s.m. iacus (university of milan) - welcome to rmetrics ... · from the sdepackage to the yuima...

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From the sde package to the Yuima Project

S.M. Iacus (University of Milan)

Meielisalp, R/RMetrics Workshop, June 28th - July 2nd, 2009

Overview of the sde package

Overview of the sdepackage

Diffusions

Exact likelihood

Pseudo-likelihood

Simulated likelihoodmethod

Hermite expansion

The Yuima Project

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Why SDE’s?


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

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Continuous time models are not just a matter of taste, particularly in economics

and finance.

The sde package, is focused on simulation and inference for continuous time

models solutions to SDE’s observed at discrete times.

Why continuous time matters if observations are always discrete?

Why SDE’s?


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

3 / 41


and finance.




An example: according to McCrorie & Chambers (2006, J. of Econ.) and

others, “spurious Granger causality [tested with VAR models] is only a

consequence of the intervals in which economic data are generated being finer

than the econometrician’s sampling interval.”

Conclusions: assume a continuous time model (SDE). Discretize that, build a

VAR from the discretized SDE and the spurious Granger causality vanishes!

Why SDE’s?


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

3 / 41


and finance.




An example: according to McCrorie & Chambers (2006, J. of Econ.) and

others, “spurious Granger causality [tested with VAR models] is only a

consequence of the intervals in which economic data are generated being finer

than the econometrician’s sampling interval.”

Conclusions: assume a continuous time model (SDE). Discretize that, build a

VAR from the discretized SDE and the spurious Granger causality vanishes!

Rephrasing: why using a binomial distribution if your underlying model is a

Gaussian?

A few examples of SDEs


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

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� gBm : dXt = µXtdt + σXtdWt

� CIR : dXt = (θ1 + θ2Xt)dt + θ3

√XtdWt

� CKLS : dXt = (θ1 + θ2Xt)dt + θ3Xθ4

t dWt

� nonlinear mean reversion (Aıt-Sahalia)

dXt = (α−1X−1t + α0 + α1Xt + α2X

2t )dt + β1X

ρt dWt

� double Well potential (bimodal behaviour, highly nonlinear)

dXt = (Xt − X3t )dt + dWt

� Jacobi diffusion (political polarization):

dXt = −θ(

Xt − 12

)

dt +√

θXt(1 − Xt)dWt

� radial Ornstein-Uhlenbeck : dXt = (θX−1t − Xt)dt + dWt

� hyperbolic diffusion : dXt = σ2

2

[

β − γ Xt√δ2+(Xt−µ)2

]

dt + σdWt

Diffusion processes solutions to SDEs


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

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From the statistical point of view, we are interested in the parametric family of

diffusion process solutions of the SDE

dXt = b(Xt, θ)dt + σ(Xt, θ)dWt, X0 = x0, t ∈ [0, T ]

θ = (α, β) ∈ Θα × Θβ = Θ, where Θα ⊂ Rp and Θβ ⊂ Rq .

Observations always come in discrete time form at some times ti = i∆n,

i = 0, 1, 2, ..., n, where ∆n is the length of the steps. We denote the

observations by Xn := {Xi = Xti}0≤i≤n.

Diffusion processes solutions to SDEs


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

5 / 41

From the statistical point of view, we are interested in the parametric family of

diffusion process solutions of the SDE

dXt = b(Xt, θ)dt + σ(Xt, θ)dWt, X0 = x0, t ∈ [0, T ]

θ = (α, β) ∈ Θα × Θβ = Θ, where Θα ⊂ Rp and Θβ ⊂ Rq .

Observations always come in discrete time form at some times ti = i∆n,

i = 0, 1, 2, ..., n, where ∆n is the length of the steps. We denote the

observations by Xn := {Xi = Xti}0≤i≤n.

