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Stochastic Modeling: Definitions, Examples, and Decision-Making

Explore the fundamentals of stochastic modeling with examples like population dynamics and applications in physics and decision-making problems. Learn about Markov processes, simulations, and Bayesian techniques. Delve into strategies, states, gain calculations, and consulting costs to make informed decisions.

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Stochastic Modeling: Definitions, Examples, and Decision-Making

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  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

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