Stochastic Processes

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Stochastic Processes

• Shane Whelan
• L551

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Review of Last 3 Lectures Chapter 0

• Real ModellingNot Mathematics
• Classifying Models
• Components of Model
• Building a Model 10 Helpful Steps
• Advantages of Modelling
• Drawbacks of Modelling (that must be guarded
  against)
• Key points to assess the suitability of a model.
• Some further considerations in modelling
  example/exercise on correlations.
• Case Study Lessons from econometric modelling in
  UK over last 4 decades.

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Next 2½ Lectures Part I of Chapter 1

• Basic terminology
• Stochastic process sample path m-increment,
  stationary increment.
• Foundational concepts
• Stationary process weak stationarity Markov
  property martingale discrete time stopping
  time
• Some elementary examples
• White noise random walk moving average (MA).
• Some important practical examples
• Poisson process compound Poisson process
  Brownian motion (or Wiener Process).

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Chapter 1

• Basic terminology Foundational concepts of
  Stochastic Processes

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Definition of Stochastic Process

• Definition A stochastic process is a sequence 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.
  I prefer ltXtgt, t?T, so as to avoid confusion with
  the state space.
• Examples
• Quick Question with Surprising Answer Let ltXtgt,
  t?Z such that ltXtgt is iid with EXt0 and
  EXt2lt?. Prove that the correlation between Xt
  and Xt-1 is -(2)-½.

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Defining a Given Stochastic Process

• Defining (or wholly understanding) ltXtgt, for all
  t?T amounts to defining the joint distribution
  Xt1, Xt2,,Xtn for all t and all n.
• Not easy to do and very cumbersome.
• But generally use indirect means, e.g., by
  defining the transition process.
• Sample path of process is a joint realisation of
  the random variables Xt, for all t?T.
• Sample path is a function from T to state space
• Each sample path has an associated probability.

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2-Dimensional Distribution
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Stationarity

• Definition A stochastic process is said to be
  stationary if the joint distributions of Xt1,
  Xt2,,Xtn and Xk1, Xk2,,Xkn are the same
  for all t, k and all n.
• Hence 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.
• Is the stochastic process of life stationary?
• Try to think of a stationary process which is not
  iid.

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Weak Stationarity

• Definition A stochastic process is said to be
  weakly stationary if
• EXtEXk for all t and k.
• CovXt , Xtm is a function only of m, for all
  t and m.
• Remarks
• Strong stationarity implies weak stationarity.
• Concept used extensively in time series analysis
• Remark Weak stationarity is not a foundational
  concept it says little enough about the
  underlying distribution and relationship
  structure. It is more practical, though.

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Increments

• Consider Xtm Xt . This is known as an
  m-increment of the process.
• Often defining how the process evolves through
  time is easier to get a handle onand a more
  natural description of the process (e.g.,
  evolution, many games, etc.)
• A process is said to have independent increments
  if Xtm Xt is independent of the past of the
  process for all t and m.
• A process is said to have stationary increments
  if the increments have the same distribution.

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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 ltXtgt, t? ? (the natural numbers)
  be a (discrete time) stochastic process. Then
  ltXtgt, is said to have the Markov property if, ?t
• PXt1 Xt, Xt-1,Xt-2,,X0PXt1 Xt.
• Definition Let ltXtgt, t? ? (the real numbers) be
  a (continuous time) stochastic process. Then
  ltXtgt, is said to have the Markov property if, ?t,
  and all sets A
• PXt?A Xs1x1, Xs2x2,,XsxPXt?AXsx
• Where s1lts2ltltsltt.

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Markov Processes

• Definition A stochastic process that has the
  Markov property is known as a Markov process.
• If state space and time is discrete then process
  known as Markov chain (see Chapter 2).
• When state space is discrete but time is
  continuous then known as Markov jump process (see
  Chapter 3).

