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Introduction to Stochastic Calculus
A beginner-friendly introduction to stochastic calculus, focusing on intuition and calculus-based derivations instead of heavy probability theory formalism.
Unlike regular differential equations (e.g., \(\frac{dx}{dt} = -kx\)) that describe smooth dynamics, SDEs blend deterministic behavior with stochastic noise, fitting phenomena like stock prices or diffusing particles. Stratonovich ensures the diffusion term reflects physical conservation laws, matching experimental data in systems like bacterial motility better than Itô, which alters the drift. \[X_{n+1} = X_n - a \left(\frac{X_n + X_{n+1}}{2}\right) \Delta t + \sigma \Delta W_n\] This implicit scheme leverages the midpoint rule, reducing numerical artifacts in models like chemical kinetics compared to Itô’s explicit steps.
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