Stochastic Calculus 7: Itos Diffusion 1

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Stochastic Calculus 7Itos Diffusion - 1
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Recap (1)

• A Stochastic Differential Equation (SDE) of the
  form-
• dXt b(t,xt).dt s(t,xt).dBt has a unique
  t-continuous, Ft adapted and finite expected
  energy solution if
• b(t,x) s(t,x) lt C(1 x) for all t x
• Ensures the solution doe not blow-up
• b(t,x) - b(t,y) s(t,x) - s(t,y) lt Dx -
  y for t x
• Ensures the Solution is unique
• Ft is the filtration generated by Bt

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Uniqueness for Stochastic Processes

• What does uniqueness mean in terms of stochastic
  processes (2 instances of the same stochastic
  process may have different trajectories)
• Before that, we need to know about weak and
  strong solutions.

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Strong vs Weak Solution to a SDE

• A strong solution Xt is Ft adapted.
• Bt , hence Ft is given and Xt can be constructed.
• A weak solution may not be a strong solution.
• Some SDEs may only have a weak solution.

• A weak solution is the pair (Xt,Bt) to a given
  SDE.
• We construct a Bt and a Xt.
• The filtration Ht generated by Bt would be
  different from Ft.
• Every Strong solution is a weak solution

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Weak and Strong Uniqueness(1)

• In deterministic world,
• The solution of an equation f(x,t) 0 is unique
  if
• f(Xt,t) f(Yt,t) 0 Xt Yt for
  all t.
• i.e. they are point-wise identical (or their
  paths are the
• same).
• In Stochastic Realm, the above definition of
  uniqueness has no meaning.

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Weak and Strong Uniqueness(2)

• Recall Presentation 1.
• A rule (for continuity or bounded-ness) in
  deterministic world is translated to stochastic
  rule by attaching an expectation or probability
  to the same rule.
• Uniqueness is no exception!
• We look for the Prob(Xt Yt) to determine
  uniqueness.

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Weak and Strong Uniqueness(3)

• The SDE dXt b(t,xt).dt s(t,xt).dBt
  has a strongly unique solution if-
• Xt Yt satisfy the same SDE P(Xt Yt)
  1 for all t.
• This is equivalent to path uniqueness.
• What if P(Xt Yt) lt 1, but Xt Yt share the
  same probability law/distribution?
• In such a case, the solution is said to be
  weakly unique.
• The uniqueness in slide 1 is strong uniqueness

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Itos Diffusion(1)

• The solution to following SDE is called an Itos
  diffusion- dXt b(t,xt).dt
  s(t,xt).dBt Xt, b(t,xt) ? Rn, Bt ? Rm,
  s(t,xt) ? Rn ? m
• For a n m 3, the solution Xt is modeled as
  the position of a suspended particle in a liquid
  moving with a velocity b(t,x).
• Xt is time homogeneous if b and s are only
  functions of Xt.
• A time homogenous Xt satisfies the Markov -
  Property

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Time Homogeneous Itos Diffusion

• We define some Notation first-
• X(s,x,t) Itos diffusion at time t, starting
  from position x at time s.
• We also note the following-
• X(0,x,t h) X(t,Xt,t h) and X(0,Xt,h) are
  weakly unique.
• X(0,x,t h), X(t,Xt,t h) and X(0,Xt,h) have
  the same probability distribution.

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Markov property of a Time-Homogenous Itos
Diffusion(1)

• For a bounded borel function f(x), the markov
  property states that-
• Ex f(X(0,x,s h))Fs Exs f(X(0,Xs,h)) .
• The property is due to the following-
• X(0,x,s h) X(s,Xs,s h) is independent of Fs
• X(s,Xs,s h) X(0,Xs,h) are weakly unique.

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Stopping Time

• Stopping time ? is a random variable s.t.
• The set of events ? ? lt t is measurable by a
  filtration Nt.
• We are able to decide if ? lt t by the
  information available up-to time t alone.
• e.g. The Time when a brownian motion Bt first
  reaches a value/ crosses a boundary is a stopping
  time.
• The time when Bt crosses a boundary last in
  0,T time interval is not a stopping time.

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Strong Markov property for Itos Diffusion

• Ex f(X(0,x,? h))F? Ex? f(X(0,X?,h))
  where ? is a stopping time wrt Ft
• For a motion Xt , F? is sigma-algebra generated
  by Bs s min(t, ?)
• Again, if given F?, We know ?, and Brownian
  motion is always strongly Markov, the result
  follows as before.

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Generator Of Itos Diffusion(1)

• Operator A is the generator of Itos diffusion.
  It is defined as follows- A.f(x) lim h ? 0
  Ex(f(Xh)) f(x)/h
• Question is What does A signify, or What does A
  measure.
• It is the rate of change of g(t) Exf(Xt) with
  time.
• It measures Both the derivative of f(x) wrt x as
  well as the double derivative.
• If Xt Bt, A.f(x) d2f/dx2

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Generator Of Itos Diffusion(2)

• Domain of A wrt f
• The set of functions s.t. A.f(x) exists at x is
  called DA(x).
• The set of functions s.t. A.f(x) exists for all x
  is called DA
• DA is the set C2Rn ? Rn
• Also, it can be proved easily (using f(Xt)
  f(x) ?df) that -
• A.f(x) b.df/dx (½)ssTd2f/dx2. and
• Ef(Xt) f(x) E ?A.f(x) for all t

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The Characteristic Operator

• The characteristic Operator is similar to the
  generator Operator. It is given as A. And is
  defined as -
• A.f(x) lim u ? x Ex(f(X?u))
  f(x)/E?u where ?u is the first exit time
  from a open set U.
• If E?u 8 for all U containing x, A.f(x) 0
• i.e. x is a trap for Xt.
• Domain of function for which A.f(x) exists for
  all x is called DA. .
• DA lt DA. for all f in DA