Filtration,Martingales

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Filtration,Martingales Brownian Motion
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A Quick Recap (1) s Algebra of a Sample Space
O

• s algebra is a collection of events in O.
• Properties-
• F ? F.
• If ? ? F, then ? ? F. (? O/ ?)
• If ?1, ?2,, ?n, ? F, then ( U(i gt 1) ?i ) ? F.
• Probability is defined over elements of s
  algebra.

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A Quick Recap (2) Measurable Function

• A function f is F measurable (F is a s algebra
  ) if it can assign a value to each element in F.
• f is a gauge/meter for elements in F. e.g.
  Speedometer is speed-measurable but tacometer is
  not.
• We may have some elements in sample space that
  are not in the sigma algebra.
• Such events cannot be measured at all.
• e.g. x x0 ? R is not a part of sigma algebra of
  the real line, but is a part of the real number
  line. There is no continuous pdf that can assign
  a probability to x x0

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Sample Space and s algebra Illustration
Any open/closed interval is in the s - algebra
The Real Number line is the Sample Space O
A particular point on the real line is not in s -
algebra
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Quick recap(3) Random variables and processes

• We can view s algebra as the collection of all
  possible events that can be measured.
• A random variable is a s algebra measurable
  function.
• All events in the s algebra can be assigned a
  value in Rn.
• A random/stocahstic process is a collection of
  random variables wrt. some continuous parameter,
  like time.
• A stochastic process X(t,?) is the state of the
  outcome ? of a probabilistic experiment at time t.

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Stochastic Process - Illustration
Y2 X(t2, ?)
Y1 Y2 are 2 different random variables.
Y1 X(t1, ?)
X(t,?1)
X(t,?2)
X(t,?3)
Stochastic Process X(t, ?) is a collection of
these Yis
Time
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Filtration(1) A concept for Stochastic Process

• A stochastic process is a random variable at time
  t.
• Hence, it must correspond to some event or
  history of events that can be measured.
• This is a problem as the amount of information
  available increases with time.
• As a result, the s-algebra over which X(t,?) is
  defined changes with time t.

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Stochastic Process - Filtration
Y2 X(t2, ?)
Y1 Y2 are 2 different random variables over
different s-algebras.
Y1 X(t1, ?)
X(t,?1)
X(t,?2)
X(t,?3)
Time
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Filtration(2) A concept for Stochastic Process

• Hence, at each time t, we filter F to get a
  s-algebra Ft.
• Ft represents all the events that can take place
  up-to time t total history of the process up-to
  t.
• This sequence Ft is called filtration of F -
• Ft is increasing. Ft1 lt Ft2 when t1 lt t2.
  (History of events increases with time)
• Ft ? F as t ? 8. (We have all the information
  after infinite time period)

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Filtration(3)

• A stochastic process x(t,?) is called Ft adapted
  when x(t,?) is Ft measurable for all t.
• i.e. x(t,?) can never predict the future.
• or the present value of x(t,?) is decided only by
  its past values.
• e.g. X(t,?) B(t/2,?) is Ft adapted but X(t,?)
  B(2t,?) is not.

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Martingales(1) Definition

• It is class of stochastic processes.
• Useful to describe/represent fair game scenario.
• e.g. representing prices in fair markets
  (expected profit is zero)
• X(t,?) is a martingale if-
• It is Ft adapted.
• E( X(t,?) ) lt 8 for all t.
• E( X(t,?) Fs ) X(s,?) for all t gt s.

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Martingales(2) Properties

• If X(t,?) is a martingale, then
• E(X(t,?)) E(X(0,?)) (Average value does not
  change with time)
• E(X(t,?)F0) X(0,?)
• E( E(X(t,?)F0) ) E( X(0,?) ) E(X(t,?))
• If E(X(t,?)) lt M lt 8 for all t, then-
• X(t,?) ? Xlim(?) and
• E( Xlim(?) ) lt 8 Martingale Convergence
  Theorem

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Martingales(3) Some New Terms

• If X(t,?) is a finite Ft adapted stochastic
  process and if-
• X(s, ?) lt E(X(t,?)Fs) (t gt s), then X(t,?) is
  a sub-martingale.
• X(s, ?) gt E(X(t,?)Fs) (t gt s), then X(t,?) is
  a super-martingale.

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Brownian Motion(1)

• Brownian Motion is the most important stochastic
  process in applied mathematics.
• It is used for
• Goodness of Fit Tests
• Quantum Mechanics
• Modeling Stock Prices
• It is also referred to as the wiener Process

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Brownian Motion(2) Definition

• A stochastic Process B(t,?) is called a brownian
  motion if-
• B(0,?) 0
• B(t s,?) B(t,?) is independent and stationary
  (independent, stationary increments)
• B(t s,?) B(t,?) is a gaussian random variable
  with mean 0 and standard deviation cs1/2.
• Hence,
• E(B(t s,?) B(t,?)2) cs
• E(B(t1,?), B(t2,?)) c min(t1,t2)

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Brownian Motion - Illustration
B(t,?1)
B(t,?2)
B(t,?3)
Time
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Brownian Motion(2) Properties

• B(t,?) is a martingale.
• E(B(t,?)Fs) B(s,?) (s lt t)
• B(t,?) is continuous everywhere, but
  differentiable nowhere.
• By Kolomogorovs theorem
• B(t,?) will hit every point in finite time. The
  expectated time of B(t,?) hitting a fixed point
  is 8 however.

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Brownian Motion(3) Properties

• B(t,?) reflected about any hyper-plane is also a
  brownian motion.
• A scaled brownian motion c.B(t,?) is also a
  brownian motion.
• Brownian motion, when scaled in time (B(t/c,?), c
  gt 1) is also a brownian motion (It is fractal)

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Brownian Motion(4) Variations

• Brownian Motion Absorbed at a value
• When B(t,?) hits a value, it stays there.
• Such a process is also a brownian motion.
• Brownian Motion reflected about origin
• Z(t,?) B(t,?) is also a brownian motion
• Geometric Brownian Motion
• Z(t,?) exp(B(t,?))
• Is not a martingale
• Used to model cases when percentages changes in
  the process are random and independent (e.g.
  Stock Prices)

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Variations of Brownian Motion Standard Brownian
Motion
Y 0
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Variations of Brownian Motion Positive Brownian
Motion
Y 0
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Variations of Brownian Motion Standard Brownian
Motion
Y -1
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Variations of Brownian Motion Reflected Brownian
Motion
Y -1
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Variations of Brownian Motion Absorbed Brownian
Motion
Y -1