{"categories":["Stochastic Calculus"],"contentHtml":"<p>The FE-610 notes describe a martingale as an adapted process whose conditional expected future value equals its current value:</p>\n<p>$$\\mathbb E[M_t\\mid\\mathcal F_s]=M_s,\\qquad s\\leq t.$$</p>\n<p>A symmetric random walk provides the discrete model. Encode heads as \\(+1\\) and tails as \\(-1\\); independent increments give zero conditional drift. A Markov process goes further by saying the current state contains enough information about the future, while a martingale says the best conditional forecast is the present value.</p>\n<p>The notes also introduce first-order and quadratic variation. A smooth function's squared increments vanish in the limit, but a random walk's accumulated squared increments do not. That difference is why Brownian motion needs a new calculus.</p>","contentMarkdown":"The FE-610 notes describe a martingale as an adapted process whose conditional expected future value equals its current value:\n\n$$\\mathbb E[M_t\\mid\\mathcal F_s]=M_s,\\qquad s\\leq t.$$\n\nA symmetric random walk provides the discrete model. Encode heads as \\(+1\\) and tails as \\(-1\\); independent increments give zero conditional drift. A Markov process goes further by saying the current state contains enough information about the future, while a martingale says the best conditional forecast is the present value.\n\nThe notes also introduce first-order and quadratic variation. A smooth function's squared increments vanish in the limit, but a random walk's accumulated squared increments do not. That difference is why Brownian motion needs a new calculus.","dataUrl":"https://sharifhsn.dev/api/posts/martingales-and-random-walks.json","date":"2024-09-19","datePublished":"2024-09-19","description":"The FE-610 notes describe a martingale as an adapted process whose conditional expected future value equals its current value:","site":"https://sharifhsn.dev","slug":"martingales-and-random-walks","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Martingales","Random Walks"],"title":"Martingales and Random Walks","url":"https://sharifhsn.dev/blog/martingales-and-random-walks/","version":"1","wordCount":110}