Different sampling schemes, different statistical procedures:

1. Large sample asymptotics: ∆ fixed, T = n∆ → ∞ as n → ∞

2. High frequency: T = n∆n fixed, ∆n → 0 as n → ∞

3. Rapidly increasing design: T = n∆ → ∞, ∆n → 0 as n → ∞ under

the additional condition n∆kn → 0 for k > 1

Likelihood in discrete time


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

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By Markov property of diffusion processes, the likelihood has this form

Ln(θ) =n

∏

i=1

pθ (∆, Xi|Xi−1)pθ(X0)

Problem: the transition density pθ (∆, Xi|Xi−1) is often not available! Only

for OU, CIR and gBm

Likelihood in discrete time


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

6 / 41

By Markov property of diffusion processes, the likelihood has this form

Ln(θ) =n

∏

i=1

pθ (∆, Xi|Xi−1)pθ(X0)

Problem: the transition density pθ (∆, Xi|Xi−1) is often not available! Only

for OU, CIR and gBm

Solutions:

� discretization of the SDE (Euler, Milstein, Ozaki, etc)

� simulation method

� hermite polynomial expansion

� partial differential equations

� other approximations of the transition density

Local Gaussian Approximation. Euler Scheme.

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By Euler discretization of the SDE : dXt = b(Xt, θ)dt + σ(Xt, θ)dWt

Xt+∆t − Xt = b(Xt, θ)∆t + σ(Xt, θ)(Wt+∆t − Wt),

we get an approximate transition density which is Gaussian. This is widely seen in applied

contexts. But is this approximation good or not? In general no!

For example, for gBm, the true transition density is a log-normal and the Euler schemes provides

only a Gaussian approximation!

It is possible to prove that estimators are not even consistent for non negligible ∆.

Euler, ∆ and bias


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

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Consider OU model

dXt = (θ1 − θ2Xt)dt + θ3dWt, X0 = x0

Both true and Euler approximation are Gaussian respectively with mean and

variance

m(∆, x) = xe−θ2∆ +θ1

θ2

(

1 − e−θ2∆)

, v(∆, x) =θ23

(

1 − e−2θ2∆)

2θ2,

and (Euler)

mEuler(∆, x) = x(1 − θ2∆) + θ1∆ , vEuler(∆, x) = θ23∆ ,

Only under high-frequency setting, i.e. ∆ → 0, the approximation is

acceptable.

Simulated likelihood method


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

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Let pθ(∆, y|x) be the true transition density of Xt+∆ at point y given

Xt = x. Consider a δ << ∆, for example δ = ∆/N for N large enough,

and then use the Chapman-Kolmogorov equation as follows:

pθ(∆, y|x) =

∫

pθ(δ, y|z)pθ(∆ − δ, z|x)dz = Ez{pθ(δ, y|z)|∆− δ} ,

It means that pθ(∆, y|x) is seen as the expected value over all possible

transitions of the process from time t + (∆ − δ) to t + ∆, taking into account

that the process was in x at time t.

So we need simulations!

What about N? We need many simulations

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Example: approximation for the CIR model

0.00 0.05 0.10 0.15 0.20

05

1015

2025

x

cond

ition

al d

ensi

ty

exactN=2N=5N=10

What about N? We need many simulations

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Example: approximation for the CIR model

0.00 0.05 0.10 0.15 0.20

05

1015

2025

x

cond

ition

al d

ensi

ty

exactN=2N=5N=10

We need many simulations (N ) for each time points (Xti , Xti+∆). But not all simulation

schemes are stable for all models

Numerical instability. Up |Down ∆ = 0.1|0.25

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Aıt-Sahalia process dXt = (5 − 11Xt + 6X2t − X3

t )dt + dWt, X0 = 5

Aıt-Sahalia’s approximation


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

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True likelihood (continuous line), Euler approximation (dashed line), Aıt-Sahalia

approximation (dotted line). Where is the dotted line? Coincides with the

continuous line! Model dXt = βXtdt + dWt

−3 −2 −1 0 1 2 3

26.0

26.5

27.0

27.5

β

log−

likel

ihoo

d

no need to have ∆ small, but (was) very difficult to implement!

