Filtering for discretely observed jump diffusions
Murray Pollock, Adam Johansen, Gareth Roberts
Dept. Of Statistics, University Of Warwick
murray.pollock@warwick.ac.uk
www.warwick.ac.uk/mpollock
September 20th, 2012
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Problem Outline
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Framework I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Framework II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Framework III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Framework IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Framework V
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? V
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? VI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? VII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? VIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? IX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? X
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? XI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? XII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? XIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What is a Diffusion? XIV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
The Problem
- Key Problem
- Not possible to simulate entire diffusion sample paths. . .
- . . . Finite representation.
- . . . Transition density inaccessible.
- What can we simulate?
- ...
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
The Problem
- Key Problem
- Not possible to simulate entire diffusion sample paths. . .
- . . . Finite representation.
- . . . Transition density inaccessible.
- What can we simulate?
- ...
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
The Problem
- Key Problem
- Not possible to simulate entire diffusion sample paths. . .
- . . . Finite representation.
- . . . Transition density inaccessible.
- What can we simulate?
- ...
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
The Problem
- Key Problem
- Not possible to simulate entire diffusion sample paths. . .
- . . . Finite representation.
- . . . Transition density inaccessible.
- What can we simulate?
- ...
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
The Problem
- Key Problem
- Not possible to simulate entire diffusion sample paths. . .
- . . . Finite representation.
- . . . Transition density inaccessible.
- What can we simulate?
- ...
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? V
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? VI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? VII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? VIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? IX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? X
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XIV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XVI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XVII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XVIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XIX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
What can we simulate? XX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Discretisation
- What about everything else?
- “Discretize” diffusion dynamics.
- Euler discretization, for instance (PJS09, Jav05).
(
Xt +∆t =
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t )
µ (Xt ) ∆t + σ (Xt ) N(0, ∆t ) + ν (Xt − )
w.p. e −λ(Xt − )∆t
w.p. 1 − e −λ(Xt − )∆t
- Miltstein, Shoji & Ozaki,. . .
- Drawbacks?
-
Implicit error.
Computationally expensive.
Moving beyond ‘bootstrap’?
Performance evaluated using (finely discretised) simulated data.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Talk Outline
- Problem Outline & Motivation.
- Area Estimator
- Exact Algorithm (BPRF06, BPR08, PJR12a)
- Jump Exact Algorithm (CR10, Gon11)
- Filtering for Jump Diffusions (FPR08, FPRS10, PJR12b)
- Other (Related) Work. (PJR12a, PJR12b)
- Questions (& Hopefully) Answers.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Talk Outline
- Problem Outline & Motivation.
- Area Estimator
- Exact Algorithm (BPRF06, BPR08, PJR12a)
- Jump Exact Algorithm (CR10, Gon11)
- Filtering for Jump Diffusions (FPR08, FPRS10, PJR12b)
- Other (Related) Work. (PJR12a, PJR12b)
- Questions (& Hopefully) Answers.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Talk Outline
- Problem Outline & Motivation.
- Area Estimator
- Exact Algorithm (BPRF06, BPR08, PJR12a)
- Jump Exact Algorithm (CR10, Gon11)
- Filtering for Jump Diffusions (FPR08, FPRS10, PJR12b)
- Other (Related) Work. (PJR12a, PJR12b)
- Questions (& Hopefully) Answers.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Talk Outline
- Problem Outline & Motivation.
- Area Estimator
- Exact Algorithm (BPRF06, BPR08, PJR12a)
- Jump Exact Algorithm (CR10, Gon11)
- Filtering for Jump Diffusions (FPR08, FPRS10, PJR12b)
- Other (Related) Work. (PJR12a, PJR12b)
- Questions (& Hopefully) Answers.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Talk Outline
- Problem Outline & Motivation.
- Area Estimator
- Exact Algorithm (BPRF06, BPR08, PJR12a)
- Jump Exact Algorithm (CR10, Gon11)
- Filtering for Jump Diffusions (FPR08, FPRS10, PJR12b)
- Other (Related) Work. (PJR12a, PJR12b)
- Questions (& Hopefully) Answers.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Talk Outline
- Problem Outline & Motivation.
- Area Estimator
- Exact Algorithm (BPRF06, BPR08, PJR12a)
- Jump Exact Algorithm (CR10, Gon11)
- Filtering for Jump Diffusions (FPR08, FPRS10, PJR12b)
- Other (Related) Work. (PJR12a, PJR12b)
- Questions (& Hopefully) Answers.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Talk Outline
- Problem Outline & Motivation.
- Area Estimator
- Exact Algorithm (BPRF06, BPR08, PJR12a)
- Jump Exact Algorithm (CR10, Gon11)
- Filtering for Jump Diffusions (FPR08, FPRS10, PJR12b)
- Other (Related) Work. (PJR12a, PJR12b)
- Questions (& Hopefully) Answers.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Area Estimator
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Expectations of Positive RVs
Goal. . .
Evaluate E(P ) for some r.v. P ∈ R+
Suppose for now. . .
P ∈ [0, 1]
u ∼ U [0, 1]
Consider a biased coin Cp . . .
Heads (1) if u ≤ P
Tails (0) if u > P
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Expectations of Positive RVs
Goal. . .
Evaluate E(P ) for some r.v. P ∈ R+
Suppose for now. . .
P ∈ [0, 1]
u ∼ U [0, 1]
Consider a biased coin Cp . . .
Heads (1) if u ≤ P
Tails (0) if u > P
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Expectations of Positive RVs
Goal. . .
Evaluate E(P ) for some r.v. P ∈ R+
Suppose for now. . .
P ∈ [0, 1]
u ∼ U [0, 1]
Consider a biased coin Cp . . .
Heads (1) if u ≤ P
Tails (0) if u > P
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Expectations of Positive RVs
Why is this interesting. . . ?
P Cp = 1 = E 1 u ≤ P
!
= E E 1 u ≤ P P
!
= E P Cp = 1P
=EP
- P(X ) = E(1X )
- Tower Property
- P(X ) = E(1X )
To estimate E P unbiasedly we can simply throw a Cp coin
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator
Let’s consider an example. . . Suppose we want to evaluate,
E

