Online drift estimation for jump-diffusion processes by Bhudisaksang, T, Cartea, A

“…We decompose the deviation of the stochastic descent direction from the deterministic descent direction into four terms: the weak solution of the non-local Poisson equation, a Riemann integral, a stochastic integral, and a covariation term. This decomposition is employed to prove the convergence of the online estimator and we use simulations to illustrate the performance of the online estimator.…”

ESO‐based output‐feedback regulation control of nonlinear systems with SiISS inverse dynamics by Chunxiao Wang, Yang Gao, Yuliang Sun, Yuqiang Wu

“…Abstract This paper considers the output‐feedback controller for a class of stochastic cascade nonlinear systems with stochastic integral input‐to‐state stable inverse dynamics, unknown control coefficients, and matching disturbances. …”
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Effective dynamics of microorganisms that interact with their own trail by Golestanian, R, Gelimson, A, Kranz, W, Zhao, K, Wong, G

“…The effective dynamics of each microorganism takes on the form of a stochastic integral equation with the trail interaction appearing in the form of short-term memory. …”

Integral representation of martingales motivated by the problem of endogenous completeness in financial economics by Kramkov, D, Predoiu, S

“…Motivated by the problem of endogenous completeness in financial economics we present conditions which guarantee that every local martingale under Q is a stochastic integral with respect to the J-dimensional martingale Stδ EQ[ψ|Ft]. …”

Threshold behaviour of a stochastic SIRS Lévy jump model with saturated incidence and vaccination by Yu Zhu, Liang Wang, Zhipeng Qiu

“…Due to the introduction of $ \mathrm {L\acute{e}vy} $ jump, the jump stochastic integral process is a discontinuous martingale. …”
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Particle representations for measure-valued population models by Donnelly, P, Kurtz, T

“…The particle model gives a simple representation of the Dawson-Perkins historical process and Perkins's historical stochastic integral can be obtained in terms of classical semimartingale integration. …”

Anticipating random periodic solutions I: SDEs with multiplicative linear noise by Feng, C, Wu, Y, Zhao, H

“…We identify them as solutions of coupled forward–backward infinite horizon stochastic integral equations (IHSIEs), using the “substitution theorem” of stochastic differential equations with anticipating initial conditions. …”

A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales. by Barndorff-Nielsen, O, Graversen, S, Jacod, J, Podolskij, M, Shephard, N

“…If further is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with W and by a Poisson random measure, we prove a central limit theorem, in the sense that \sqrt(n) (V(Y;r,s)^n-V(Y;r,s)) converges in law to a process which is the stochastic integral with respect to some other Brownian motion W', which is independent of the driving terms of Y and \sigma. …”

Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation by Cristina-Liliana Pripoae, Iulia-Elena Hirica, Gabriel-Teodor Pripoae, Vasile Preda

“…Using this geometrization, we characterize the equivalence between the Gibbs–Helmholtz entropy and the Boltzmann–Gibbs–Shannon, Tsallis, and Kaniadakis entropies, respectively, by means of three stochastic integral equations. We prove that some specific (infinite) families of normal probability distributions are solutions for these equations. …”
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Efficient integral equation based algorithms for parasitic extraction of interconnects with smooth or rough surface by Zhu, Zhenhai, 1970-

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The Ito calculus: a vector-integral approach by Ling, PD

“…The Itô calculus is the theory of stochastic integrals ∫^t₀ X_u dS_u, where S is a semimartingale, and X is a suitable previsible process. …”

Analysis of various estimators for multi-dimensional Zakai equations by Wang, Z

“…</p> <p>Then, we propose an implicit finite difference scheme for a two-dimensional parabolic SPDE of Zakai type, based on a Milstein approximation to the stochastic integral and an alternating direction implicit (ADI) discretisation of the elliptic term. …”

Fifth-stage stochastic runge-kutta method for stochastic differential equations by Noor Amalina Nisa, Ariffin

“…The relative paucity of numerical methods in SDEs is due to the complexity of approximating high-order multiple stochastic integrals. A stochastic integral provides information of the Wiener process, which then contributes to the order of the methods. …”
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Stein approximation for Ito and Skorohod integrals by Edgeworth type expansions by Privault, Nicolas

“…As a consequence we obtain Stein approximation bounds for stochastic integrals, which apply to SDE solutions and to multiple stochastic integrals.…”
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Girsanov identities for Poisson measures under quasi-nilpotent transformations by Privault, Nicolas

“…The proofs use combinatorial identities for the central moments of Poisson stochastic integrals.…”
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ANGOL NYELVŰ MAGYAR TANKÖNYV OXFORDBAN by ZSÓFIA BAK

“… A Budapesti Corvinus Egyetem Közgazdaságtudományi Kara Matematika tanszékén oktató Medvegyev Péter egyetemi docens Stochastic Integration Theory című könyvét az Oxford University Press jelentette meg 2007-ben. …”
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Boundedness and convergence analysis of stochastic differential equations with Hurst Brownian motion by Davood Ahmadian, Omid Farkhondeh Rouz

“…By the simple analysis and using the mean value theorem for stochastic integrals we conclude that in case of decreasing diffusion function, the solution of FBM is bounded for any H ∈ (0,1). …”
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Asymptotic expansions and distribution properties for diffusion processes by She, Qihao

“…We then consider multiple stochastic integrals, where we obtain the Stein approximation bounds using the derived conditional Edgeworth-type expansions. …”
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Studying Some Stochastic Differential Equations with trigonometric terms with Application by Abdulghafoor Jasim Salim , Ali A. Asmael

“…Then we find the exact solution for the different trigonometric stochastic differential equations by the use of stochastic integrals. Ilustrate the approach with various examples. …”
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The functional Itō formula under the family of continuous semimartingale measures by Oberhauser, H

“…By revisiting old work of Bichteler and Karandikar we show that one can construct pathwise versions of complex functionals like the quadratic variation, stochastic integrals or Itō processes that are still regular enough such that a functional Itō-formula applies.…”