8.2 Key concepts and results used throughout Part II
8.2.1 Brownian motion and some related probability distributions
Definition 8.2. Brownian motion with drift
A real-valued stochastic process W = (Wt)t∈[0,∞) is called Brownian motion with driftν ∈Rif it holds
• W0 = 0,
• W has independent, stationary increments,
• W has Gaussian increments Wt+u−Wt∼N(ν·u, u) and
• W has continuous sample paths: t→Wt is continuous almost surely.
W is a standard Brownian motion if ν= 0.
The first passage time distribution of a positive border by a Brownian motion with positive drift has been studied since more than 100 years – it was first reported by Schr¨odinger (1915) – and been given the nameinverse Gaussian distribution due to its similarity to the Gaussian distribution (Tweedie, 1945). For a historical survey about research related to this distribution, we refer to Chapter 1 in Seshadri (1993) and for a broad overview about application areas to Part II in Seshadri (1999). Here, we first state the definition of the inverse Gaussian distribution and recall results about the first moments. As assumed distribution of dominance times this distribution will be of major importance for the Hidden Markov as well as for the Hierarchical Brownian Model.
Definition 8.3. Inverse Gaussian distribution
The inverse Gaussian (IG) distribution is for the mean parameter µ > 0 and the shape parameterλ >0 given by the density
f(x) = ( λ
2πx3
1/2
exp−λ(x−µ)2
2µ2x
, if x >0,
0, else.
Proposition 8.4. Moments of the IG distribution
Let X be IG distributed with parameters µ, λ >0. Then it holds E[X] =µ, Var(X) =µ3/λ.
Proof: These results are, for example, given by Seshadri (1993).
Remark 8.5. Reparametrization of the IG distribution used in this thesis
To get a direct overview of the moments of the IG distribution, we use the standard deviation σ=p
µ3/λinstead of λ as second parameter when parametrizing the IG distribution in the rest of the thesis: we denote an IG distribution with mean µ and standard deviation σ by IG(µ, σ) with the density fµ,σIG.
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8. Introduction
Figure 8.5 gives an impression of the behavior of the inverse Gaussian distribution for different values of µandσ. With µincreasing the peak of the distribution naturally shifts to the right and the density curve gets more and more bell shaped as the variance can be distributed around both sides ofµ and not only to the right side as for smallµ. This fact also explains why for a largeσ the width of the distribution increases withµ.
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
0 0.1 0.2
0 100 200 300
x
fµ, σIG (x)
5 30 80
5
50
100
200
µ σ
Figure 8.5: Density of the inverse Gaussian distribution fµ,σIG(x) for different mean µ (rows) and standard deviation σ (columns).
8. Introduction
Now we state the relation of the inverse Gaussian distribution to the Brownian motion.
Proposition 8.6. Relation of the IG distribution to the Brownian motion
Leta >0be a fixed level and W := (Wt)t≥0 be a Brownian motion with drift ν >0. Then, the first passage time ofaby W is distributed according to an inverse Gaussian random variable Ta:
Ta:= inf{t >0 :Xt=a} ∼IG(a/ν,p a/ν3).
Proof: Compare, e.g., Seshadri (1993) or M¨orters and Peres (2010).
Further, the inverse Gaussian distribution can be expressed as a member of the exponential family, which we will need to state results about asymptotic normality of estimators later on.
Note that the density of members of the exponential family with parameters Θ can be written as
f(x|Θ) =h(x) exp (η(Θ)t(x)−A(Θ)), (8.1) whereh(x), t(x), η(Θ) and A(Θ) are known functions (e.g., Lehmann and Casella, 1998).
Corollary 8.7. IG distribution as member of the exponential family
The IG(µ, σ) distribution is a representative of the exponential family with Θ = (µ, σ) and h(x) = 1
√
2πx3, A(Θ) =−1 2log
µ3 σ2
−µ2 σ2, t(x) =
x,1
x T
, η(Θ) =
− µ
2σ2,− µ3 2σ2
.
Proof: Follows by rearranging the density of the inverse Gaussian distribution.
Connection of the IG distribution to the Gamma distribution As we use both – the Gamma and the IG distribution – to model dominance times, we are interested in properties these two probability distributions share. The IG and the Gamma distribution are both special cases of the generalized inverse Gaussian distribution (Johnson et al., 1994) which cannot be converted into each other. However, in particular for a small CV=σ/µthe densities of the Gamma and the IG distributions are difficult to distinguish per eye (panels A and D in Figure 8.6). With an increasing CV the differences between the two densities tend to be more obvious, where the behavior for a small and largeµis comparable (compare A-C with D-F).
