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Stochastic Modeling
Presented by: Zhenhuan Sui
Nov. 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
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
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 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!
http://mcm.sdu.edu.cn/Files/class_file
NO!
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):
Results
S1
180,000
S2
-150,000
A2
120,000
-50,000
A3
100,000
-10,000
A1
Strategies
States
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?
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
Resources
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http://baike.baidu.com/view/1456851.html?fromTaglist
http://zh.wikipedia.org/wiki/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A
8%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%A
E%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