( Z


P := E exp −
T
n
If 0 ≤ f (t ) ≤ M then P := exp −
(P ) = P ⇒ find and flip Cp
E
RT
0
0
)

f (t ) dt 
o
f (t ) dt ∈ [0, 1]
Consider a Poisson process with instantaneous rate f (t ) on [0, T ],
( Z
P N = 0 = exp −
T
)
f (t ) dt
0
PN =0 ≡P=EP
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator
Let’s consider an example. . . Suppose we want to evaluate,
E

( Z


P := E exp −
T
n
If 0 ≤ f (t ) ≤ M then P := exp −
(P ) = P ⇒ find and flip Cp
E
RT
0
0
)

f (t ) dt 
o
f (t ) dt ∈ [0, 1]
Consider a Poisson process with instantaneous rate f (t ) on [0, T ],
( Z
P N = 0 = exp −
T
)
f (t ) dt
0
PN =0 ≡P=EP
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator
Let’s consider an example. . . Suppose we want to evaluate,
E

( Z


P := E exp −
T
n
If 0 ≤ f (t ) ≤ M then P := exp −
(P ) = P ⇒ find and flip Cp
E
RT
0
0
)

f (t ) dt 
o
f (t ) dt ∈ [0, 1]
Consider a Poisson process with instantaneous rate f (t ) on [0, T ],
( Z
P N = 0 = exp −
T
)
f (t ) dt
0
PN =0 ≡P=EP
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator
Let’s consider an example. . . Suppose we want to evaluate,
E

( Z


P := E exp −
T
n
If 0 ≤ f (t ) ≤ M then P := exp −
(P ) = P ⇒ find and flip Cp
E
RT
0
0
)

f (t ) dt 
o
f (t ) dt ∈ [0, 1]
Consider a Poisson process with instantaneous rate f (t ) on [0, T ],
( Z
P N = 0 = exp −
T
)
f (t ) dt
0
PN =0 ≡P=EP
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator
Let’s consider an example. . . Suppose we want to evaluate,
E

( Z


P := E exp −
T
n
If 0 ≤ f (t ) ≤ M then P := exp −
(P ) = P ⇒ find and flip Cp
E
RT
0
0
)

f (t ) dt 
o
f (t ) dt ∈ [0, 1]
Consider a Poisson process with instantaneous rate f (t ) on [0, T ],
( Z
P N = 0 = exp −
T
)
f (t ) dt
0
PN =0 ≡P=EP
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator
Let’s consider an example. . . Suppose we want to evaluate,
E