The sde package

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The sde package implements Aıt-Sahalia method. It also implements the following methods

� local Gaussian (dcEuler), Elerian (dcElerian), Ozaki (dcOzaki) and Shoji-Ozaki

(dcShoji) approximations

� Simulated Likelihood Method (dcSim), Kessler’s (dcKessler) and Aıt-Sahalia

(HPloglik) approximations

all of them can be passed to the mle function in R or used to build appropriate likelihood

functions.

The sde package

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The sde package also implements many simulation schemes, including: Euler, Milstein,

Milstein2, Elerian, Ozaki, Ozaki-Shoji, Exact Simulation Scheme, Simulation from conditional

distribution, Predictor-Correction scheme, etc via the unique sde.sim function

sde.sim(t0 = 0, T = 1, X0 = 1, N = 100, delta, drift, sigma,

drift.x, sigma.x, drift.xx, sigma.xx, drift.t,

method = c("euler", "milstein", "KPS", "milstein2",

"cdist","ozaki","shoji","EA"),

alpha = 0.5, eta = 0.5, pred.corr = T, rcdist = NULL,

theta = NULL, model = c("CIR", "VAS", "OU", "BS"),

k1, k2, phi, max.psi = 1000, rh, A, M=1)

The sde.sim function

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For the OU process, dXt = −5Xtdt + 3.5dWt, it is as easy as

> d <- expression(-5 * x)

> s <- expression(3.5)

> sde.sim(X0=10,drift=d, sigma=s) -> X

> str(X)

Time-Series [1:101] from 0 to 1: 10 9.32 8.79 8.89 8.48 ...

The sde.sim function

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For the CIR model dXt = (6 − 3Xt)dt + 2√

XtdWt

d <- expression( 6-3*x )

s <- expression( 2*sqrt(x) )

sde.sim(X0=10,drift=d, sigma=s) -> X

or, via model name

sde.sim(X0=10, theta=c(6, 3, 2), model="CIR") -> X

or, via exact conditional distribution rcCIR (also implemented in sde)

sde.sim(X0=10, theta=c(6, 3, 2), rcdist=rcCIR, method="cdist") -> X

Also in the sde package

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The package also implements other estimation procedures

� estimating functions (linear, quadratic, martingale)

� GMM (but be careful, not really what you want to use with SDE!)

� approximate AIC statistics for model selection (sdeAIC)

� φ-divergence test statistics for parametric hypotheses testing (not in the book)

� change point (cpoint) analysis; both parametric and nonparametric

� non parametric estimation of drift (ksdrift) and diffusion (ksdiff) coefficients

� Markov Operator distance (MOdist) for clustering of SDE paths

The companion book: Simulation and Inference for Stochastic Differential Equations, with R

Examples, Springer (2008).

Limits of the sde package


Diffusions

Exact likelihood

Pseudo-likelihood


Hermite expansion

The Yuima Project

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� only deals with 1-dimensional SDE’s

� SDE only driven by a single Wiener noise

� no jumps

� essentially only Markovian SDE’s (via specification of cdist)

� (last but not least) written by a single person!

The Yuima Project


The Yuima Project

Multidimensional SDE:theoretical model

The general Levy SDE

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The Yuima Project Team


The Yuima Project



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M. Fukasawa (Osaka Univ.)

H. Hino (Waseda Univ., Tokyo)

S.M. Iacus (Milan Univ.)

K. Kengo (Tokyo Univ.)

H. Masuda (Kyushu Univ.)

Y. Shimitzu (Osaka Univ.)

M. Uchida (Osaka Univ.)

N. Yoshida (Tokyo Univ.)

. . . more to come

The yuima package1 is written by people working in mathematical statistics

and finance, who actively publish results in the field, have some knowledge of

R, and have the feeling on “what’s next” in the field.

