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Category: Probability

Markov Chains: a functional view

  • Laplacian; Adjoints; Harmonic fn; Green’s fn; Forward Eqn; Backward Eqn.
  • Markov Chains and Martingales; Green’s Functions and occupancy; Potential functions; time-reversal and adjoints.

Continue reading “Markov Chains: a functional view”

Author appliedprobabilityPosted on August 30, 2017August 30, 2017Categories ProbabilityLeave a comment on Markov Chains: a functional view

Spitzer’s Lyapunov Ergodicity

We show that relative entropy decreases for continuous time Markov chains.

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Author appliedprobabilityPosted on August 8, 2017August 8, 2017Categories Optimization, ProbabilityLeave a comment on Spitzer’s Lyapunov Ergodicity

A Mean Field Limit

We consider a system consisting of N interacting objects. As we let the number of objects increase, we can characterize the limiting behaviour of the system.

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Author appliedprobabilityPosted on July 5, 2017July 5, 2017Categories Probability, Stochastic NetworksLeave a comment on A Mean Field Limit

Cross Entropy Method

In the Cross Entropy Method, we wish to estimate the likelihood

Here X is a random variable whose distribution is known and belongs to a parametrized family of densities f( , v). Further S(X) is often a solution to an optimization problem.

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Author appliedprobabilityPosted on July 5, 2017July 5, 2017Categories Optimization, ProbabilityLeave a comment on Cross Entropy Method

Sanov’s Theorem

Sanov’s asks how likely is it that the empirical distribution some IIDRV’s is far from the distribution. And shows that the relative entropy determines the likelihood of being far.

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Author appliedprobabilityPosted on July 5, 2017July 5, 2017Categories ProbabilityLeave a comment on Sanov’s Theorem

Entropy and Boltzmann’s Distribution

Entropy and Relative Entropy occur sufficiently often in these notes to justify a (somewhat) self-contained section. We cover the discrete case which is the most intuitive.

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Author appliedprobabilityPosted on July 5, 2017July 5, 2017Categories ProbabilityLeave a comment on Entropy and Boltzmann’s Distribution

Ito’s Formula: a heuristic derivation

  • A heuristic look at the stochastic integral.
  • heuristic derivation of Itô’s formula.

Continue reading “Ito’s Formula: a heuristic derivation”

Author appliedprobabilityPosted on April 12, 2017May 18, 2017Categories Control for Finance, ProbabilityLeave a comment on Ito’s Formula: a heuristic derivation

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