Array-RQMC for Markov Chains with Random Stopping Times

1

Array-RQMC for Markov Chains with Random Stopping Times

Pierre L’Ecuyer Maxime Dion

Adam L’Archevˆ eque-Gaudet

Informatique et Recherche Op´ erationnelle, Universit´ e de Montr´ eal

1. Markov chain setting, Monte Carlo, classical RQMC.

2. Array-RQMC: preserving the low discrepancy of the chain’s states.

3. Least-squares Monte Carlo for optimal stopping times.

4. Examples.

2

Monte Carlo for Markov Chains

Setting: A Markov chain with state space X ⊆ R

, evolves as X

= x

, X

= ϕ

(X

, U

), j ≥ 1,

where the U

are i.i.d. uniform r.v.’s over (0, 1)

. Want to estimate µ = E[Y ] where Y =

τ

X

j=1

g

(X

)

and τ is a stopping time w.r.t. the filtration F {(j , X

), j ≥ 0}.

Ordinary MC: For i = 0, . . . , n − 1, generate X

= ϕ

(X

, U

), j = 1, . . . , τ

, where the U

’s are i.i.d. U(0, 1)

. Estimate µ by

ˆ µ

= 1

n

n

X

i=1 τi

X

j=1

g

(X

) = 1 n

n

X

i=1

Y

.

2

Monte Carlo for Markov Chains

Setting: A Markov chain with state space X ⊆ R

, evolves as X

= x

, X

= ϕ

(X

, U

), j ≥ 1,

where the U

are i.i.d. uniform r.v.’s over (0, 1)

. Want to estimate µ = E[Y ] where Y =

τ

X

j=1

g

(X

)

and τ is a stopping time w.r.t. the filtration F {(j , X

), j ≥ 0}.

Ordinary MC: For i = 0, . . . , n − 1, generate X

= ϕ

(X

, U

), j = 1, . . . , τ

, where the U

’s are i.i.d. U(0, 1)

. Estimate µ by

ˆ µ

= 1

n

n

X

i=1 τ_i

X

j=1

g

(X

) = 1 n

n

X

i=1

Y

.

3

Classical RQMC for Markov Chains

Put V

= (U

, U

, . . . ). Estimate µ by ˆ

µ

= 1 n

n

X

i=1 τi

X

j=1

g

(X

)

where P

= {V

, . . . , V

} ⊂ (0, 1)

has the following properties:

(a) each point V

has the uniform distribution over (0, 1)

; (b) P

has low discrepancy.

Dimension is s = inf{s

: P [d τ ≤ s

] = 1}.

For a Markov chain, the dimension s is often very large!

4

Array-RQMC for Markov Chains

[L´ ecot, Tuffin, L’Ecuyer 2004, 2008]

Simulate n chains in parallel. At each step, use an RQMC point set P

to advance all the chains by one step, while inducing global negative

dependence across the chains.

Intuition: The empirical distribution of S

= {X

, . . . , X

}, should be a more accurate approximation of the theoretical distribution of X

, for each j , than with crude Monte Carlo. The discrepancy between these two distributions should be as small as possible.

Then, we will have small variance for the (unbiased) estimators:

µ

= E [g

(X

)] ≈ 1 n

n−1

X

i=0

g

(X

) and µ = E [Y ] ≈ 1 n

n−1

X

i=0

Y

.

How can we preserve low-discrepancy of X

, . . . , X

when j increases?

Can we quantify the variance improvement?

4

Array-RQMC for Markov Chains

[L´ ecot, Tuffin, L’Ecuyer 2004, 2008]

Simulate n chains in parallel. At each step, use an RQMC point set P

to advance all the chains by one step, while inducing global negative

dependence across the chains.

Intuition: The empirical distribution of S

= {X

, . . . , X

}, should be a more accurate approximation of the theoretical distribution of X

, for each j , than with crude Monte Carlo. The discrepancy between these two distributions should be as small as possible.

Then, we will have small variance for the (unbiased) estimators:

µ

= E [g

(X

)] ≈ 1 n

n−1

X

i=0

g

(X

) and µ = E [Y ] ≈ 1 n

n−1

X

i=0

Y

.