Aims at filling the gap between theory and practice!1

The Yuima project is funded by the Japan Science Technology (JST) Basic Research Programs PRESTO, Grants-in-Aid for ScientificResearch No. 19340021, and the Global COE program “The research and training center for new development in mathematics” of GraduateSchool of Mathematical Sciences, University of Tokyo.

The yuima package goal: fill the gap between theory and practice


The Yuima Project



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The Yuima Project aims at implementing, via the yuima package, a very

abstract framework to describe probabilistic and statistical properties of

stochastic processes in a way which is the closest as possible to their

mathematical counterparts but also computationally efficient.

� it is an S4 package, where the basic class extends to SDE’s with jumps,

but soon to semimartingales (non Markov), Markov switching regime

processes, fBm driven SDE’s, etc.

� separates the data description from the inference tools and simulation

schemes

� the design allows for multidimensional, multi-noise processes

specification

� it includes a variety of tools useful in finance, like asymptotic expansion

of functionals of Levy processes via Malliavin calculus

The yuima package


The Yuima Project



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� we plan to add soon I/O functions to connect to other R packages

(including those presented at this workshop!) to avoid replications

� the package is designed to have three layers

� layer 1: is the very basic, low-level framework, (occasional) user

unfriendly. You have to specify the correct model and arguments;

� layer 2: macros and short-cuts for the common user (e.g. “please,

fit this Levy process with NIG structure”)

� layer 3: point & click interface

Layers 1 and 2 almost completed. Layer 3...

Let’s see how it is supposed to work.

The yuima package


The Yuima Project



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The basic object is an S4 object of class yuima, which contains the slots

yuima object

@ modelof class yuima.model: an abstractdescritiption of the model (e.g. SDE)

@ dataof class yuima.data is a wrapper forany data type zoo, xts, tseries,etc.

@ sampling

of class yuima.sampling: infor-mation about the sampling scheme(random, Poisson, deterministic, ticktimes)

@ characteristicof class yuima.characteristic:the characteristic triplet and more

The yuima package


The Yuima Project



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Each slots of the yuima object can be a list of yuima.* objects or an

empty slot.

For example, one can have a yuima object “foo” with only the slot data, and

other two yuima objects “bar” (and SDE without jumps) and “baz” (and SDE

with jumps) with only the description of two different models.

Then, we want to fit the data foo to each model bar and baz separately, for

comparisons, model selection, etc.

The yuima package offers several constructors to build each object and merge

them together.

The idea: first describe the model, then manipulate it.

The yuima.model constructor


The Yuima Project



25 / 41

To describe the following SDE: dXt = −3Xtdt + 11+X2

t

dWt, we use

> mod1 <- set.model(drift = "-3*x", diffusion = "1/(1+x^2)")

Solution variable (lhs) not specified. Trying to use state variables

> str(mod1)

Formal class ’yuima.model’ [package "yuima"] with 9 slots

..@ drift : expression((-3 * x))

..@ diffusion :List of 1

.. ..$ : expression(1/(1 + x^2))

..@ parameter :Formal class ’model.parameter’ [package "yuima"] with 4 slots

.. .. ..@ all : chr(0)

.. .. ..@ common : chr(0)

.. .. ..@ diffusion: chr(0)

.. .. ..@ drift : chr(0)

..@ state.variable : chr "x"

..@ time.variable : chr "t"

..@ noise.number : num 1

..@ equation.number: int 1

..@ dimension : int [1:4] 0 0 0 0

..@ solve.variable : chr "x"

Automatic parameter extraction


The Yuima Project



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To describe the following SDE: dXt = −θXtdt + 11+Xγ

t

dWt, we use

> mod2 <- set.model(drift = "-theta*x", diffusion = "1/(1+x^gamma)")

Solution variable (lhs) not specified. Trying to use state variables

> str(mod2)


..@ drift : expression((-theta * x))


.. ..$ : expression(1/(1 + x^gamma))


.. .. ..@ all : chr [1:2] "theta" "gamma"

.. .. ..@ common : chr(0)

.. .. ..@ diffusion: chr "gamma"

.. .. ..@ drift : chr "theta"

