偏微分方程和非線性編程在金融領域是怎麼應用的？

11-26

想了解一下在金工領域這兩方面的知識都是怎麼應用的呢。美本數學系選課，這兩門課下學習2選1，不知道哪個會比較有用。
課程介紹：
偏微分方程：Introduces partial differential equations, emphasizing the wave, diffusion and potential (Laplace) equations. Focuses on understanding the physical meaning and mathematical properties of solutions of partial differential equations. Includes fundamental solutions and transform methods for problems on the line, as well as separation of variables using orthogonal series for problems in regions with boundary. Covers convergence of Fourier series in detail.

非線性編程：Iterative and analytical solutions of constrained and unconstrained problems of optimization; gradient and conjugate gradient solution methods; Newton"s method, Lagrange multipliers, duality and the Kuhn-Tucker theorem; and quadratic, convex, and geometric programming.

謝過各位大神們了

Won"t be using knowledge from either of the courses much in the industry. At least in my risk analytics group, we mostly use proprietary/internal models to analyze exposure. I graduated from Duke math, but feel that unless you aim for research, math beyond calculus can only provide you a sense of analytical logic when it comes to real projects in the industry. From the course description you posted, PDE focuses upon theories and pencil-and-paper application of tricks. NLP teaches you essentials tools in optimization problems but I bet they focus on theory more than industrial application.
In all, if you are really interested in math or financial engineering, take both. But you should expect the least that classroom knowledge can be directly applied to your non-research career. If you are really interested in Financial Mathematics, take stochastic calculus and read books by Shreve on the topic. Also, programming is an essential part of daily jobs thus I encourage you to know programming especially Python and C++. Linux is a plus too.
Hope it helps.

我對PDE在期權定價上的應用比較熟, 說下我的淺見, 拋磚引玉.
要給一個期權定價, 比較常見的方法有closed-formula, monte-carlo simulation, PDE, lattice approach等等. 使用PDE對期權進行定價, 我覺得有幾個好處.
1. 數學推導相對比較簡單.
以普通歐式期權為例, 基於no-arbitrage原理,可以很快推導出歐式期權的PDE．而相對應的，要推倒BS公式則複雜的多．　
2. 程序實現比較有優勢.
PDE數值解法, 比如Finite Difference Method, 的程序比較容易實現. 同時在 consistency, stability, convergence等方面都可以進行直接的評估. 由此可以在保證stability的前提下, 儘可能的提高速度.

綜合而言, 我個人還是比較喜歡PDE的方法來對期權進行定價.

UIUC的MATH442和MATH484是吧，沒上過484不是很清楚，下學期的話442Nikolaos Tzirakis是個很好的教授，但是提醒一句442如果物理基礎非常薄弱的話慎選，否則很痛苦。

無所謂的，上學時按照興趣來…其實工作了可以再學，上手也不難

pde樓上有兩位講過了，針對題主的選擇，我覺得pde更值得學，不過這兩門都不簡單。
nlp那個偏理論，對於應用，很多演算法都已有現成的工具（cpp，r等），專門學優化理論我個人覺得沒必要

feynman kac 定理把偏微分和隨機微分聯繫了起來。某個非線性的contingent claim就可以用數值偏微分的方法得到現值。

第二個應該叫非線性規劃吧。應用範圍更廣。p quant類各種統計學習問題都可以歸結為函數擬合問題。函數擬合問題就會遇到非線性規劃。或者在進行各類統計推斷時，最優化也會用到。
q quant類在進行calibration時會經常遇到非線性規劃。

只能說這兩個課都很重要