Introduction to Stochastic Processes and Calculus

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Introduction to Stochastic Processes and Calculus
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Preliminaries (1)

• What is sigma-algebra of a set O?
• A sigma algebra F is a set of subsets ? of O
  s.t.-
• F ? F.
• If ? ? F, then ? ? F. (? O/ ?)
• If ?1, ?2,, ?n, ? F, then U(I gt 1) ?i ? F.
• The set (O,F ) is called a measurable space.
• There may be be certain elements in O that are
  not in F
• The smallest sigma algebra generated by the real
  space (O Rn) is called the Borel Sigma Algebra
  ß.

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Preliminaries (2)

• What is a probability Measure ?
• A probability measure is the triplet (O,F,P)
  where P F ? 0,1 is a function from F to 0,1.
• P(F) 0 and P(O) 1 always.
• The elements in O that are not in F have no
  probability.
• We can extend the probability definition by
  assigning a probability of zero to such elements.

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Preliminaries (3)

• What is a random variable x wrt (O,F,P) ?
• x F ? Rn is a measurable function.
• i.e. x-1(z) ? F for all z in Rn
• Hence, P F ? 0,1 is translated to an
  equivalent function µx Rn ? 0,1, which is the
  distribution of x.
• What is a stochastic Process X(t, ?)?
• It is a parameterized collection of random
  variables x(t), or X(t, ?) x(t)t.
• Normally, t is taken as time.
• Think of ? as one outcome from a set of possible
  outcomes of an experiment. Then, X(t, ?) is the
  state of an outcome ? of the 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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Stochastic Process A few considerations (1)

• A stochastic Process is a function of a
  continuous variable (most oftentime).
• The question now becomes how to determine the
  continuity and differentiability of a stochastic
  process?
• It is not simple as a stochastic process is not
  deterministic.

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Stochastic Process A few considerations (2)

• We use the same definitions of continuity, but
  now look at the expectations and probabilities.
• A deterministic function f(t) is cts if-
  f(t1) f(t2) lt Ct1 t2.
• To determine if a stochastic process X(t,?) is
  cts, we need to determine P( X(t1, ?)
  X(t2, ?) lt Ct1 t2) or E( X(t1, ?)
  X(t2, ?))

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Stochastic Process Kolomogorov Continuity Theorem

• If for all T gt 0, there exist a,b,D gt 0 such
  that E( X(t1, ?) X(t2, ?)a) lt D.t1
  t2(1 b) Then X(t, ?) can be considered as a
  continuous stochastic process.
• Brownian motion is a continuous stochastic
  Process.

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Stochastic Processes Applications (1)

• In Real world, several systems can be expressed
  as differential equations
• e.g. Population growth ( dN/dt a(t)N(t) )
• However, in real world several factors introduce
  a random factor in such models.
• E.g. a(t) b(t) s(t). Noise. b(t) s(t)
  W(t)
• W(t) is a stochastic Process that represents the
  source of randomness (e.g. White Noise)
• A simple differential Equation becomes a
  stochastic differential equation.

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Stochastic Processes Applications (2)

• Other Applications where stochastic processes are
  used -
• Filtering problems (Kalman-bucy filter)
• Minimize the expected estimation error for a
  system state.
• Dirichlet Boundary Value Problem
• Optimal Stopping Theorem
• Financial Mathematics
• Theory of option pricing uses differential heat
  equation applied to a geometric brownian motion.

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Stochastic calculus Introduction(1)

• Let us consider-
• dx/dt b(t,x) s(t,x) W(t)
• White noise assumptions on W(t) would make W(t)
  discontinuous.
• This is bad news.
• Hence, we consider the discrete version of the
  equation-
• xk 1 - xk b(tk,xk)?tk s(tk,xk)W(tk)?tk
  (xk x(tk,?) )
• We can make white noise assumptions on Bk where
  ?Bk W(tk)?tk.
• It turns out that Bk can only be the brownian
  motion

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Stochastic calculus Introduction(2)

