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

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.

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

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) = 1/4 0 0 P01 = P(Xt+1 = 1 | Xt = 0) = 3/4 0 1 P10 = P(Xt+1 = 0 | Xt = 1) = 1/2 1 0 P11 = P(Xt+1 = 1 | Xt = 1) = 1/2 1 1 \ P = 30

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).

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.]

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

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

Example of Stationary Process

Example of Stationary Process

Stochastic Process II

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).

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

Probability measure Probability Measure

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).

Brownian Motion

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

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.

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.

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

Memoryless process are independent

Invariance The distribution of depends only on t.

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

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

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)

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

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

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

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.

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.

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

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

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.

跳躍擴散模型

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.

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.

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.

Simulating Poisson process

Simulating Brownian Motion

Simulating Stock Price by Brownian Motion

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

dlnS=ds/s=

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