Exact Simulation in a Nutshell
1
Murray Pollock , Adam Johansen & Gareth Roberts
1 - Overview
. . . in a nutshell. . .
2 - Summary of Key Methodology
1.1 - The Goal. . . ??? Evaluate with certainty whether or not a given
2.1 - Exact Algorithm (EA). . . A diffusion path space retrospective
jump diffusion sample path crosses a barrier.
rejection sampler which characterises entire (accepted) sample paths in the
form of a finite dimensional skeleton, composed of the sample path at a
finite collection of intermediate points and spatial information.
2.2 - -Strong Simulation (SS). . . Methodology for constructing upper and lower convergent dominating processes (X ↓ and X ↑), which enfold
1.2 - Main Difficulties. . . ??? Sample paths are infinite dimensional ran-
almost surely sample paths over some finite interval.
dom variables. Discretisation schemes introduce error and don’t sufficiently
2.3 - Sufficient Conditions. . . β ∈ C 1, σ ∈ C 2 and strictly positive, λ
characterise sample paths to determine barrier crossing.
locally bounded, linear growth and Lipschitz continuity coefficient conditions.
1.3 - Applications. . . Monte Carlo Integration, Option Pricing, Simulat-
2.4 - Further Details. . . arXiv 1302.6964 or scan QR code!
ing First Hitting Times, Killed Diffusions, Rare Events. . .
3 - Key Ideas
Inversion Sampling
Retro. Bernoulli Sampling
1
1
Bernoulli Sampling
1
Rejection Sampling
Over Estimate
w1 ≤
u ∼ U [0, 1]
π
π(X1 )
q(X1 )M
Probability
π(X1 )
q(X1 )M
(Unknown) p
w2 >
Probability
u ∼ U [0, 1]
Fπ (X)
Density
q·M
No Need To Evaluate Further
u<p
Fπ−1 (u)
Cauchy Sequence Terms (k)
Cauchy Sequence Terms (k)
“Regular” EA Skeleton
Adaptive EA Skeleton
X
Key Idea: Find and simulate some finite dimensional
4 - The Exact Algorithm
F, such that an
auxiliary random variable F := F(X) ∼
Idealised Exact Algorithm
unbiased estimator of the acceptance probability can be
x .
0,T
constructed which can be evaluated using only a finite
x
d
T
0,T
1
x (X) :=
2 – With probability PP0,T
x (X) ∈ [0, 1]
M dP0,T
x .
3 – X|(I = 1) ∼ T0,T
X
Implementable Exact Algorithm
1 – Simulate XT := y ∼ h.
X
yx
dimensional subset of the proposal sample path. . .
set I = 1.
1 –T
P
x,y
0,T
F.
y
3 – Simulate X f ∼
x – target law.
0,T
x
2 – Simulate F ∼ F.
Key Points
2 –P
u∈
/ (S2k , S2k+1 )
X2
X1
X
1 – Simulate X ∼ P
0
0
0
Under Estimate
x
4 – With probability PP0,T
| F (X) accept, else reject and
x – equivalent (+ tractable) proposal law.
0,T
return to 1.
x
dT0,T
3 – dPx (X) bounded (by M < ∞).
0,T
s
x,y
5 – *** Simulate X c ∼ P0,T (X f , F) as required. ***
s
t
t
Time
Time
5 - -Strong Simulation
Intermediate Augmentation
↑
123456
X
y
y
ℓ↑s,t
9
1
3
s
t
↑
↓
X − X ≤ 0.2
Idealisation
Time
ψ2
s
t
Time
t
4
2
−4
−2
0
X
2
−2
−4
ψ1 ψ2
0
Time
ψ3 ψ42π
ψ1
0
Time
6.3 - Example 3: Jump Diffusion Intersection
ψ2
2π
ψ12π
0
Time
Time
Barrier Not Crossed
Barrier Crossed
6.4 - Example 4: Circular Barrier
Barrier Not Crossed
Barrier Crossed
Idealisation
1.5
0.0
X2
0.0
X2
0.0
X2
0
0
X
X
0
−1.5
−1.5
−1.5
−2
−2
−2
−4
X
2
2
2
1.5
4
4
1.5
4
Idealisation
Barrier Crossed
0
X
2
−2
0
X
−2
ψ1
s
t
Time
Barrier Not Crossed
4
1
Time
−4
ψ2
t
−3
−3
−3
ψ1
s
t
6.2 - Example 2: Nonlinear Two Sided Barrier
−1
X
−2
−1
X
−1
−2
X
s
x
67
4
2
0
1
34
1
8
Time
0
1
78
q
s
6.1 - Example 1: -Strong Simulation of Jump Diffusions
0
5
2
↕
ℓs,t
45
9
ℓ↓s,t
6
Time
↑
↓
X − X ≤ 0.4
4
3
4
x = υ↓ υ↑
2
x
y
Time
y
1
X
w
X
ℓ↑∗
s,t
12
ℓ↓s,t
x
ℓ↑
ℓ↓
Sample path skeleton
456
x
Density
y
X
3
t
789
123
789
s
Augmented Skeleton
Layer Refinement
↑
↓∗
υs,t
υs,t
Layer Dissection
↓
↕
υs,t
υs,t υs,t
Adaptive EA Skeleton
0
ψ1u2
ψ1l
Time
u
0 ψ1
2
Time
0
ψ1l ψ1u
ψ2l
Time
2
−1.5
0.0
X1
1.5
−1.5
0.0
1.5
−1.5
X1
“O God, I could be bounded in a nut shell and count myself a king of infinite space, were it not that I have bad dreams,” — William Shakespeare, Hamlet Act II, Scene II
1
Corresponding & Presenting Author — a: Department of Statistics, University of Warwick, Coventry, UK, CV4 7AL — e: m.pollock@warwick.ac.uk — w: www.warwick.ac.uk/mpollock
0.0
X1
1.5