1. 水分子不時受撞,跳格子(c.p. 車行)
投骰子 (最穩定)
股票 (價格是不穏定,但報酬過程略穩定)
地震的次數 (不穩定)
Y1, y2, y3-> y2,y3,y4-> y3,y4,y5
Moving windows

2. Definition of Stochastic Process
Definition: A stochastic process is a sequence (family) or continuum of random variables indexed by an ordered set T.
Generally, of course, T records time.
A stochastic process is often denoted {Xt, tT}.
the set of values that the random variables Xt are capable of taking is called the state space of the process, S.

3. Stochastic Process Example Coin-tossing
Define a random variable
It is easy to show that it has the following properties:
& are independent

4. Stochastic Process Example Markov Process
t = day index 0, 1, 2, ...
Xt = 0 high defective rate on tth day
= 1 low defective rate on tth day
two states ===> n = 1 (0, 1)
P00 = P(Xt+1 = 0 | Xt = 0) = 1/
P01 = P(Xt+1 = 1 | Xt = 0) = 3/
P10 = P(Xt+1 = 0 | Xt = 1) = 1/
P11 = P(Xt+1 = 1 | Xt = 1) = 1/
\ P = 30

5. Comment
A stochastic process is used as a generic model for a time-dependent random phenomenon. So, just as a single random variable describes a static random phenomenon, a stochastic process is a collection of random variables Xt, one for each time t in some set J. The process is denoted {Xt : tJ}.
We are interested in modelling <Xt> at each time t, which in general depends on previous values in sequence (i.e., there is a path-dependency).

6. Stationarity
Definition: A stochastic process is said to be stationary if the joint distributions of Xt+1, Xt+2,…,Xt+n and Xk+1, Xk+2,…,Xk+n are the same for all t, k and all n.
Remarks
Stationarity means that statistical properties unaffected by a time shift.
In particular, Xt and Xk have the same distribution
A stringent requirement, difficult to test
[The assumption of stationarity ‘sweats’ the data – allows max. use of available data.]

7. Stochastic Processes
First order to n-order distribution can characterize the stochastic process.
First order:
Second order:
Stationary
Strict stationary
For all n,k and N

8. Stochastic Processes Stationary
Wide-sense Stationary (weak stationary)
If just its mean and autocovariance are invariant with time.

9. Example of Stationary Process

10. Example of Stationary Process

11. Stochastic Process II

12. Stochastic Process
In the mathematics of probability, a stochastic process is a random function. In practical applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a time series in applications) or a region of space (a stochastic process being called a random field).

13. Stochastic Definition
Mathematically, a stochastic process is usually defined as an indexed collection of random variables
fi : W → R,
where i runs over some index set I and W is some probability space on which the random variables are defined.
This definition captures the idea of a random function in the following way. To make a function
f : D → R

14. Probability measure
Probability Measure

15. Stochastic Definition
With domain D and range R into a random function, means simply making the value of the function at each point of D, into a random variable with values in R.
The domain D becomes the index set of the stochastic process, and a particular stochastic process is determined by specifying the joint probability distributions of the various random variables f(x).

16. Brownian Motion

20. Path regularity (i) is continuous a.s.
(ii) is nowhere differentiable a.s.
a.s. almost surely

21. Brownian Motion Model
A Brownian motion process with drift µ and variance σ2 can be written as: X(t)=σB(t) + µt So this process X(t) is normal with mean µt and variance σ2t.

22. Description
Mathematically, Brownian motion is a Wiener process in which the conditional probability distribution of the particle's position change at time t+dt, given that its position at time t is p,
Itis a normal distribution with a mean ofμdt and a variance of σ2dt; the parameter μ is the drift velocity, and the parameter σ2 is the power of the noise.

23. Definition of Brownian motion
Brownian motion is the unique process
with the following properties:
(i) No memory
(ii) Invariance
(iii) Continuity
(iv)

24. Memoryless process
are independent

25. Invariance
The distribution of
depends only on t.

26. Path regularity (i) is continuous a.s.
(ii) is nowhere differentiable a.s.

27. Why Brownian motion? Brownian motion belongs to several families
of well understood stochastic processes:
Markov processes
Martingales
Gaussian processes
Levy processes

28. Markov processes The theory of Markov processes uses
tools from several branches of analysis:
Functional analysis (transition semigroups)
Potential theory (harmonic, Green functions)
Spectral theory (eigenfunction expansion)
PDE’s (heat equation)

29. Martingales Martingales are the only family of processes
for which the theory of stochastic integrals is
fully developed, successful and satisfactory.

30. Gaussian processes is multidimensional normal (Gaussian)
Excellent bounds for tails
Second moment calculations
Extensions to unordered parameter(s)

31. Notations: Ω, μ, σ, t, T, Field,It, Ft , pn ,St

32. Markov property
Informally, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state, depends only upon the current state, and conditionally independent of the past states (the path of the process) given the present state. A process with the Markov property is usually called a Markov process, and may be described as Markovian.

33. Concept: The Markov Property
When the future evolution of the system depends only on its current state – it is not affected by the past – the system has the Markov property.
Definition: Let <Xt>, t  (the natural numbers) be a (discrete time) stochastic process. Then <Xt>, is said to have the Markov property if, t
P[Xt+1 | Xt, Xt-1,Xt-2,…,X0]=P[Xt+1 | Xt].
Definition: Let <Xt>, t  (the real numbers) be a (continuous time) stochastic process. Then <Xt>, is said to have the Markov property if, t, and all sets A
P[XtA| Xs1=x1, Xs2=x2,…,Xs=x]=P[XtA|Xs=x]
Where s1<s2<…<s<t.

34. Wiener process
In the Wiener process, the probability distribution of Xs − Xt is normal with expected value 0 and variance t.

35. 股票Geometric Brownian Motion
Model stock prices as geometric Brownian motion:
( a standard normal RV)

36. Information Filter F(t)
There are two possibilities for the filtration F(t) for a Brownian motion：
F(t) contain only the information obtained by observing the Brownian motion itself up to time t.
F(t) information obtained by observing the Browning motion and one or more other process.

37. 跳躍擴散模型

38. stationary
To call the increments stationary means that the probability distribution of any increment Xs − Xt depends only on the length s − t of the time interval; increments with equally long time intervals are identically distributed.

39. Poisson process
In the Poisson process, the probability distribution of Xt is a Poisson distribution with expected value λt, where λ > 0 is the "intensity" or "rate" of the process.

40. Poisson process
The number of arrivals in each interval of time or region in space is a random variable with a Poisson distribution, and
The number of arrivals in one interval of time or region in space and the number of arrivals in another disjoint (non-overlapping) interval of time or region in space are independent random variables.
Technically, and perhaps more precisely, one should say each set of finite measure is assigned such a Poisson-distributed random variable.

41. Simulating Poisson process

43. Simulating Brownian Motion

46. Simulating Stock Price by Brownian Motion

47. S(t)=S(0)e^ ( (u-1/2^2)t + sqrt(t) )
Bt = (BT-B0)= sqrt(t) , (t=T-0)

48. dlnS=ds/s=

49. S(t)=S(0)e^ ( (u-1/2^2)t + sqrt(t) )