Abstract: We study nonlinear heat and wave equations on a Lie group. The noise is assumed to be a spatially homogeneous Wiener process. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise.
TL;DR: In this article, a class of Markov processes whose transition probability densities are defined by multifractional pseudodifferential evolution equations on compact domains with variable local dimension was introduced.
Abstract: In this article, we introduce a class of Markov processes whose transition probability densities are defined by multifractional pseudodifferential evolution equations on compact domains with variable local dimension. The infinitesimal generators of these Markov processes are given by the trace of strongly elliptic pseudodifferential operators of variable order on such domains. The results derived provide a pseudomultifractal version of some existing special classes of multifractional Markov processes. In particular, pseudostable processes are defined on domains with variable local dimension in this framework. In the case where the local dimension of the domain and the local Holder exponents of the transition probability densities are constant, the existing results on fractal versions of Levy processes are recovered.
TL;DR: In this article, multistep Bayesian betting strategies in coin-tossing games were studied in the framework of game-theoretic probability of Shafer and Vovk.
Abstract: We study multistep Bayesian betting strategies in coin-tossing games in the framework of game-theoretic probability of Shafer and Vovk [12]. We show that by a countable mixture of these strategies, a gambler or an investor can exploit arbitrary patterns of deviations of nature's moves from independent Bernoulli trials. We then apply our scheme to asset trading games in continuous time and derive the exponential growth rate of the investor's capital when the variation exponent of the asset price path deviates from two.
TL;DR: In this paper, a stochastic Taylor expansion of some functional applied to the solution process of an Ito or Stratonovich Stochastic differential equation with a multi-dimensional driving Wiener process is given, which is similar to the B-series expansion for solutions of ordinary differential equations in the deterministic setting.
Abstract: In this article, a stochastic Taylor expansion of some functional applied to the solution process of an Ito or Stratonovich stochastic differential equation with a multi-dimensional driving Wiener process is given. Therefore, the multi-colored rooted tree analysis is applied in order to obtain a transparent representation of the expansion which is similar to the B-series expansion for solutions of ordinary differential equations in the deterministic setting. Further, some estimates for the mean-square and the mean truncation errors are given.
TL;DR: In this article, the authors consider a stochastic inventory system that has been operated under a policy different from the one that will be implemented in the future and develop three models under different assumptions that describe the demand during the disposal period and present analytical results characterizing their optimal solutions.
Abstract: We consider a stochastic inventory system that has been operated under a policy different from the one that will be implemented in the future. Such a situation may arise as a result of changes in model assumptions leading to the implementation of a different policy. Before the new policy is implemented, there may be some units on hand which may exceed the optimal order-up-to level. Hence, one needs to evaluate a one-time inventory disposal decision immediately before the new policy replaces the policy in use. For this purpose, we develop three models under different assumptions that describe the demand during the disposal period and present analytical results characterizing their optimal solutions.
TL;DR: In this article, a time-homogeneous one-dimensional diffusion process defined in I⊂ℝ, starting at x ∈ I and let c∈ I a barrier with c ǫx, and for the first-exit time of from an open interval containing c, when the infinitesimal coefficients are allowed to change as crosses the boundary c, was studied, whenever it exists.
Abstract: Let X(t) be a time-homogeneous one-dimensional diffusion process defined in I ⊂ ℝ, starting at x ∈ I and let c ∈ I a barrier with c x, and for the first-exit time of from an open interval containing c, when the infinitesimal coefficients are allowed to change as crosses the boundary c. Moreover, the stationary distribution of is studied, whenever it exists. Some explicit examples are also reported.
TL;DR: In this paper, the existence of an admissible investment strategy for any given consumption rate process in a Markov, regime-switching Black-Scholes-Merton economy is discussed.
Abstract: We discuss the existence of an admissible investment strategy for any given consumption rate process in a Markov, regime-switching Black–Scholes–Merton economy. A martingale representation for a double martingale generated by the Brownian motion and the Markov chain is used to establish the existence of the admissible investment strategy. We also employ the martingale representation to prove the attainability of a European contingent claim in the regime-switching environment under a pricing kernel specified by the Esscher transform based on the Laplace cumulant process.