..@ state.variable : chr "x"


..@ noise.number : num 1


..@ dimension : int [1:4] 2 0 1 1

..@ solve.variable : chr "x"

Multidimensional SDE: theoretical model

27 / 41

dX1t = X2

t

∣

∣X1t

∣

∣

2/3dW 1

t ,

dX2t = g(t)dX3

t ,

dX3t = X3

t (µdt + σ(ρdW 1t +

√

1 − ρ2dW 2t ))

, (X10 , X2

0 , X30 ) = (1.0, 0.1, 1.0)

(1)

with µ = 0.1, σ = 0.2, ρ = −0.7 and g(t) = 0.4 + (0.1 + 0.2t)e−2t, for example, where

W = (W 1, W 2) is a 2-dim standard Brownian motion. Then, it corresponds to

Xt = X0 +

∫ t

0b(s, Xs)ds +

∫ t

0c(s, Xs)dWs (2)

with

b(s, x) =

0g(s)µx3

µx3

, c(s, x) =

x2|x1|2/3 0

g(s)σρx3 g(s)σ√

1 − ρ2x3

σρx3 σ√

1 − ρ2x3

for x = (x1, x2, x3).

Multidimensional SDE: R code

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> mu <- 0.1; sig <- 0.2; rho <- -0.7

> g <- function(t) 0.4 + (0.1 + 0.2*t)* exp(-2*t)

> diff.coef.1 <- function(t, x1, x2, x3) {

+ ret <- 0

+ if(x1 > 0 && x2 > 0) ret <- x2*exp(log(x1)*2/3)

+ ret

+ }

> diff.coef.2 <- function(t, x1, x2, x3=0) {

+ ret <- 0

+ if(x3 > 0) ret <- rho*sig*x3

+ ret

+ }

> diff.coef.3 <- function(t, x1, x2, x3) {

+ ret <- 0

+ if(x3 > 0) ret <- sqrt(1-rho^2)*sig*x3

+ ret

+ }

> diff.coef.matrix <- matrix(c("diff.coef.1(t,x1,x2,x3)", "diff.coef.2(t,x1,x2,x3) * g(t)",

+ "diff.coef.2(t,x1,x2,x3)", "0", "diff.coef.3(t,x1,x2,x3)*g(t)", "diff.coef.3(t,x1,x2,x3)"),

+ 3, 2)

> sabr.mod <- set.model(drift = c("0", "mu*g(t)*x3", "mu*x3"), diffusion = diff.coef.matrix,

+ state.variable = c("x1", "x2", "x3"), solve.variable = c("x1", "x2", "x3"))

Multidimensional SDE

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> str(sabr.mod)


..@ drift : expression((0), (mu * g(t) * x3), (mu * x3))


.. ..$ : expression(diff.coef.1(t, x1, x2, x3), 0)

.. ..$ : expression(diff.coef.2(t, x1, x2, x3) * g(t), diff.coef.3(t, x1, x2, x3) * g(t))

.. ..$ : expression(diff.coef.2(t, x1, x2, x3), diff.coef.3(t, x1, x2, x3))


.. .. ..@ all : chr "mu"

.. .. ..@ common : chr(0)

.. .. ..@ diffusion: chr(0)

.. .. ..@ drift : chr "mu"

..@ state.variable : chr [1:3] "x1" "x2" "x3"


..@ noise.number : int 2


..@ dimension : int [1:4] 1 0 0 1

..@ solve.variable : chr [1:3] "x1" "x2" "x3"

Simulation of paths

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# Sets the sampling scheme

> yuima.samp <- set.sampling(Terminal=1,division=1000)

# build the yuima object: put together model and sampling

> my.yuima <- set.yuima(model=sabr.mod, sampling=yuima.samp)

# Solve/Simulate SDE using Euler-Maruyama method

# add the simulated data to the yuima object

> my.sde <- simulate(my.yuima, xinit=c(1.0,0.1,1.0))

# and a plot method exists inherited from ‘zoo’ package

> plot(my.sde)

The newly generated yuima object

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> str(my.sde)

Formal class ’yuima’ [package "yuima"] with 4 slots

..@ data :Formal class ’yuima.data’ [package "yuima"] with 2 slots

.. .. ..@ original.data: mts [1:1001, 1:3] 1 0.998 0.999 0.996 1.001 ...

