{"categories":["Stochastic Calculus"],"contentHtml":"<p>Itô's formula is the stochastic analogue of the multivariable chain rule. If</p>\n<p>$$dX_t=a(t,X_t)\\,dt+b(t,X_t)\\,dW_t,$$</p>\n<p>then a smooth function \\(f(t,X_t)\\) acquires a second-order term because \\((dW_t)^2=dt\\):</p>\n<p>$$df=f_t\\,dt+f_x\\,dX_t+\\tfrac12 f_{xx}(dX_t)^2.$$</p>\n<p>The FE-610 notes apply the formula to Itô processes, generalized geometric Brownian motion, and short-rate examples such as Vasicek and Cox–Ingersoll–Ross. The extra curvature term is the practical difference from ordinary calculus; dropping it produces the wrong drift for a transformed process.</p>","contentMarkdown":"Itô's formula is the stochastic analogue of the multivariable chain rule. If\n\n$$dX_t=a(t,X_t)\\,dt+b(t,X_t)\\,dW_t,$$\n\nthen a smooth function \\(f(t,X_t)\\) acquires a second-order term because \\((dW_t)^2=dt\\):\n\n$$df=f_t\\,dt+f_x\\,dX_t+\\tfrac12 f_{xx}(dX_t)^2.$$\n\nThe FE-610 notes apply the formula to Itô processes, generalized geometric Brownian motion, and short-rate examples such as Vasicek and Cox–Ingersoll–Ross. The extra curvature term is the practical difference from ordinary calculus; dropping it produces the wrong drift for a transformed process.","dataUrl":"https://sharifhsn.dev/api/posts/ito-calculus.json","date":"2024-10-10","datePublished":"2024-10-10","description":"Itô's formula is the stochastic analogue of the multivariable chain rule. If","site":"https://sharifhsn.dev","slug":"ito-calculus","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Ito Lemma","Geometric Brownian Motion"],"title":"Itô Calculus","url":"https://sharifhsn.dev/blog/ito-calculus/","version":"1","wordCount":68}