• Now we have another problem
• x(t) ? b(tk,xk)?tk ?s(tk,xk) ?Bk
• As ?tk ? 0, ? b(tk,xk)?tk ? time integral of
  b(t,x)
• What about ?s(tk,xk) ?Bk ?
• Hence, we need to find expressions for integral
  and differentiation of a function of stochastic
  process.
• Again, we have a problem.
• Brownian motion is continuous, but not
  differentiable (with a high probability)

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Stochastic calculus Introduction(3)

• Stochastic Calculus provides us a mean to
  calculate integral of a stochastic process but
  not differentiation.
• It makes sense as most stochastic processes are
  not differentiable.

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Stochastic calculus Introduction(4)

• We use the definition of integral of
  deterministic functions as a base-
• ? s(t,?) dB ? s(tk,?) ?Bk , where tk ? tk,tk
  1) as tk 1 tk ? 0.
• However, we cannot chose any tk ? tk,tk 1.
• E.g. if tk tk, then E(? Bk ?Bk) 0.
• E.g. if tk tk 1, then E(? Bk ?Bk) t.
• Hence, we need to be careful in choosing tk

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Stochastic calculus Ito and Stratonovich

• Two choices for tk are popular-
• If tk tk , then it is called Itos integral.
• If tk ( tk tk 1 )/2, then it is called
  Stratonovich integral.
• We will concentrate on Itos integral as it
  provides computational and conceptual simplicity.
• Itos and Stratonovich integrals differ by a
  simple time integral only.

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Stochastic calculus Itos Theorem (1)

• For a given f(t,?) if
• f(t,?) if Ft adapted
• f(t,?) can be determined by t and values of
  brownian motion Bt(?) upto t.
• Bt/2(?) is Ft adapted but B2t(?) is not.
• E(?f2(t,?) dt ) lt 8 (Expected energy)
• Then ? f(t, ?) dBt(?) ? F(tk, ?) (Bk
  1 - Bk) and E(? f(t, ?) dBt(?)2 ) E(?f2(t,?)
  dt )
• F(t , ?) are called elementary functions.
• Their values are constant in the interval tk,tk
  1
• E(? f(t, ?) - F(t,?)2dt ) ? 0 (Difference in
  expected energy is insignificant)

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Stochastic calculus Itos Theorem (2)

• If f(t,?) Bt(?) B(t,?), then-
• Select F(t,?) B(tk,?) when t ? tk,tk 1
• We can see that E(? f(t, ?) - F(t,?)2dt )
  ?t3/2 ? 0 as ?t tk 1 - tk ? 0.
• This gives us ?B(t,?) dB(t,?) ? B(tk,?)
  (B(tk 1,?) - B(tk,?))
• This gives us-
• ?B(t,?) dB(t,?) B2(t,?)/2 t/2.
• Note that Itos integral gives us more than the
  expected B2(t,?)/2. This is due to the
  time-variance of the brownian motion.

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Stochastic calculus Itos Process (1)

• For a general process x(t,?), how do we define
  integral ? f(t,x) dx ?.
• If x can be expressed by a stochastic
  differential equation, we can calculate ?f(t,x).
• x(t,?) is called an Itos process if
• dx udt vdBt , where u has a finite integral
  with probability 1 and v has finite energy with
  probability 1.

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Stochastic calculus Itos Process (2)

• In such a case
• ?f(t,x) (df/dt)dt (df/dx)dx ½d2f/dx2 (dx)2
• where dt.dt dBt.dt dt. dBt 0, dBt. dBt
  dt.

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Stochastic calculus Itos Process (3)

• If f(t,?) Bt(?) B(t,?) (u 0, v 1), then-
• Define g(t,?) B2(t,?)/2.
• Now ?(B2(t,?)/2) 0.dt B(t,?).dBt
  ½.1.(dBt)2 ?(B2(t,?)/2) 0.dt B(t,?).dBt
  ½.dt
• Hence, B2(t,?)/2 ? B(t,?).dBt ? ½.dt
• ? B(t,?).dBt B2(t,?)/2 - t/2