{"categories":["Stochastic Calculus"],"contentHtml":"<p>The FE-610 notes introduce Poisson processes as counting processes with independent increments and a constant arrival intensity. For a rate \\(\\lambda\\), the number of arrivals by time \\(t\\) has distribution</p>\n<p>$$\\mathbb P(N_t=k)=e^{-\\lambda t}\\frac{(\\lambda t)^k}{k!}.$$</p>\n<p>The waiting time to the first arrival is exponential, and independent waiting times produce the full process. Adding random jump sizes gives a compound Poisson process; combining it with Brownian motion gives a jump-diffusion model.</p>\n<p>The notes return to quadratic variation: jumps contribute their squared sizes, while the continuous part contributes its usual diffusion variation. That decomposition is the starting point for an Itô formula with jumps.</p>","contentMarkdown":"The FE-610 notes introduce Poisson processes as counting processes with independent increments and a constant arrival intensity. For a rate \\(\\lambda\\), the number of arrivals by time \\(t\\) has distribution\n\n$$\\mathbb P(N_t=k)=e^{-\\lambda t}\\frac{(\\lambda t)^k}{k!}.$$\n\nThe waiting time to the first arrival is exponential, and independent waiting times produce the full process. Adding random jump sizes gives a compound Poisson process; combining it with Brownian motion gives a jump-diffusion model.\n\nThe notes return to quadratic variation: jumps contribute their squared sizes, while the continuous part contributes its usual diffusion variation. That decomposition is the starting point for an Itô formula with jumps.","dataUrl":"https://sharifhsn.dev/api/posts/poisson-processes.json","date":"2024-11-28","datePublished":"2024-11-28","description":"The FE-610 notes introduce Poisson processes as counting processes with independent increments and a constant arrival intensity. For a rate \\(\\lambda\\), the number of arrivals by t…","site":"https://sharifhsn.dev","slug":"poisson-processes","source":"FE-610 | Stochastic Calculus","sourceUrl":null,"tags":["Stochastic Calculus","Poisson Processes","Jump Diffusion"],"title":"Poisson Processes","url":"https://sharifhsn.dev/blog/poisson-processes/","version":"1","wordCount":101}