stochastic differential equation

随机微分方程

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Other names / variants: SDE

Translation evidence: Literature usage

Meaning in context

In the Brownian Itô setting, an equation $dX_t=b(X_t)dt+\sigma(X_t)dB_t$ means an adapted continuous process satisfies $X_t=X_0+\int_0^t b(X_s)ds+\int_0^t\sigma(X_s)dB_s$, with the integrals well defined.

在布朗运动的Itô框架下，要求适应连续过程满足相应漂移积分与随机积分之和的积分恒等式，且各积分有定义的演化方程。

Example

The Itô solution of $dX_t=cX_t\,dB_t$, $X_0=1$, is $\exp(cB_t-c^2t/2)$.

Usage and distinctions

Itô and Stratonovich conventions generally produce different drift terms; strong and weak solutions specify different relationships to the prescribed noise.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gregory F. Lawler — Stochastic Calculus: An Introduction with Applications§3.5 Diffusions, equation (3.21) and integral solution construction · Checked 2026-10-14

Translation sources

1. 北京大学数学科学学院 — 随机过程课程内容提要，条目4：多维扩散过程的描述方式，随机微分方程 · Checked 2026-10-14

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