.. .. .. ..- attr(*, "dimnames")=List of 2

.. .. .. .. ..$ : NULL

.. .. .. .. ..$ : chr [1:3] "Series 1" "Series 2" "Series 3"

.. .. .. ..- attr(*, "tsp")= num [1:3] 0 1 1000

.. .. .. ..- attr(*, "class")= chr [1:2] "mts" "ts"

.. .. ..@ zoo.data :List of 3

.. .. .. ..$ Series 1:zooreg series from 0 to 1

Data: num [1:1001] 1 0.998 0.999 0.996 1.001 ...

Index: num [1:1001] 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 ...

Frequency: 1000


Data: num [1:1001] 0.1 0.104 0.106 0.106 0.103 ...

Index: num [1:1001] 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 ...

Frequency: 1000


Data: num [1:1001] 1 1.01 1.01 1.01 1.01 ...

Index: num [1:1001] 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 ...

Frequency: 1000

..@ model :Formal class ’yuima.model’ [package "yuima"] with 9 slots

.. .. ..@ drift : expression((0), (mu * g(t) * x3), (mu * x3))

.. .. ..@ diffusion :List of 3

.. .. .. ..$ : expression(diff.coef.1(t, x1, x2, x3), 0)

.. .. .. ..$ : expression(diff.coef.2(t, x1, x2, x3) * g(t), diff.coef.3(t, x1, x2, x3) * g(t))

.. .. .. ..$ : expression(diff.coef.2(t, x1, x2, x3), diff.coef.3(t, x1, x2, x3))

.. .. ..@ parameter :Formal class ’model.parameter’ [package "yuima"] with 4 slots

.. .. .. .. ..@ all : chr "mu"

.. .. .. .. ..@ common : chr(0)

.. .. .. .. ..@ diffusion: chr(0)

.. .. .. .. ..@ drift : chr "mu"

.. .. ..@ state.variable : chr [1:3] "x1" "x2" "x3"

Motivation

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It is better than the sde package because it allows for multidimensional specification but also

provide a formal definition of an SDE.

But we need more that this!

Motivation

32 / 41

It is better than the sde package because it allows for multidimensional specification but also

provide a formal definition of an SDE.

But we need more that this! We need at least Levy processes

Asset prices (left) do not evolve continuously, they exhibit jumps or spikes and log returns (right)

are non-gaussian, skewed, heavy tails

100

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Oct 1997 Oct 1998 Oct 1999 Oct 2000 Oct 2001 Oct 2002 Oct 2003 Oct 2004

USD/JPY

!0.02 !0.01 0.0 0.01 0.02

020

40

60

80

The general L evy SDE

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In general the sde.model describes the following d-dimensional SDE:

dXt = a(Xt)dt + b(Xt)dWt

+

∫

|z|>1c(Xt−, z)µ(dt, dz) +

∫

0<|z|≤1c(Xt−, z){µ(dt, dz)− ν(dz)dt},

X0 = x0, (3)

where the ingredients are given as follows.

� the initial random variable X0 ∈ Rd;

� an rW -dimensional standard Wiener process W ;

� a random measure µ on (0,∞) × Rrµ associated with jumps of X , that is,

µ(dt, dz) =∑

s>0

1{∆Zs 6=0}δ(s,∆Zs)(dt, dz), (4)

where δ denotes the Dirac measure

The general L evy SDE

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� Z the driving pure-jump Levy process of the form

Zt =

∫ t

0

∫

|z|≤1z{µ(ds, dz) − ν(dz)ds} +

∫ t

0

∫

|z|>1zµ(ds, dz); (5)

� a Levy measure ν, which is any measure satisfying ν({0}) = 0 and∫

(1 ∧ |z|2)ν(dz) < ∞;

� the coefficient functions a : Rd → R

d, b : Rd → R

d ⊗ Rrw , and c : R

d × Rrµ → R

d.

