{"categories":["Stochastic Calculus"],"contentHtml":"<p>The SDE section writes a state process as a drift plus a diffusion term,</p>\n<p>$$dX_t=b(t,X_t)\\,dt+\\sigma(t,X_t)\\,dW_t.$$</p>\n<p>The coefficients describe deterministic motion and random shocks. Unlike an ordinary differential equation, an SDE is interpreted through an integral equation and a chosen filtration. The notes compare continuous diffusions with jump processes and use Itô's formula to transform solutions.</p>\n<p>For pricing, this notation is valuable because a model can be specified by its local characteristics even when there is no closed-form path. Existence, integrability, and the chosen measure determine whether the process is usable as a financial model.</p>","contentMarkdown":"The SDE section writes a state process as a drift plus a diffusion term,\n\n$$dX_t=b(t,X_t)\\,dt+\\sigma(t,X_t)\\,dW_t.$$\n\nThe coefficients describe deterministic motion and random shocks. Unlike an ordinary differential equation, an SDE is interpreted through an integral equation and a chosen filtration. The notes compare continuous diffusions with jump processes and use Itô's formula to transform solutions.\n\nFor pricing, this notation is valuable because a model can be specified by its local characteristics even when there is no closed-form path. Existence, integrability, and the chosen measure determine whether the process is usable as a financial model.","dataUrl":"https://sharifhsn.dev/api/posts/stochastic-differential-equations.json","date":"2024-11-21","datePublished":"2024-11-21","description":"The SDE section writes a state process as a drift plus a diffusion term,","site":"https://sharifhsn.dev","slug":"stochastic-differential-equations","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","SDEs","Diffusions"],"title":"Stochastic Differential Equations","url":"https://sharifhsn.dev/blog/stochastic-differential-equations/","version":"1","wordCount":94}