Stability of Stochastic Functional Differential Equations with Regime-Switching: Analysis Using Dupire’s Functional Itô Formula

Potential Analysis - This work focuses on almost sure and Lp stability of stochastic functional differential equations by using Lyapunov functionals with the help of the recently developed...     

On Itô s formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X = (X ¹,…,X ^d ) and a function F which belongs locally to the Sobolev space W ^1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f _k (X), X ^k ] involving the weak first partial derivatives f _k of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X ^k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Received: 16 March 1998 / Revised version: 4 April 1999     

Dupire It\^{o}’s formula for the exponential synchronization of stochastic semi-Markov jump systems with mixed delay under impulsive control

This paper emphasizes the exponential synchronization for a class of stochastic semi-Markov jump systems with mixed delay via stochastic hybrid impulsive control. The impulsive sequence includes synchronous and asynchronous impulses with the impulsive gains being a sequence of stochastic variables. Inspired by the idea of average, a concept of ``average stochastic impulsive gain" is used to qualify the impulse intensity. Our approach expands Dupire functional It\^{o}$"s formula to the stochastic semi-Markov jump systems with mixed delay for the first time. Moreover, in view of the established Lyapunov functional, graph theory, and stochastic analysis theory, some exponential synchronization criteria for the systems are derived. The theoretical results are applied to a class of Chua"s circuit systems with semi-Markov jump and mixed delay. Some synchronization criteria for the circuit systems are provided. The simulation results verify the effectiveness of the theoretical results.     

Itô’s formula for the L p -norm of stochastic {W^{1}_{p}} -valued processes

We prove Itô’s formula for the L _p -norm of a stochastic ${W^{1}_{p}}$ -valued processes appearing in the theory of SPDEs in divergence form.     

New explicit stabilized stochastic Runge-Kutta methods with weak second order for stiff Itô stochastic differential equations

Numerical Algorithms - This paper introduces a new class of weak second-order explicit stabilized stochastic Runge-Kutta methods for stiff Itô stochastic differential equations. The...     

A “direct” method to prove the generalized Itô–Venttsel’ formula for a generalized stochastic differential equation

For the first time we present a complete proof (from the standpoint of stochastic analysis) of the generalized Itô–Venttsel’ formula whose ideas were adduced in [8]. The proposed proof is an approach to construct the generalized Itô–Venttsel’ formula based on the direct application of the generalized Itô formula and the theory of stochastic approximation in contrast to the proof presented in [17] and based on the method of the integral invariants of a stochastic differential equation.     

Stability of linear impulsive Itô differential equations with bounded delays

We use the method of “model” equations to study the exponential p-stability (2 ≤ p < ∞) of the trivial solution with respect to the initial function for a linear impulsive system of Itô differential equations with bounded delays. The specific form of the equation and the method used permit one to analyze the stability of solutions starting from an arbitrary point of the half-line [0,∞) and obtain constructive sufficient conditions in terms of the parameters of the equations to be studied.     

Trotter’s formula for transition semigroups

Continuity properties and Trotter's formula for the transition semigroups corresponding to Markov processes are discussed. Applications to stochastic invariance are given as well. The research was motivated by the semigroup approach to invariance proposed by P. Kotelenez in [9]. November 15, 1998     

Wick–Itô formula for regular processes and applications to the Black and Scholes formula

We derive a Wick–Itô formula, that is, an Itô-type formula based on Wick integration. We derive it in the context of regular Gaussian processes which include Brownian motion and fractional Brownian motion with Hurst parameter greater than 1/2. We then consider applications to the Black and Scholes formula for the pricing of a European call option. It has been shown that using Wick integration in this context is problematic for economic reasons. We show that it is also problematic for mathematical reasons because the resulting Black and Scholes formula depends only on the variance of the process and not on its dependence structure.     

Remark on Itô’s Diffusion in Multidimensional Scattering with Sign-Indefinite Potentials

This paper extends some results of Denisov and Kupin (Int Math Res Not, doi:10.1093/imrn/rnr131, 2011) to the case of sign–indefinite potentials by applying methods developed in Denisov (J Funct Anal 254:2186–2226, 2008). This enables us to prove the presence of a.c. spectrum for the generic coupling constant.     

Existence of strong solutions for Itô's stochastic equations via approximations

Summary Given strong uniqueness for an Itô's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on any probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ^d and in domains in ^d are considered.Research supported by the Hungarian National Foundation of Scientific Research No. 2990.Supported in part by NSF Grant DMS-9302516     

Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.     

Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution

Abstract

In a ring with involution, we prove that a Drazin invertible element is pseudo core invertible if and only if its spectral idempotent is {1, 4}-invertible. As its applications, we obtain necessary and sufficient conditions for 1???ba (resp., ba) being pseudo core invertible while 1???ab (resp., ab) has pseudo core inverse, and the pseudo core inverse of 1???ba (resp., ba) is given in terms of 1???ab (resp., ab). Inspired by the above idea, Jacobson’s lemma for Moore-Penrose inverse is considered.     

A closed form solution for vulnerable options with Heston’s stochastic volatility

Over-the-counter stock markets in the world have been growing rapidly and vulnerability to default risks of option holders traded in the over-the-counter markets became an important issue, in particular, since the global finance crisis and Eurozone crisis. This paper studies the pricing of European-type vulnerable options when the underlying asset follows the Heston dynamics. In this paper, we obtain a closed form analytic formula of the option price as a stochastic volatility extension of the classical Heston formula and find how the stochastic volatility effect on the Black–Scholes price as well as on the decreasing speed of the option price with credit risk depends on moneyness.