Introduction to Stochastic Calculus with Applications

Klebaner, Fima C.

doi:10.1142/p821

Cited by 395 publications

(339 citation statements)

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“…The equations for γ and β are developed using the theory of stochastic differential equations based on Ito calculus (Klebaner, 1998) . We only give the relevant derivations for our model development in this section.…”

Section: Methodsmentioning

confidence: 99%

“…If

y (t)

is a stochastic process, and the related stochastic differential equation (SDE) in the integral form can be written as follows:

y true(t true) = {true \int}_{t_{0}}^{t} μ d t + {true \int}_{t_{0}}^{t} σ d w (t) .

where

t

is time,

t_{0}

is initial time,

μ

is the drift coefficient,

σ

is the diffusion coefficient and

w (t)

is the standard Wiener process with zero mean and variance t . Equation is usually written in the differential form but it can only be interpreted in the integral form . As the Weiner process is Markovian (memory less property), y ( t ) is Markovian with martingale properties .…”

Section: Methodsmentioning

confidence: 99%

“…Equation is usually written in the differential form but it can only be interpreted in the integral form . As the Weiner process is Markovian (memory less property), y ( t ) is Markovian with martingale properties . The differential form of Eq.…”

Section: Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Computational modeling and experimental validation of odor detection behaviors of classically conditioned parasitic wasp, Microplitis croceipes

Zhou

Kulasiri

Samarasinghe

et al. 2014

Biotechnology Progress

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A prototype chemical sensor named Wasp hound® that utilizes five classically conditioned parasitoid wasps, Microplitis croceipes (Cresson) (Hymenoptera: Braconidae), to detect volatile odors was successfully implemented in a previous study. To improve the odor-detecting ability of Wasp Hound®, searching behaviors of an individual wasp in a confined area are studied and modeled through stochastic differential equations in this paper. The wasps are conditioned to 20 mg of coffee when associated with food and subsequently, tested to 5, 10, 20, and 40 mg of coffee. A stochastic model is developed and validated based on three positive behavioral responses (walking, rotation around odor source, and self-rotation) from conditioned wasps at four different test dosages. The model is capable to reproducing the behaviors of conditioned wasps, and can be used to improve the ability of Wasp Hound® to assess changes in odor concentration. The model simulation results show the behaviors of conditioned wasps are significantly different when tested at different coffee dosages. We conjecture that the searching behaviors of conditioned wasps are based on the temporal and spatial neuron activity of olfactory receptor neurons and glomeruli, which are strongly correlated to the training dosages. The overall results demonstrate the utility of mathematical models for interpreting experimental observations, gaining novel insights into the dynamic behavior of classically conditioned wasps, as well as broadening the practical uses of Wasp Hound.

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Section: Methodsmentioning

confidence: 99%

“…If

y (t)

is a stochastic process, and the related stochastic differential equation (SDE) in the integral form can be written as follows:

y true(t true) = {true \int}_{t_{0}}^{t} μ d t + {true \int}_{t_{0}}^{t} σ d w (t) .

where

t

is time,

t_{0}

is initial time,

μ

is the drift coefficient,

σ

is the diffusion coefficient and

w (t)

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Computational modeling and experimental validation of odor detection behaviors of classically conditioned parasitic wasp, Microplitis croceipes

Zhou

Kulasiri

Samarasinghe

et al. 2014

Biotechnology Progress

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“…Definition (Isometry property) If

X false(t, ω false)

is a function on

false[a, b false] \times normalΩ

, the Itô isometry is defined as follows:

double-struckE ([], {((), \int_{a}^{b} X false(t, ω false) d B false(t false))}^{2}) = double-struckE ([], \int_{a}^{b} X^{2} false(t, ω false) d t) .

…”

Section: Preliminariesmentioning

confidence: 99%

Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations

Saffarzadeh

Heydari

Loghmani

2020

Math Methods in App Sciences

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In this paper, an efficient and accurate numerical iterative algorithm based on the linear spline interpolation for solving the system of nonlinear stochastic Itô-Volterra integral equations is presented. The most important merit of this method is that it does not need to solve any system of nonlinear algebraic equations. An upper bound for the linear spline approximation of the stochastic function is provided. Using this upper bound and under the Lipschitz and linear growth conditions, the convergence analysis of the suggested method is studied.Finally, to verify the efficiency of the proposed scheme, some problems in the finance, physics, and biology are investigated, and the obtained results are compared with the stochastic -method. KEYWORDSlinear spline interpolation, multiple stock models, pendulum problem, predator-prey models, successive approximations method, system of stochastic Volterra integral equations MSC CLASSIFICATION 60H10; 60H20; 65C30Math Meth Appl Sci. 2020;43:5212-5233. wileyonlinelibrary.com/journal/mma

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“…It is proved under rather general hypotheses by referring the equation back to the definition of Ito integral. More complete details on Ito integrals and stochastic calculus can be found in a number of texts, including Refs .…”

Section: Solutions Of Sdesmentioning

confidence: 99%

Computational solution of stochastic differential equations

Sauer

2013

WIREs Computational Stats

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Stochastic differential equations (SDEs) provide accessible mathematical models that combine deterministic and probabilistic components of dynamic behavior. This article is an overview of numerical solution methods for SDEs. The solutions are stochastic processes that represent diffusive dynamics, a common modeling assumption in many application areas. We include a description of fundamental numerical methods and the concepts of strong and weak convergence and order for SDE solvers. In addition, we briefly discuss the extension of SDE solvers to coupled systems driven by correlated noise.Under appropriate conditions, ordinary differential equations have a unique solution for each initial condition. SDEs, on the other hand, have solutions that are continuous-time stochastic processes. Methods for the computational solution of SDEs are based on techniques for ordinary differential equations, but adapted to account for stochastic dynamics.Some fundamental concepts from stochastic calculus are needed to describe the numerical methods. A set of random variables X t indexed by real numbers t ≥ 0 is called a continuous-time stochastic process. Each instance, or realization of the stochastic process is a choice from the random variable X t for each t, and is therefore a function of t.Any (deterministic) function f (t) can be trivially considered as a stochastic process, with variance V(f (t)) = 0. An archetypal example that is ubiquitous in models from physics, chemistry, and finance is the Wiener process W t , a continuous-time stochastic process with the following three properties: (1) For each t, the random variable W t is normally distributed with mean 0 and variance t.(2) For each t 1 < t 2 , the normal random variable W t 2 − W t 1 is independent of the random variable W t 1 , and in fact independent of 362

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Introduction to Stochastic Calculus with Applications

Cited by 395 publications

References 0 publications

Computational modeling and experimental validation of odor detection behaviors of classically conditioned parasitic wasp, Microplitis croceipes

Computational modeling and experimental validation of odor detection behaviors of classically conditioned parasitic wasp, Microplitis croceipes

Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations

Computational solution of stochastic differential equations

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