Input of the jump part

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> set.model( drift = "a(x,theta)", diffusion = "b(x,sigma)",

+ jump.coeff = "c(x,z,xi)", measure = list(*),

+ measure.type = string(**),

+ state.var = c("x"), jump.var = c("z"), time.var = "t",

+ solve.var="c("x") )

where measure and measure.type are strictly related. For example, suppose that Z is a

gamma process with Levy density

f(z) =a

ze−bz

1(0,∞)(z).

Then Zt ∼ Γ(at, b), therefore one uses

> set.model(measure="dgamma(a*t,b)", measure.type="increment", ...)


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Suppose that νη(dz) = η0 × Fη(dz), where η0 ≥ 0 is a constant and Fη(z) is a distribution

function (d.f.) of jump size with a parameter vector η = (η1, η2, . . .). This is the Compound

Poisson formulation of a Levy measure.

In this case users can input the intensity η0 and the d.f. Fη(z) separately as follows:

> my.meas <- list( intensity=eta0, df=list(F(z,eta1,eta2,...))

where eta0 and F() are defined somewhere in the R workspace and

> set.model( measure = my.meas, measure.type = "CP", ... )


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Or one can directly input the density of the Levy measure ν, i.e.

νη(dz) = f(z, η) dz.

In this case users can input the intensity η0 and the d.f. Fη(z) separately as follows:

> set.model(measure = list(density=f(z,eta)), measure.type = "density", ... )

or

> F <- function(z,eta) f(z,eta)

> set.model(measure = list(df=F(z,eta)), measure.type = "density", ... )

Many other options and already coded typical Levy measures are implemented in the yuima

package (e.g. NIG, stable, tempered stable, bilateral Gamma, etc.)

Simulation and Inference

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Once a yuima model is available, it can be passed to the simulate method or to some

estimation methods along with the data.

When passed to simulate, the output is a new yuima.object with both the yuima.model,

yuima.data and yuima.sampling slots.

There is a function called simulate.levy which is a shortcut function, used to simulate Levy

paths according to the above specifications.

For the inference and simulation, the methods are supposed to work, by default, using the best

theory available for the data and the model, up to current literature, unless the user specifies

differently.

e.g. in the high frequency case the limiting distributions are of mixed-normal type, so confidence

intervals and standard errors are more delicate to handle

Toy examples

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Just to prove different aspects and flexibility of this approach, we have implemented several

methods

� the covariance estimator of Yoshida-Hayashi for multidimensional Ito processes with

asynchronous data.

� quasi-likelihood estimation for multidimensional diffusions with jumps

� asymptotic expansion of functionals of the solutions of SDE’s with small noise (useful in

option pricing)

� change point estimation for multidimensional Ito processes

� Bayes type estimators

� asymptotic expansion of the distribution of estimators (for small sample asymptotics)

A formula-type interface

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We also plan to add the formula interface to mimic model specification in R. A few working

examples are

dXt = b(Xt)dt + σ(Xt)dWt

passed to set.model as

dX ~ b(X)*dt + s(X)*dW

or, the following stochastic volatility model

dXt = b(Xt, θ, t)dt + σ(Xt, Yt, β)dWt

dYt = d(Yt, τ)dZt

as

dX ~ b(X,theta,t)*dt + s(X,Y,beta)*dW

dY ~ d(Y,tau)*dZ

First release?


The Yuima Project



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Roadmap

� we plan to release openly the basic infrastructure by the end of this year

� implement several basic methods in early 2010

� open development to potential new contributors (if there is enough

interest) mid 2010

� write the point & click layer 3 (...one day!)

s.m. iacus (university of milan) - welcome to rmetrics ... · from the sdepackage to the yuima...

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