( Z


P := E exp −
T
n
If 0 ≤ f (t ) ≤ M then P := exp −
(P ) = P ⇒ find and flip Cp
E
RT
0
0
)

f (t ) dt 
o
f (t ) dt ∈ [0, 1]
Consider a Poisson process with instantaneous rate f (t ) on [0, T ],
( Z
P N = 0 = exp −
T
)
f (t ) dt
0
PN =0 ≡P=EP
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator I
The algorithm. . .
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator V
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator VI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator VII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator VIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator IX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator X
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XIV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XVI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XVII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XVIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XIX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XXI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Retrospective Area Estimator XXII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Denisty
- First consider no jumps
- Transform Diffusion - Lamperti Transform
dXt = α(Xt ) dt + dBt
- Propose sample paths (which can be simulated) - Brownian motion.
- Absolutely continuous.
- Accept or reject.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Denisty
- First consider no jumps
- Transform Diffusion - Lamperti Transform
dXt = α(Xt ) dt + dBt
- Propose sample paths (which can be simulated) - Brownian motion.
- Absolutely continuous.
- Accept or reject.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Denisty
- First consider no jumps
- Transform Diffusion - Lamperti Transform
dXt = α(Xt ) dt + dBt
- Propose sample paths (which can be simulated) - Brownian motion.
- Absolutely continuous.
- Accept or reject.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Denisty
- First consider no jumps
- Transform Diffusion - Lamperti Transform
dXt = α(Xt ) dt + dBt
- Propose sample paths (which can be simulated) - Brownian motion.
- Absolutely continuous.
- Accept or reject.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Denisty
- First consider no jumps
- Transform Diffusion - Lamperti Transform
dXt = α(Xt ) dt + dBt
- Propose sample paths (which can be simulated) - Brownian motion.
- Absolutely continuous.
- Accept or reject.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density V
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density VI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density VII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density VIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density IX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density X
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density XI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density XII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density XIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density XIV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density XV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density XVI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Transition Density XVII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm
Exact Algorithm
1 Simulate end point - y ∼ h
2 Propose sample bridge
3 Accept or Reject proposed sample bridge (and GOTO 1)
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm V
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm VI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm VII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm VIII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm IX
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm X
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Exact Algorithm XI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Random Weights I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Random Weights II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Random Weights III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Poisson Estimator I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Poisson Estimator
- Key Idea: Poisson Estimator - Simulate unbiased importance weight,
EWsx,,ty
"
t
( Z
exp −
)#
φ(Xu ) du
s
- What if φ(Xu ) ∈ [0, M ]?
- Broader class of Poisson Estimators
- Unbiased
- Finite Variance (Generalised Poisson Estimators (FPR08))
- Positivity (Wald Generalised Poisson Estimator (FPRS10))
- Auxiliary Poisson Estimators (APES (PJR12b))
- Generalised APES (GRAPES (PJR12b) - Jump Diffusions)
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Poisson Estimator
- Key Idea: Poisson Estimator - Simulate unbiased importance weight,
EWsx,,ty
"
t
( Z
exp −
)#
φ(Xu ) du
s
- What if φ(Xu ) ∈ [0, M ]?
- Broader class of Poisson Estimators
- Unbiased
- Finite Variance (Generalised Poisson Estimators (FPR08))
- Positivity (Wald Generalised Poisson Estimator (FPRS10))
- Auxiliary Poisson Estimators (APES (PJR12b))
- Generalised APES (GRAPES (PJR12b) - Jump Diffusions)
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Poisson Estimator
- Key Idea: Poisson Estimator - Simulate unbiased importance weight,
EWsx,,ty
"
t
( Z
exp −
)#
φ(Xu ) du
s
- What if φ(Xu ) ∈ [0, M ]?
- Broader class of Poisson Estimators
- Unbiased
- Finite Variance (Generalised Poisson Estimators (FPR08))
- Positivity (Wald Generalised Poisson Estimator (FPRS10))
- Auxiliary Poisson Estimators (APES (PJR12b))
- Generalised APES (GRAPES (PJR12b) - Jump Diffusions)
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Poisson Estimator
- Key Idea: Poisson Estimator - Simulate unbiased importance weight,
EWsx,,ty
"
t
( Z
exp −
)#
φ(Xu ) du
s
- What if φ(Xu ) ∈ [0, M ]?
- Broader class of Poisson Estimators
