Stochastic Modeling

Presented by: Zhenhuan SuiNov. 30th, 2009

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

Stochastic Model

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

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

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

Decision-making Problem In Consulting

http://en.wikipedia.org/wiki/Law_of_total_probability

Law of Total Probability

Conditional Probability

Bayes Theorem

http://en.wikipedia.org/wiki/Conditional_probability

http://en.wikipedia.org/wiki/Bayes%27_theorem

Useful Formulas:

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

Model ContinuedMax 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

ExampleThere 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

S1 S2

A1

A2

A3

180,000

120,000

100,000

-150,000

-50,000

-10,000

Results

Strategies

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)=0.6Cost of consulting and surveying is 5000 dollars. Should the company go for consulting?

Solutionv11=180000, v12=-150000, v21=120000v22=-50000, v31=100000, v32=-10000

Expected gain of the strategies:E(A1)=0.6×180000+0.4×(－ 150000)=48000E(A2)=0.6×120000+0.4×(－ 50000)=52000E(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

Resources

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%B

F%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