MONTE CARLO SIMULATION Presented by Zhenhuan Sui Introduction

MONTE CARLO SIMULATION Presented by: Zhenhuan Sui

Introduction From A Simple Application 1 M S=N/M N 1

Facts • • • It is the computational algorithm that rely on repeated random sampling to compute its result Investigating radiation shielding and the distance that neutrons would likely travel through various materials at Los Alamos Scientific Laboratory in Manhattan Project John von Neumann and Stanislaw Ulam: modeling the experiment on a computer using chance The name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble 1950 s’ hydrogen bomb

General Steps For MCS • • Define a domain of possible inputs Generate inputs randomly from the domain using a certain specified probability distribution Perform a deterministic computation using the inputs Aggregate the results of the individual computations into the final result suited to calculation by a computer statistical simulation method http: //en. wikipedia. org/wiki/Monte_Carlo_method#Monte_Carlo_and_random_numbers

Elements • • • probability density function random number generator sampling rules simulation results uncertainties estimation technology to reduce the variance

Buffon's Needle The uniform probability density function of x between 0 and t /2 is 2/t dx The uniform probability density function of θ between 0 and π/2 is 2/π d θ http: //en. wikipedia. org/wiki/Buffon%27 s_needle http: //upload. wikimedia. org/wikipedia/commo ns/f/f 6/Buffon_needle. gif

Further For π http: //en. wikipedia. org/wiki/Buffon%27 s_needle

Steps For Applications • based on the properties of the systems we want to solve, we construct theoretical models which can describe the properties. • try to get the probability density functions of some properties for the models. • From the probability density functions, we can produce some random samplings and get some simulation results. • analyze the results and do predictions for some properties of the systems.

Applications Physics: • quantum chromodynamics • statistical physics • particle physics Mathematics: • Monte Carlo integration: algorithms for the approximate evaluation of definite integrals, multidimensional ones • numerical optimization Finance and business: • calculate the value of companies • evaluate investments in projects • evaluate financial derivatives • risk management, there would be a lot of variables in the equations • Quasi-Monte Carlo methods in finance

RESOURCES � � � http: //en. wikipedia. org/wiki/Monte_Carlo_method#Finance_and_busines s http: //en. wikipedia. org/wiki/Monte_Carlo_method http: //en. wikipedia. org/wiki/Quasi-Monte_Carlo_methods_in_finance http: //baike. baidu. com/view/7775. html http: //www. virtualphantoms. org/egs 4/temp/mchome. htm http: //www. linuxsir. org/bbs/thread 288992. html