- Unbiased
- Finite Variance (Generalised Poisson Estimators (FPR08))
- Positivity (Wald Generalised Poisson Estimator (FPRS10))
- Auxiliary Poisson Estimators (APES (PJR12b))
- Generalised APES (GRAPES (PJR12b) - Jump Diffusions)
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Poisson Estimator
- Key Idea: Poisson Estimator - Simulate unbiased importance weight,
EWsx,,ty
"
t
( Z
exp −
)#
φ(Xu ) du
s
- What if φ(Xu ) ∈ [0, M ]?
- Broader class of Poisson Estimators
- Unbiased
- Finite Variance (Generalised Poisson Estimators (FPR08))
- Positivity (Wald Generalised Poisson Estimator (FPRS10))
- Auxiliary Poisson Estimators (APES (PJR12b))
- Generalised APES (GRAPES (PJR12b) - Jump Diffusions)
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Particle Filter I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Particle Filter II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Particle Filter III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Particle Filter IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Transition Density
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Transition Density I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Transition Density II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Exact Algorithm I
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Exact Algorithm II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Exact Algorithm III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Exact Algorithm IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Exact Algorithm V
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Exact Algorithm VI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Jump Exact Algorithm VII
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Example
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Example: Particle Filtering for Jump Diffusions I
An Example. . .
Exact Particle Filtering for Jump Diffusions In Practice!
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Example: Particle Filtering for Jump Diffusions II
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Example: Particle Filtering for Jump Diffusions III
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Example: Particle Filtering for Jump Diffusions IV
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Example: Particle Filtering for Jump Diffusions V
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Example: Particle Filtering for Jump Diffusions VI
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Other (Related) Work
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Other (Related) Work
Current Work
- Poisson Estimator
- As discussed. . .
- minu∈[s ,t ] α2 (Xu ) + α0 (Xu )
- Exact Algorithm
- Alternative to EA3 (EA4 - ‘Mondrian’ EA)
- EA3 Implementation
- Jump EA
- -Strong Simulation (BPR12)
- Initialisation
- Various Sampling Steps
- Extensions
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Other (Related) Work
Current Work
- Poisson Estimator
- As discussed. . .
- minu∈[s ,t ] α2 (Xu ) + α0 (Xu )
- Exact Algorithm
- Alternative to EA3 (EA4 - ‘Mondrian’ EA)
- EA3 Implementation
- Jump EA
- -Strong Simulation (BPR12)
- Initialisation
- Various Sampling Steps
- Extensions
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
Other (Related) Work
Current Work
- Poisson Estimator
- As discussed. . .
- minu∈[s ,t ] α2 (Xu ) + α0 (Xu )
- Exact Algorithm
- Alternative to EA3 (EA4 - ‘Mondrian’ EA)
- EA3 Implementation
- Jump EA
- -Strong Simulation (BPR12)
- Initialisation
- Various Sampling Steps
- Extensions
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
References I
[BPR08] A. Beskos, O. Papaspiliopoulos, and G. O. Roberts. A
factorisation of diffusion measure and finite sample path
constructions. Methodology and Computing in Applied
Probability, 10:85–104, 2008.
[BPR12] A. Beskos, S. Peluchetti, and G. O. Roberts. -strong
simulation of the brownian path. Bernoulli., 2012.
[BPRF06] A. Beskos, O. Papaspiliopoulos, G.O. Roberts, and
P. Fearnhead. Exact and computationally effcient
likelihood-based estimation for discretely observed diffusion
processes (with discussion). Journal of the Royal Statistical
Society, Series B (Statistical Methodology)., 68(3):333–382,
2006.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
References II
[CR10] B. Casella and G. O. Roberts. Exact simulation of
jump-diffusion processes with monte carlo applications.
Methodology and Computing in Applied Probability,
13(3):449–473, 2010.
[FPR08] P. Fearnhead, O. Papaspiliopoulos, and G. O. Roberts. Particle
filters for partially-observed diffusions. Journal of the Royal
Statistical Society, Series B (Statistical Methodology).,
70(4):755–777, 2008.
[FPRS10] P. Fearnhead, O. Papaspiliopoulos, G. Roberts, and A. Stuart.
Random-weight particle filtering of continuous time processes.
Journal of the Royal Statistical Society, Series B (Statistical
Methodology)., 72(4):497–512, 2010.
[Gon11] F. Gonçalves. Exact simulation and Monte Carlo inference for
jump-diffusion processes. PhD thesis, Dept. Of Statistics,
University of Warwick., 2011.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick
References III
[Jav05] A. Javaheri. Inside Volatility Arbitrage. Wiley., 1st edition, 2005.
[PJR12a] M. Pollock, A. M. Johansen, and G. O. Roberts. EA4. Technical
report, CRISM, Department of Statistics, University of
Warwick., 2012.
[PJR12b] M. Pollock, A. M. Johansen, and G. O. Roberts. Exact Particle
Filters. Technical report, CRISM, Department of Statistics,
University of Warwick., 2012.
[PJS09] N. Polson, M. Johannes, and J. Stroud. Optimal filtering of
jump-diffusions: Extracting latent states from asset prices.
Review of Financial Studies, 22(7):2259–2299, 2009.
Murray Pollock (University Of Warwick)
Filtering for jump diffusions
SMC 2012 – Warwick