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8. Introduction
fΓ(x) fIG(x)
0 5 10 15 20 25 0.0
0.2 0.4 0.6 0.8
density
CV=0.1
A
0 5 10 15 20 25 0.0
0.2 0.4 0.6 0.8
CV=0.5
B
0 5 10 15 20 25 0.0
0.2 0.4 0.6 0.8
CV=1.0
µ=5
C
0 400 800 1200
0.00 0.01 0.02
density
x
D
0 400 800 1200 0.00
0.01 0.02
x
E
0 400 800 1200 0.00
0.01 0.02
x
µ=180
F
Figure 8.6: Comparison between the densities of the IG (green) and the Gamma (cyan) distribution. The mean is µ= 5(A-C) and µ= 180 (D-F). The CV is 0.1 (A,D), 0.5 (B,E) and 1 (C,F). In A and D the lines are not drawn continuously as otherwise only one density would be visible.
8. Introduction
Reciprocal inverse Gaussian distribution To derive results about the exact and asymp-totic distribution of estimators of parameters of the IG distribution, we require the reciprocal of the IG distribution. IfX is IG(µ, σ) distributed, the distribution of 1/X is called reciprocal inverse Gaussian(RIG) with parameters µ and σ (Seshadri, 1993). We state its density and expectation in the next proposition. These are given in Proposition 2.18 and Table 2.4 of Seshadri (1993), respectively.
Proposition 8.8. Density and expectation of the RIG distribution Let X be IG(µ, σ)-distributed. The density of 1/X is given by
f(x) =f1/µ,µ/σIG 2∗fµΓ3/(4σ2),µ6/(8σ4)(x), if x >0,
and 0 otherwise. The sign∗ denotes the convolution of the inverse Gaussian and the Gamma density. The expected value of 1/X is
E 1
X
= 1 µ +σ2
µ3.
Normal-inverse Gaussian distribution In the context of the HBM, the position of a Brownian motion with drift at a point in time determined by a normal distributed random variable is important to derive transition probabilities between hidden states. The distribution of this position is given by thenormal-inverse Gaussian distribution(NIG) as is stated in Corollary 8.10. First, we introduce this distribution, which is a special case of the generalized hyperbolic distributions (Barndorff-Nielsen, 1978). The NIG distribution has four parameters and is fittable to a wide range of data including fat tails and skewness. Its application area mainly is in finance, e.g., for stochastic volatility modeling (Barndorff-Nielsen, 1997). Second, Corollary 8.10 is shown.
Definition 8.9. Normal-inverse Gaussian distribution Let Y be an IG
√ δ α2−β2,
√ δ (α2−β2)3/4
-distributed random variable. A random variable X follows the normal-inverse Gaussian (NIG) distribution with parametersξ, α, β, δ if
X|Y =y∼N(ξ+βy, y), where the parameters satisfy the conditions α≥ |β|and δ≥0.
The density of a NIG(ξ, α, β, δ)-distributed random variable is given by
f(x) = αδ π exp
δp
α2−β2+β(x−ξ) K1
αp
δ2+ (x−ξ)2 pδ2+ (x−ξ)2 , with K1(x):= 12
∞
R
0
exp −12x t+t−1
dt as the modified Bessel function of the third kind and order 1 (e.g., Watson, 1995). We term the density in this thesis as Ψξ,α,β,δ(x).
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8. Introduction
Corollary 8.10. Relation of the NIG distribution to the Brownian motion
Let(Yt)t≥0 be a Brownian motion with drift νY >0 and starting position Y0 = 0. Moreover, bY >0 is a border. Let (Xt)t≥0 be another Brownian motion independent of(Yt)t≥0 with drift νX >0 andX0 = 0. Define T := inf{t≥0 :Yt=bY}. It holds
XT ∼NIG(0, q
νX2 +νY2, νX, bY).
Proof: Due to Proposition 8.6, T is IG distributed with parameters bY/νY and q
bY/νY3. Reparameterizing this in terms ofξ, α, β, δ of Definition 8.9 yields the following equations
(I) δ/(α2−β2)1/2 =bY/νY, (II) δ/(α2−β2)3/2 =bY/νY3, (III) β =νX.
As X starts in 0, it holds ξ= 0. By (III) we directly have β =νX. Dividing (I) by (II) yields νY2 =α2−β2 (IV). Plugging this in (I) we obtain δ =bY. Plugging (III) in (IV) we moreover obtainα=
q
νX2 +νY2.
In the thesis the normal-inverse Gaussian distribution generally emerges from its relation to the Brownian motion. Hence, the parameterξ vanishes, and for simplification we term for eachx∈R: Ψα,β,δ(x):= Ψ0,α,β,δ(x) unless stated otherwise.