1 / 11

Stochastic Modeling

Presented by: Zhenhuan Sui Nov. 30 th , 2009. Stochastic Modeling. Definitions. Stochastic : having a random variable Stochastic process ( random process) : counterpart to a deterministic process. some uncertainties in its future evolution described by probability distributions.

caldwell-mullins
Download Presentation

Stochastic Modeling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Presented by: Zhenhuan Sui Nov. 30th, 2009 Stochastic Modeling

  2. Definitions • Stochastic: having a random variable • Stochastic process(random process): • counterpart to a deterministic process. • some uncertainties in its future evolution described by probability distributions. • even if the initial condition is known, the process still has many possibilities(some may be more probable) • Mathematical Expression: • For a probability space, a stochastic process with state space X is a collection of X-valued random variables indexed by a set time T • where each Ft is an X-valued random variable. • http://en.wikipedia.org/wiki/Stochastic_process

  3. StochasticModel • Stochastic model: • tool for estimating probability distributions of potential outcomes • allowing for random variation in one or more inputs over time • random variation is from fluctuations gained from historical data • Distributions of potential outcomes are from a large number of simulations • Markov property

  4. Markov Property • Andrey Markov: Russian mathematician • Definition of the property: the conditional probability distribution of future states only depends upon the present state and a fixed number of past states(conditionally independent of past states) • Mathematical Expression: • X(t): state at time t,t > 0; x(s): history of states, time s < t • probability of state y at time t+h, when having the particular state x(t) at time t • probability of y when at all previous times before t. • future state is independent of its past states. • http://en.wikipedia.org/wiki/Markov_process

  5. Simple Examples and Application • Examples: • Population: town vs. one family • Gambler’s ruin problem • Poisson process: the arrival of customers, the number of raindrops falling over an area • Queuing process: McDonald's vs. Wendy’s • Prey-predator model • Applications: • Physics: Brownian motion: random movement of particles in a fluid(liquid or gas) • Monte Carlo Method • Weather Forecasting • Astrophysics • Population Theory • Decision Making

  6. Decision-making Problem In Consulting Useful Formulas: Law of Total Probability http://en.wikipedia.org/wiki/Law_of_total_probability Conditional Probability http://en.wikipedia.org/wiki/Conditional_probability Bayes Theorem http://en.wikipedia.org/wiki/Bayes%27_theorem

  7. Decision-making Problem In Consulting Model: Set of strategies: A ={A1,A2,…,Am} Set of states: S={S1,S2,…,Sn}, and its Probability distribution is P{Sj}=pj Function of decision-making: vij=V(Ai,Sj), which is the gain (or loss) at state Sj taking strategy Ai Set of the consulting results: I={I1,I2,…,Il}, the quality of consulting is P(Ik|Sj)=pkj, cost of consulting: C

  8. Model Continued Max gain before consulting By Law Of Total Probability and Bayes Theorem Max expected gain when the result of consulting is I k Expected gain after consulting YES! NO! http://mcm.sdu.edu.cn/Files/class_file

  9. Example There are A1, A2 and A3 three strategies to produce some certain product. There are two states of demanding, High S1, Low S2. P(S1)=0.6, P(S2)=0.4. Results for the strategies are as below (in dollars): States Results S1 S2 A1 180,000 120,000 100,000 -150,000 -50,000 -10,000 Strategies A2 A3 If conducting survey to the market, promising report: P(I1 )=0.58 Not promising report: P(I2)=0.42 Abilities to conduct the survey: P(I1|S1)=0.7, P(I2|S2)=0.6 Cost of consulting and surveying is 5000 dollars. Should the company go for consulting?

  10. Solution v11=180000, v12=-150000, v21=120000 v22=-50000, v31=100000, v32=-10000 Expected gain of the strategies: E(A1)=0.6×180000+0.4×(－150000)=48000 E(A2)=0.6×120000+0.4×(－50000)=52000 E(A3)=0.6×100000+0.4×(－10000)=56000 q11=P(S1|I1)=0.72, q21=P(S2|I1)=0.28, q12=P(S1|I2)=0.43, q22=P(S2|I2)=0.57 Result is I1, max expected gain is Result is I2, max expected gain is Expected gain after consulting: ER–E(A3)=67202–56000=11202>C=5000YES!!! http://mcm.sdu.edu.cn/Files/class_file

  11. http://baike.baidu.com/view/1456851.html?fromTaglist http://zh.wikipedia.org/wiki/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B http://baike.baidu.com/view/18964.htm http://www.hudong.com/wiki/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B http://en.wikipedia.org/wiki/Markov_process http://zh.wikipedia.org/wiki/%E8%B4%9D%E5%8F%B6%E6%96%AF%E5%AE%9A%E7%90%86 http://en.wikipedia.org/wiki/Law_of_total_probability http://en.wikipedia.org/wiki/Stochastic_modelling_(insurance) http://en.wikipedia.org/wiki/Markov_chain Resources

More Related