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To Prove

• Lemma 1.1 A process with independent increments
  has the Markov Property.
• Proof On Board
• Lemma 1.2 Our definition of the Markov property
  (discrete time) is equivalent to
• PXt1 Xs, Xs-1,Xs-2,,X0PXt1 Xs, where
  s?t.
• Proof On Board

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Examples of Stochastic Processes

• Discrete White Noise
• A sequence of independent identically distributed
  random variables, Z0, Z1,Z2,
• Important sub-classifications include zero-mean
  white noise, i.e., EZi0 symmetric white
  noise etc. have the obvious meaning.
• General random walk
• Let Z0, Z1,Z2,be white noise and define
• Xn?nZt, with X0x0. Then ltXngt is a random walk.
• It is a discrete time Markov process that is not
  weakly stationary.
• When Zt can only take values ?1 then process
  known as a simple random walk. Generally we set
  X00.

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More Special Processes MA(p)

• Let Z1,Z2,Z3, be white noise and let ?i be real
  numbers. Then Xn is a moving average process of
  order p iff
• Note process is stationary but not iid.
• Moving average processes are stationary but not,
  in general, Markovian.

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Poisson Process

• Definition A Poisson process with rate ? is a
  continuous-time process Nt, t?0 such that
• N00
• ltNtgt has independent increments
• ltNtgt has Poisson distributed increments, i.e.,
• where n??

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Remarks on Poisson Process

• Poisson Process is a Markov jump process, i.e.,
  Markovian with a discrete state space in
  continuous time.
• It is not even weakly stationary.
• Think of it as the stochastic generalisation of
  the deterministic natural numbers stochastic
  counting.
• A central process in insurance and finance due to
  role as the the natural stochastic counting
  process, e.g., number of claims.

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Compound Poisson Process

• Definition Let ltNtgt be a Poisson process and
  let Z1,Z2,Z3,be white noise. Then Xt is said to
  be a compound Poisson process where
• With convention when Nt0 then Xt0.

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Remarks on Compound Poisson Process

• We are stochastically counting incidences of an
  event with a stochastic payoff.
• Markov property holds.
• Important as model for cumulative claims on
  insurance companythe Cramér-Lundberg model
  after Lundbergs Uppsala thesis of 1903the
  basis of classical risk theory
• Key problem in classical risk theory is
  estimating the probability of ruin,
• i.e., ? s.t. ?(u)Puct-Xtlt0, for some tgt0.

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Brownian Motion (or Wiener Process)

• Definition Brownian motion, Bt, t?0, is a
  stochastic process with state space ? (the real
  line) such that
• B00
• Bt has independent increments
• Bt-Bs is distributed N(?(t-s), ?2(t-s))
• Bt has continuous sample paths.

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Remarks on Brownian Motion

• Guassian Normal
• ? is known as the drift.
• Standard Brownian motion is when B00, ?0, and
  ?21.
• Sample paths have no jumps.
• This is the continuous time analogue of a random
  walk (as well see in Semester 2).
• By CLT, Bt is the limiting continuous stochastic
  process for a wide class of discrete time
  processes.
• Simpler definition Brownian motion is a
  continuous process with independent Guassian
  increments.

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Question 1-A

• Let Xt be a simple random walk with prob. of an
  upward move given by p. Calculate
• P(X22,X53X00)
• P(X20, X42X00)
• Is the random walk stationary?
• What is the joint distribution of X2,X4, given
  X00
• Prove that Xt has the Markov property

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Martingales Stopping Times

• (Part II of Chapter 1)

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Martingales in Discrete Time

• A discrete time stochastic process Xt , t?0, is
  said to be a martingale if
• EXtlt? for all t.
• EXnX0,,Xm-1, XmXm for all mltn
• Explanation the current value Xm is the optimal
  estimator of all future values. All known
  information by time m on the future of the
  process is factored into Xm
• A generalisation of the notion of a fair game.
• Useful concept in probability theory as many
  important limit theorems can be proved for
  martingales.
• The building block of much of capital market
  theory.

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Conditional Expectation Recap
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Conditional Expectation Key Properties

• Property of iterative expectations,
• EEXYEX
• which generalises to EEXY1,Y2,,YnEX.
• If X is a constant then EX X
• If C is a constant then ECXCEX
• If X is independent of Y then EX?YEX

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Simple Property of Martingales

• Lemma 1.3 If ltXtgt is a martingale then
• EXtEX0
• Proof Immediate.

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Two Lemmas

• Lemma 1.4 For every function, f(.),
• Lemma 1.5 is the optimal
  estimator of X based on Y1,..Yn in the sense that
  for every function f(.),