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Title: Introduction to Models Stochastic Models Chapter 1 Continued Lectures 5


1
Introduction to Models- Stochastic Models
Chapter 1 ContinuedLectures 5 6
  • Shane Whelan
  • L527

2
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.

3
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.
  • When state space is discrete but time is
    continuous then known as Markov jump process.

4
More Special Processes MA(p)
  • Let Z1, Z2, Z3, be white noise and let ?i be
    real numbers. Then ltXngt is a moving average
    process of order p iff
  • Notes
  • An MA(p) process is stationary but not iid.
  • Moving average processes are stationary but not,
    in general, Markovian.

5
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??

6
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 weakly stationary.
  • Think of the Poisson Process as the stochastic
    generalisation of the deterministic natural
    numbers stochastic counting.
  • It is a central process in insurance and finance
    modelling due to role as a natural stochastic
    counting process, e.g., number of claims.
  • Uppsala Thesis of 1903 of Filip Lundberg.

7
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.

8
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
    building on Lundbergs Uppsala Thesisthe 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.

9
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 stationary increments
  • And either
  • Bt-Bs is distributed N(?(t-s), ?2(t-s))
  • Or
  • ltBtgt has continuous sample paths.

10
Remarks on Brownian Motion
  • Standard Brownian motion is when B00, ?0, and
    ?21.
  • Simpler definition Brownian motion is a
    continuous process with independent Guassian
    increments.
  • Guassian Normal
  • ? is known as the drift.
  • Sample paths have no jumps deep result.
  • This is the continuous time analogue of a random
    walk.
  • By Central Limit Theorem, ltBtgt is the limiting
    continuous stochastic process for a wide class of
    discrete time processes important result.

11
Question
  • Let ltXtgt be a simple random walk with prob. of
    an upward move given by p.
  • Calculate P(X22X00)
  • Calculate P(X20, X42X00)
  • Is the random walk stationary?
  • What is the joint distribution of X2,X4, given
    X00?
  • Prove that ltXtgt has the Markov property

12
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

13
Review of Chapter 1
  • Basic terminology
  • Stochastic process sample path m-increment,
    stationary increment.
  • Foundational concepts
  • Stationary process weak stationarity Markov
    property
  • Some elementary examples
  • White noise random walk moving average (MA).
  • Some less-elementary examples
  • Poisson process compound Poisson process
    Brownian motion (or Wiener Process).

14
Completes Chapter 1
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