Models Stochastic Models

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Models - Stochastic Models

• Shane Whelan
• L527

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Review of Last 3 Lectures Chapter 1

• Introduction to (Actuarial) Modelling
• 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.
• Case Study Lessons from econometric modelling in
  UK over last four decades.

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Next 3 -5 Lectures Chapter 2

• Basic terminology
• Stochastic process sample path m-increment,
  stationary increment.
• Foundational concepts
• Stationary process weak stationarity Markov
  property filtrations
• 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 2

• 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.
• Recall the set of values that the random
  variables Xt are capable of taking is called the
  state space of the process, S.

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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 ltXtgt at each time
  t, which in general depends on previous values in
  sequence (i.e., there is a path-dependency).

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Examples

• Example 1 Discrete White Noise
• A sequence of independent identically distributed
  random variables, Z0, Z1,Z2,is known as white
  noise.
• Important sub-classifications include zero-mean
  white noise, i.e., EZi0 symmetric white
  noise where the distribution of the random
  variable is symmetric.
• Example 2 General random walk
• Let Z0, Z1,Z2,be white noise and define
• Xn?nZt, with X0x0 (the initial value). Then
  ltXngt is a random walk.
• When Zt can only take values ?1 then process
  known as a simple random walk. Generally we set
  X00.

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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.

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2-Dimensional Distribution
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Increments

• Consider Xtm Xt . This is known as an
  m-increment of the process.
• A 1-increment is simply known as an 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.)
• Hence stochastic processes often defined by their
  initial value and how each increment is
  generated. See how random walk was defined.
• 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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Sample Path

• The sample path of process is a joint realisation
  of the random variables Xt, for all t?T.
• Remarks
• Sample path is a function from T to state space
• Each sample path has an associated probability.
• Example 3 Consider model of salary
• Progression, where salary at future
• time t is modelled as
• Equally likely sample paths from
• this model are graphed alongside.

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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.
• 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.

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Five Quick Questions

• Is white noise stationary?
• Is a random walk stationary?
• Is the Salary(t) model a stationary model?
• 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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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 is said to possess 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 3).
• When state space is discrete but time is
  continuous then known as Markov jump process (see
  Chapter 4).

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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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More Examples of Stochastic Processes

• Example 4 An MA(p) process
• Let Z1, Z2, Z3, be white noise and let ?i be a
  real number for each i. Then Xn is a moving
  average process of order p iff (iff if and only
  if)
• Remarks
• Note process is stationary but not independent
  and identically distributed (iid).
• Moving average processes are stationary but not,
  in general, Markovian.

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More Examples of Stochastic Processes

• 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.,
  has Markov property 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 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 ltXtgt 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 company
• the 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., ?(u) such that ?(u)Puct-Xtlt0, for some
  tgt0.

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Brownian Motion (or Wiener Process)

• Definition Brownian motion, ltBtgt, 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))

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Remarks on Brownian Motion

• Guassian Normal distribution.
• ? is known as the drift.
• Standard Brownian motion is Brownian motion when
  B00, ?0, and ?21.
• Sample paths have no jumps.
• This is the continuous time analogue of a random
  walknot obvious but true
• By CLT,ltBtgt 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 Deep result.

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Question 1

• Let ltXtgt 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 ltXtgt has the Markov property

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Question 2

• Let ltXtgt, t?Z be a white noise process with
  EXt0 and EXt2lt?.
• Prove that the correlation between Xt-1 and
• (Xt-Xt-1) is -(2)-½.

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Filtrations

• Let ltXtgt be a stochastic process
• Associated with each stochastic process we have a
  sample space ? each point (or outcome) in ?
  corresponds to a sample path (i.e., a single
  realisation of the process).
• Also associated with each stochastic process we
  have a set of events F a collection of subsets
  of ? (forming a sigma algebra) each of which have
  an associated probability.
• In our case, for each time t, we define Ft (Ft
  ?F), which is the subset of events known at time
  t,
• i.e., A?Ft iff Af(X1,,Xt). So, as each X1,Xt
  take known values at time t, A also takes a known
  value by time t.
• The family of (nested) sets, (Ft), t?0 is known
  as the natural filtration associated with the
  stochastic process ltXtgt.
• It describes the information gained from
  observing the process up to time t.

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Markov Property Defined Again

• Definition The stochastic process ltXtgt is said
  to have the Markov property iff
• PXtx FsPXtx Xs
• for all ts
• where (Ft)t0 is the natural filtration
  associated with ltXtgt.

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Models based on Markov Chains

• Model 1 The No Claims Discount (NCD) system is
  where the motor insurance premium depends on the
  drivers claims record. It is a simple example of
  a Markov chain.
• Instance Three states 0 discount 25
  discount and 50 discount. A claim-free year
  results in a transition to a higher discount (or
  remain at the highest). A claim moves to the next
  lower discount level (or remain at 0).

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Model 2

• Consider the 4-state NCD model given by
• State 0 0 Discount
• State 1 25 Discount
• State 2 40 Discount
• State 3 60 Discount
• Here the transition rules are move up one
  discount level (or stay at max) if no claim in
  the previous year. Move down one-level if claim
  in previous year but not the year before move
  down 2 levels if claim in two immediately
  preceding years.

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Model 2

• This is not a Markov chain
• PXn10Xn2, Xn-11? PXn10Xn2, Xn-13
• But
• We can simply construct a Markov chain from
  Model 2. Consider the 5-state model with states
  0,1,3 as before but define
• State 2 40 discount and no claim in previous
  year.
• State 2- 40 discount and claim in the previous
  year.
• This is now a 5 state Markov chain.
• Check!

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Model 3 Another Example of Making A Process into
a Markov Chain

• Let us suppose that whether it rains today
  depends on the weather in the last two days.
  Specifically
• If it rained in each of last two days then
  probability that it will rain today is 0.5
• If it rained yesterday but not the day before, it
  will rain today with probability 0.4.
• If it did not rain yesterday but rained the day
  before that, the probability of rain today is
  0.3.
• If it did not rain in the last two days then the
  probability of rain today is 0.2
• The process above can be transformed into a 4
  state Markov chain model.

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Models - Stochastic Models

• Completes Chapter 2
• Shane Whelan
• L527