Apr 2016-Mar 2017

Math-Fi seminar on 24 Nov.

2016.11.21 Mon up
  • Date : 24 Nov. (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room
  • Time : 16:30-18:00
  • Speaker: Reiichiro Kawai (Sydney)
  • Title: Computable Bounding Functions for Expectation, Boundary Value and Obstacle Problems
  • Abstract: 
​The computation of expectations involving stochastic processes has long been one of the central issues, in one form or another, in various fields of natural and social sciences, such as the Fokker-Planck equation, financial derivatives pricing, the assessment of ruin probabilities of an insurance company, to name just a few.

In this talk, we propose novel methods for obtaining hard bounding functions, without recourse to sample path simulation, without truncating the naturally unbounded domain that arises in this problem, and without discretizing the time and state variables.

Unlike accurate approximate solutions via the existing discretization-based methods, our hard bounding functions are free from statistical error and act as pointwise 100% confidence intervals within which the unknown solution is guaranteed to exist.

The proposed approaches can be applied to a variety of problem settings, such as mixed boundary conditions, stochastic volatility, stochastic processes with jumps, regime-switching and obstacle problems.

Numerical results are presented throughout to support our theoretical developments and to illustrate the effectiveness of the proposed approaches.

This talk consist of two parts; (i) optimization and (ii) perturbation.

 

(i)

We propose a novel method for obtaining and tightening hard bounding functions for the expected value on stochastic differential equations with the help of the mathematical programming and the Dynkin formula.

In a single implementation of semi-definite programming, the proposed approach obtains explicit bounds in the form of piecewise polynomial functions, which bound the expectation over the whole domain both in time and state.

As a consequence, these global bounds store a continuum of bounding information in the form of a finite number of polynomial coefficients.

In this talk, we pay particular attention to the American style option pricing problem.

 

(ii)

It is often the case that expectations are easy to compute for a simple model, while small perturbations make the computation of expectation suddenly prohibitive.

We propose a novel method for measuring the impact of such small perturbations in expectations without significant computing effort.

Our hard bounding functions are deterministic in the form of Markov-type inequalities, parametrized by the perturbation parameter, so that the upper and lower bounds converge to each other when the perturbation tends to vanish.

The proposed method requires only well-developed numerical methods for boundary value problems for partial differential equations and elementary numerical integration of smooth functions.

 

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