{"categories":["Stochastic Calculus"],"contentHtml":"<h4>Lecture Notes</h4><p>Came late because my brother couldn’t get up 🙁</p><h5>Stochastic Differential Equations</h5><p>A SDE is of the form</p><p><math><mi>𝑑</mi><mi>𝑋</mi><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑢</mi><mo form=\"postfix\" stretchy=\"false\">)</mo><mo>=</mo><mi>𝛽</mi><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑢</mi><mo>,</mo><mi>𝑋</mi><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑢</mi><mo form=\"postfix\" stretchy=\"false\">)</mo><mo form=\"postfix\" stretchy=\"false\">)</mo><mi>𝑑</mi><mi>𝑢</mi><mo>+</mo><mi>𝛾</mi><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑢</mi><mo>,</mo><mi>𝑋</mi><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑢</mi><mo form=\"postfix\" stretchy=\"false\">)</mo><mo form=\"postfix\" stretchy=\"false\">)</mo><mi>𝑑</mi><mi>𝑊</mi><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑢</mi><mo stretchy=\"false\">)</mo></math></p><p>where <math><mi>𝛽</mi></math> is the drift and <math><mi>𝛾</mi></math> is the diffusion.</p><p>Diffeqs always need a boundary condition, given by</p><p><math><mi>𝑋</mi><mo form=\"prefix\" stretchy=\"false\">(</mo><mi>𝑡</mi><mo form=\"postfix\" stretchy=\"false\">)</mo><mo>=</mo><mi>𝑥</mi><mo>,</mo><mi>𝑡</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>𝑥</mi><mo>∈</mo><mi>ℝ</mi></math></p><p>In order to solve, we want to find</p><p>Most of the time, these don’t have closed-form solutions, so you have to solve numerically with Euler’s method.</p><p>We can express SDE as PDE using Feynman-Kac, if our solution</p><p>This is the end of the book of the final.</p><p>Over the next two weeks, we will look at what we can do based on what we’ve been learning.</p><p>Next week, we will introduce the idea of non-continuous stock processes i.e.&nbsp;jumps.</p>","contentSourceFormat":"typst","contentTypst":"=== Lecture Notes\n<lecture-notes>\nCame late because my brother couldn't get up 🙁\n\n==== Stochastic Differential Equations\n<stochastic-differential-equations>\nA SDE is of the form\n\n$d X\\(u\\)= beta\\(u\\,X\\(u\\)\\)d u + gamma\\(u\\,X\\(u\\)\\)d W\\(u\\)$\n\nwhere $beta$ is the drift and $gamma$ is the diffusion.\n\nDiffeqs always need a boundary condition, given by\n\n$X\\(t\\)= x\\,t gt.eq 0\\,x in bb(R)$\n\nIn order to solve, we want to find\n\nMost of the time, these don't have closed-form solutions, so you have to solve numerically with Euler's method.\n\nWe can express SDE as PDE using Feynman-Kac, if our solution\n\nThis is the end of the book of the final.\n\nOver the next two weeks, we will look at what we can do based on what we've been learning.\n\nNext week, we will introduce the idea of non-continuous stock processes i.e.~jumps.","dataUrl":"https://sharifhsn.dev/api/posts/stochastic-differential-equations.json","date":"2024-11-21","datePublished":"2024-11-21","description":"Came late because my brother couldn’t get up 🙁 where 𝛽 is the drift and 𝛾 is the diffusion. Diffeqs always need a boundary condition, given by","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":163}