Rare-event asymptotics and estimation for dependent random sums
Exit Talk of Patrick J. Laub
University of Queensland & Aarhus University
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PhD outline
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2015 | Aarhus |
2016 Jan-Jul | Brisbane |
2016 Aug-Dec | Aarhus |
2017 | Brisbane/Melbourne |
2018 Jan-Apr (end) | China |
Supervisors: Søren Asmussen, Phil Pollett, and Jens L. Jensen
10101
Sums of random variables
Asymptotic analysis / rare-events
Monte Carlo simulation
What is applied probability?
Data
Fitted model
Decision
Statistics
App. Prob.
You have some goal..
Insurance
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Estimated financial cost of the natural disasters in the USA which cost over $1bn USD. Source: National Centers for Environmental Information
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Cramér-Lundberg model
Interested in
- Probability of ruin (bankruptcy) in the next 10 years
- Probability of ruin eventually
- Stop-loss premiums
E.g. guaranteed benefits
An investor's problems
Want to know:
- cdf values
- value at risk
- expected shortfall
Modelling stock prices
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of political economy
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Fischer Black Myron Scholes
Can you tell which is BS?
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S&P 500 from Oct 1998 to Apr 2008
Google Finance
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Geometric Brownian motion
In words, Stock Price = (Long-term) Trend + (Short-term) Noise
That's just the beginning..
General diffusion processes...
Stochastic volatility processes...
SV with jumps...
SV with jumps governed by a Hawkes process with etc...
Monte Carlo
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Quasi-Monte Carlo
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4325421/last.gif)
For free, you get a confidence interval
Sum of lognormals distributions
where
What is that?
Start with a multivariate normal
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4295034/multinorm.gif)
Then set
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4325430/NormPDF.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4325431/LogNormPDF.png)
Then add them up
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4325439/SumLogNormPDF.png)
What's known about sums?
Easy to calculate interesting things with the density
Density can be known..
Example
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4296869/SumPlot.png)
Kernel-density estimation
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Laplace transform approximation
No closed-form exists for a single lognormal
Asmussen, S., Jensen, J. L., & Rojas-Nandayapa, L. (2016). On the Laplace transform of the lognormal distribution. Methodology and Computing in Applied Probability
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Generalise to d dimensions
- Setup Laplace's method
- Find the maximiser
- Apply Laplace's method
Laub, P. J., Asmussen, S., Jensen, J. L., & Rojas-Nandayapa, L. (2016). Approximating the Laplace transform of the sum of dependent lognormals. Advances in Applied Probability
Generalise to d dimensions
- Setup Laplace's method
- Find the maximiser
- Apply Laplace's method
Solve numerically:
What is the maximiser?
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4329310/Screen_Shot_2016-05-25_at_10.40.03_PM.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4296581/simple_asymptotic.png)
Savage condition
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4304780/savage.gif)
Hashorva, E. (2005), 'Asymptotics and bounds for multivariate Gaussian tails', Journal of Theoretical Probability
Laplace's method
Find
Expand with 2nd order Taylor series about the maximiser
Example
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4295388/laplace_test.gif)
Orthogonal polynomial expansions
- Choose a reference distribution, e.g,
2. Find its orthogonal polynomial system
3. Construct the polynomial expansion
Pierre-Olivier Goffard
Asmussen, S., Goffard, P. O., & Laub, P. J. (2017). Orthonormal polynomial expansions and lognormal sum densities. Risk and Stochastics - Festschrift for Ragnar Norberg (to appear).
Orthogonal polynomial systems
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Example: Laguerre polynomials
If the reference is Gamma
then the orthonormal system is
For r=1 and m=1,
Final steps
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4312008/convergence.png)
Final final step: cross fingers & hope that the q's get small quickly...
Calculating the coefficients
1. From the moments
2. Monte Carlo Integration
3. (Dramatic foreshadowing) Taking derivatives of the Laplace transform...
Applied to sums
- Moments
- Monte Carlo Integration
- Taking derivatives
- Gauss Quadrature
Title Text
Subtitle
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4312043/ortho_sln_test_1.png)
An example test
Applications to option pricing
Dufresne, D., & Li, H. (2014).
'Pricing Asian options: Convergence of Gram-Charlier series'.
Goffard, P. O., & --- (2017). 'Two numerical methods to evaluate stop-loss premiums'. Scandinavian Actuarial Journal (submitted).
Extension to random sums
Say you don't know how many summands you have...
Imagine you are an insurance company;
there's a random amount of accidents to pay out (claim frequency),
and each costs a random amount (claim size)
Approximate S using orthogonal polynomial expansion
A simplification
where
The stuff we want to know
As
and using
we can write
Laplace transform of random sum
With we can deduce
and just take derivatives
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4325882/code.png)
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Another example test
![](https://s3.amazonaws.com/media-p.slid.es/uploads/782452/images/4312548/poisson_gamma_svf.png)
Other things I don't have time to talk about
- Andersen, L.N., ---, Rojas-Nandayapa, L. (2017) ‘Efficient simulation for dependent rare events with applications to extremes’. Methodology and Computing in Applied Probability
- Asmussen, S., Hashorva, E., --- and Taimre, T. (2017) ‘Tail asymptotics of light-tailed Weibull-like sums’. Polish Mathematical Society Annals.
In progress:
- Taimre, T., ---, Rare tail approximation using asymptotics and polar coordinates
- Salomone, R., ---, Botev, Z.I., Density Estimation of Sums via Push-Out, Mathematics and Computers in Simulation
- Asmussen, S., Ivanovs, J., ---, Yang, H., 'A factorization of a Levy process over a phase-type horizon, with insurance applications
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Thomas Taimre
Robert Salomone
Thanks for listening!
and a big thanks to UQ/AU/ACEMS for the $'s
And thanks to my supervisors
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Rare-event asymptotics and estimation for dependent random sums – an exit talk, with applications to finance and insurance
By plaub
Rare-event asymptotics and estimation for dependent random sums – an exit talk, with applications to finance and insurance
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