Approximations of the Laplace Transform of a Lognormal Random - PowerPoint PPT Presentation

Introduction Approximations of the Laplace transform Applications Approximations of the Laplace Transform of a Lognormal Random Variable Leonardo Rojas Nandayapa Joint work with Søren Asmussen & Jens Ledet Jensen The University of Queensland School of Mathematics and Physics August 1, 2011 Conference in Honour of Søren Asmussen: New Frontiers in Applied Probability Sandbjerg Gods, Denmark. Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Approximations of the Laplace transform Applications Outline Introduction 1 Sums of Lognormal Random Variables Laplace transforms in probability Approximations of the Laplace transform 2 The Laplace method The exponential family generated by a Lognormal Applications 3 Cdf of a sum of lognormal via inversion Tail probabilities and rare-event simulation Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Outline Introduction 1 Sums of Lognormal Random Variables Laplace transforms in probability Approximations of the Laplace transform 2 The Laplace method The exponential family generated by a Lognormal Applications 3 Cdf of a sum of lognormal via inversion Tail probabilities and rare-event simulation Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Sums of Lognormal random variables Due to the popularity of the Lognormal random variables sums of lognormal appear in a wide variety of applications Finance Stock prices are modeled as lognormals. Sums of 1 lognormals arise in portfolio and option pricing. Insurance Individual claims are also modeled lognormal: 2 Total claim amount is a sum of lognormals. Engineering. Sums of lognormals arise in a large amount 3 of applications. Most prominent in telecommunications. Biology, Geology,. . . 4 Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Sums of Lognormal random variables Since the distribution of the sum of lognormals is not available a large number of numerical and approximative methods have been developed. Approximating distributions. A popular approach is 1 using another lognormal distribution. More recently Pearson Type IV, left skew normal, log-shifted gamma, power lognormal distributions have been used. Transforms Inversion. 2 Bounds. 3 Monte Carlo methods. 4 Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Sums of Lognormal random variables However, most of these methods have drawbacks: Inaccuracies in certain regions. Lower regions and upper 1 tail. Poor approximations for large/low number of summands. 2 Same for extreme parameters. Difficulties arising from non-identically distributed. 3 Complicated methods. 4 Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Outline Introduction 1 Sums of Lognormal Random Variables Laplace transforms in probability Approximations of the Laplace transform 2 The Laplace method The exponential family generated by a Lognormal Applications 3 Cdf of a sum of lognormal via inversion Tail probabilities and rare-event simulation Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Laplace Transform We denote the Laplace transform of a density f � ∞ e − θ x f ( x ) dx = E [ e − θ X ] . L f ( θ ) = 0 where the domain of convergence of the transform is Θ = { θ ∈ R : θ ≥ 0 } . Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Some applications of Laplace transforms Cumulative Distribution Functions: It follows that the Laplace transform of its cdf F is L F ( θ ) = L f ( θ ) θ > 0 . , θ Thus we can compute probabilities by using any of the numerical inversion methods available in the literature. Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Common applications of Laplace transforms Example (Bromwich inversion integral) If F is supported over [ 0 , ∞ ] with no atoms then � γ + i ∞ 1 e x θ L F ( θ ) d θ, F ( x ) = γ > 0 . 2 π i γ − i ∞ Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Common applications of Laplace transforms Sums of Independent Random Variables: Let X 1 , . . . , X n be independent random variables with pdf’s f 1 , . . . , f n and let F be the cdf of S n := X 1 + · · · + X n . Then � n i = 1 L f i ( θ ) L F ( θ ) = , θ > 0 . θ n Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Common applications of Laplace transforms Exponential families generated by a random variable Let X be a random variable with distribution F . The family of distributions defined by dF θ ( x ) = e − θ x dF ( x ) θ ∈ Θ . , L f ( θ ) is known as the exponential family of distributions generated by X . Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Common applications of Laplace transforms Example In some applications (saddlepoint approximation and rare-event simulation for example) it is often required to find the solution θ to the equation E θ [ X ] = y , y fixed . Here E θ is the expectation w.r.t. F θ . Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction Sums of Lognormal Random Variables Approximations of the Laplace transform Laplace transforms in probability Applications Laplace transform of a Lognormal No closed form of the Laplace transform of a Lognormal random variable is known � � � ∞ − θ x − ( log x − µ ) 2 1 √ L f ( θ ) = exp dx 2 σ 2 x 2 πσ 0 Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction The Laplace method Approximations of the Laplace transform The exponential family generated by a Lognormal Applications Outline Introduction 1 Sums of Lognormal Random Variables Laplace transforms in probability Approximations of the Laplace transform 2 The Laplace method The exponential family generated by a Lognormal Applications 3 Cdf of a sum of lognormal via inversion Tail probabilities and rare-event simulation Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction The Laplace method Approximations of the Laplace transform The exponential family generated by a Lognormal Applications Intuitive approach We consider for k = 0 , 1 , 2 , . . . � ∞ � � x k − 1 − θ x − ( log x − µ ) 2 E [ X k e − θ X ] = √ exp dx 2 σ 2 2 π σ 0 � ∞ � � − θ e y + ky − ( y − µ ) 2 1 = √ exp dy . 2 σ 2 2 π σ −∞ The change of variable y = log x was used here . Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction The Laplace method Approximations of the Laplace transform The exponential family generated by a Lognormal Applications Intuitive approach The Laplace method suggest to replace the expression − θ e y + ky − ( y − µ ) 2 (1) 2 σ 2 by a Taylor approximation of second order around the value ρ k that maximizes this expression. That is � � 1 + ( y − ρ k ) + ( y − ρ k ) 2 + ky − ( y − µ ) 2 − θ e ρ k (2) . 2 σ 2 2 Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction The Laplace method Approximations of the Laplace transform The exponential family generated by a Lognormal Applications Intuitive approach The figures illustrate the idea Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable

Introduction The Laplace method Approximations of the Laplace transform The exponential family generated by a Lognormal Applications Intuitive approach Moreover, the method works because the resulting integral can be explicitly obtained. � ∞ � � � � 1 +( y − ρ k )+( y − ρ k ) 2 + ky − ( y − µ ) 2 1 − θ e ρ k √ exp dy . 2 σ 2 2 2 πσ −∞ (Notice that the expression in the brackets is simply a second order polynomial). Leonardo Rojas Nandayapa Laplace Transform of a Lognormal Random Variable