Abstract
A brief introduction to stochastic calculus is presented for stochastic differential equations (SDEs). The essence of technical problems in SDEs (i.e., selection of stochastic integrals) is intuitively explained from the viewpoints of \(\delta \)-functions. We first review the Itô integral, the Itô-type SDEs, and the corresponding stochastic calculus. Various SDEs are also reviewed from the viewpoint of the ordinary chain rule and the Wong-Zakai theory.
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Notes
- 1.
More precisely, Eq. (5.16) should be understood as the integral form \(\int _0^t (d\hat{W})^{2} f(\hat{v}) = \int _0^t dt f(\hat{v})\) and \(\int _0^t (d\hat{W})^{n} f(\hat{v}) = 0\) \((n\ge 3)\) in the sense of mean-square convergence. Here, \(\hat{v}\) is the solution of Eq. (5.20). See Ref. [1] for more detail.
- 2.
For \(Z'\equiv dt d\hat{W}\), we obtain \(\langle (Z'-\langle Z'\rangle )^2\rangle =0 + o(dt)\).
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Kanazawa, K. (2017). Stochastic Calculus for the Single-Trajectory Analysis. In: Statistical Mechanics for Athermal Fluctuation. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-6332-9_5
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DOI: https://doi.org/10.1007/978-981-10-6332-9_5
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