final report - probability.ca

If one considers a small time increment dt, V t+dt − V t is approximately Gaussian with mean and variance given
by the Euler-Maruyama approximation
V t+dt ≈ V t + b (V t ; θ) dt + σ (V t ; θ) √ dt · Y, Y ∼ N (0, 1) .
Higher order approximations are also available. The exact dynamics of the diffusion process are govern by its
transition density:
p t (v, w; θ) = P [V t ∈ dw|V 0 = v; θ] /dw, t > 0, w, v ∈ R. (3.3)
Suppose that there exists a finite set of observations for the diffusion process above, i.e., one has the collection of
time instances 0 < t 0 < t 1 < · · · < t n with corresponding observations given at v = {V t0 , V t1 , · · · , V tn }. Denote the
time increment between consecutive observations by ∆t i = t i − t i−1 for 1 ≤ i ≤ n.
Then the log-likelihood of the data set v is given by
n∑
(
l (θ|v) = l i (θ) , l i (θ) := log p ∆ti Vti−1 , V ti ; θ ) .
i=1
Unfortunately, other than a few special cases, (e.g., when the diffusion follows a geometric Brownian motion),
this transition density is rarely available in an analytically tractable form. Hence, deriving maximum likelihood
estimates for generic discretely observed diffusions is a challenging problem. One approach started by Aït-Sahalia
is to expand the transition density using Hermite polynomials. It is shown that this closed-form approximation is
an unbiased estimator of the true (unknown) transition density. A second approach is to use simulation to obtain
the MLE estimates. Exact simulation of diffusion sample paths is a recent approach that is feasible for a wide
class of diffusion processes, although not all diffusions can be handled by this method. The Exact Algorithm was
introduced by Beskos [BPR06], and it uses technique called retrospective sampling.
First it is necessary to apply a variance-stabilization transformation to the diffusion process (3.2). Consider the
Lamperti transform, V s ↦→ η (V s ; θ) = X s ,
η (u; θ) =
ˆ u
1
σ (z; θ) dz.
If σ (·; θ) is continuously differentiable, then we can apply Itô’s rule to obtained the transformed diffusion’s SDE:
where X 0 = x = η (V 0 ; θ) for s ∈ [0, t], and
dX s = α (X s ; θ) ds + dB s ,
α (u; θ) = b { η −1 (u; θ) ; θ }
σ {η −1 (u; θ) θ; θ} − σ′ { η −1 (u; θ) ; θ } /2, u ∈ R;
η −1 denotes the inverse of the Lamperti transformation, and σ ′ denotes the derivative w.r.t. the space variable.
Assume that α (·; θ) is continuously differentiable, ( α 2 + α ′) (·; θ) is bounded below, and the Girsanov formula for
X, given by
is a martingale w.r.t. Wiener measure. Define
{ˆ t
exp α (ω s ; θ) dω s − 1
0
2
A (u; θ) =
ˆ t
0
ˆ u
α (z; θ) dz
}
α 2 (ω s ; θ) ds ,
to be any anti-derivative of α. Then the transition density of X is defined as
˜p t (x, y; θ) = P [X t ∈ dy|X 0 = x; θ] /dy, t > 0, x, y ∈ R.
Let Q (t,x,y)
θ
denote the distribution of the process X conditioned to start at X 0 = x and finish at X t = y for some
fixed x, y, and W (t,x,y) be the probability measure for the corresponding Brownian bridge. The objective of the EA
is to obtain a rejection sampling algorithm to obtain draws from Q (t,x,y)
θ
. The following Proposition provides the
necessary methodology for this. Let N u (0, t) denote the density of a normal random variable with mean zero and
variance t evaluated at u ∈ R.
Proposition 3.1. Under the conditions above, Q (t,x,y)
θ
is absolutely continuous with respect to W (t,x,y) with density:
dQ (t,x,y)
θ
dW = N {
ˆ
y−x (0, t)
t
(t,x,y) ˜p t (x, y; θ) exp 1 (
A (y; θ) − A (x; θ) − α 2 + α ′) }
(ω s ; θ) ds .
2
0
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