Introduction to Models Stochastic Models

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Introduction to Models - Stochastic Models

• Dr Shane Whelan, FFA
• L527

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Review of Chapter 1

• Real ModellingNot Mathematics
• What a model is and what is its objective
• Classifying Models (deterministic, stochastic)
• Components of Model (structural part, parameters)
• Computers Modelling (the revolution)
• Lessons from history of modelling
• Orders of Complexity in Modelling

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From now on

• To model stochastically, we need to appreciate
  the different forms of stochastic models
• This is the key content of this short course to
  overview stochastic processes that are widely
  used in modelling by actuaries
• Introduce key concepts
• Give some straightforward examples

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Chapter 2

• Foundational Concepts in 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.
• Notes
• Generally, of course, T records time.
• A stochastic process is often denoted Xt, t?T
  or, as I prefer, ltXtgt, t?T.
• Recall
• State space discrete time process continuous
  time process.

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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, where EZi0
• symmetric white noise where the (common)
  distribution is symmetric.
• General random walk
• Let Z1, Z2, Z2, be white noise and define
• Define
• with, say, X0, a general random variable.
• 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) the stochastic
  process, 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.
• Reconsider how we defined white noise and a
  random walk.

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Picture of Joint Distribution Just two variates

• Density Function of Bivariate Normal

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Defining a Given Stochastic 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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Some Sample Paths in Stochastic Salary Model
(from before)
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Segment of Sample Path from Symmetric Random Walk
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Increments

• Consider Xtm Xt . This is known as an
  m-increment of the process.
• Xt1 Xt is simply known as an increment.
• 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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Concept 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
• In particular, the same mean and variance.
• Stationarity is a stringent requirement,
  difficult to test in practice.
• Note that the assumption of stationarity
  sweats the data allows max. use of available
  data.

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

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Concept 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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Simple Property of Martingales

• Lemma If ltXtgt is a martingale then
• EXtEX0
• Proof Use property of iterative expectations.

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Table Classifying White Noise Random Walk

• Key Y yes, always ? no, never
  sometimes, depends.

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To Prove

• Lemma A process with independent increments has
  the Markov Property.
• Proof On Board

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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 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.

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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
  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.

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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.

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

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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.

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Quick Questions

• Is the stochastic process of life stationary?
• Is White Noise stationary?
• Is a Random Walk stationary?
• Try to think of a stationary process which is not
  iid.

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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?

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Review of Chapter 2

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