{"categories":["Stochastic Calculus"],"contentHtml":"<p>The FE-610 notes define an integral such as \\(\\int_0^T\\Delta_t\\,dW_t\\) by starting with simple adapted processes. On each partition interval, the position \\(\\Delta_t\\) is chosen from information already available, while the increment comes from Brownian motion.</p>\n<p>Refining the partition gives the Itô integral. The adaptedness condition is the key financial interpretation: a trading strategy can react to the past but cannot see the next Brownian increment. The integral is itself a random variable because every Brownian path produces a different gain.</p>\n<p>Two results organize the construction. The Itô integral is a martingale under suitable integrability, and the Itô isometry relates its second moment to the ordinary time integral of the squared integrand:</p>\n<p>$$\\mathbb E\\left[\\left(\\int_0^T\\Delta_t\\,dW_t\\right)^2\\right]=\\mathbb E\\left[\\int_0^T\\Delta_t^2\\,dt\\right].$$</p>","contentMarkdown":"The FE-610 notes define an integral such as \\(\\int_0^T\\Delta_t\\,dW_t\\) by starting with simple adapted processes. On each partition interval, the position \\(\\Delta_t\\) is chosen from information already available, while the increment comes from Brownian motion.\n\nRefining the partition gives the Itô integral. The adaptedness condition is the key financial interpretation: a trading strategy can react to the past but cannot see the next Brownian increment. The integral is itself a random variable because every Brownian path produces a different gain.\n\nTwo results organize the construction. The Itô integral is a martingale under suitable integrability, and the Itô isometry relates its second moment to the ordinary time integral of the squared integrand:\n\n$$\\mathbb E\\left[\\left(\\int_0^T\\Delta_t\\,dW_t\\right)^2\\right]=\\mathbb E\\left[\\int_0^T\\Delta_t^2\\,dt\\right].$$","dataUrl":"https://sharifhsn.dev/api/posts/stochastic-integrals.json","date":"2024-10-03","datePublished":"2024-10-03","description":"The FE-610 notes define an integral such as \\(\\int_0^T\\Delta_t\\,dW_t\\) by starting with simple adapted processes. On each partition interval, the position \\(\\Delta_t\\) is chosen fr…","site":"https://sharifhsn.dev","slug":"stochastic-integrals","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Stochastic Integrals","Ito Integral"],"title":"Stochastic Integrals","url":"https://sharifhsn.dev/blog/stochastic-integrals/","version":"1","wordCount":114}