How can we preserve low-discrepancy of X

, . . . , X

when j increases?

Can we quantify the variance improvement?

4

Array-RQMC for Markov Chains

[L´ ecot, Tuffin, L’Ecuyer 2004, 2008]

Simulate n chains in parallel. At each step, use an RQMC point set P

to advance all the chains by one step, while inducing global negative

dependence across the chains.

Intuition: The empirical distribution of S

= {X

, . . . , X

}, should be a more accurate approximation of the theoretical distribution of X

, for each j , than with crude Monte Carlo. The discrepancy between these two distributions should be as small as possible.

Then, we will have small variance for the (unbiased) estimators:

µ

= E [g

(X

)] ≈ 1 n

n−1

X

i=0

g

(X

) and µ = E [Y ] ≈ 1 n

n−1

X

i=0

Y

.

How can we preserve low-discrepancy of X

, . . . , X

when j increases?

Can we quantify the variance improvement?

5

To simplify, suppose each X

is a uniform r.v. over (0, 1)

.

Select a discrepancy measure D for the point set S

= {X

, . . . , X

} over (0, 1)

, and a corresponding measure of variation V , such that

Var[ˆ µ

] = E[(ˆ µ

− µ

)

] ≤ E[D

(S

)] V

(g

).

If D is defined via a reproducing kernel Hilbert space, then, for some random ξ

(that generally depends on S

),

E [D

(S

)] = Var

" 1 n

n

X

i=1

ξ

(X

)

#

= Var

" 1 n

n

X

i=1

(ξ

◦ ϕ

)(X

, U

))

#

≤ E [D

(Q

)] · V

(ξ

◦ ϕ

)

for some other discrepancy D

over (0, 1)

, where Q

= {(X

, U

), . . . , (X

, U

)}.

Heuristic: Under appropriate conditions, we should have V

(ξ

◦ ϕ

) < ∞

and E[D

(Q

)] = O(n

) for some α ≥ 1.

5

To simplify, suppose each X

is a uniform r.v. over (0, 1)

.

Select a discrepancy measure D for the point set S

= {X

, . . . , X

} over (0, 1)

, and a corresponding measure of variation V , such that

Var[ˆ µ

] = E[(ˆ µ

− µ

)

] ≤ E[D

(S

)] V

(g

).

If D is defined via a reproducing kernel Hilbert space, then, for some random ξ

(that generally depends on S

),

E [D

(S

)] = Var

"

1 n

n

X

i=1

ξ

(X

)

#

= Var

"

1 n

n

X

i=1

(ξ

◦ ϕ

)(X

, U

))

#

≤ E [D

(Q

)] · V

(ξ

◦ ϕ

)

for some other discrepancy D

over (0, 1)

, where Q

= {(X

, U

), . . . , (X

, U

)}.

Heuristic: Under appropriate conditions, we should have V

(ξ

◦ ϕ

) < ∞

and E[D

(Q

)] = O(n

) for some α ≥ 1.

6

In the points (X

, U

) of Q

, the U

can be defined via some RQMC scheme, but the X

cannot be chosen; they are determined by the history of the chains.

The idea is to select a low-discrepancy point set

Q ˜

= {(w

, U

), . . . , (w

, U

)},

where the w

∈ [0, 1)

are fixed and the U

∈ (0, 1)

are randomized, and then define a bijection between the states X

and the w

so that the

X

are “close” to the w

(small discrepancy between the two sets).

Bijection defined by a permutation π

of S

.

State space in R

: same algorithm essentially.

6

In the points (X

, U

) of Q

, the U

can be defined via some RQMC scheme, but the X

cannot be chosen; they are determined by the history of the chains.

The idea is to select a low-discrepancy point set

Q ˜

= {(w

, U

), . . . , (w

, U

)},

where the w

∈ [0, 1)

are fixed and the U

∈ (0, 1)

are randomized, and then define a bijection between the states X

and the w

so that the

X

are “close” to the w

(small discrepancy between the two sets).

Bijection defined by a permutation π

of S

.

State space in R

: same algorithm essentially.

7

Array-RQMC algorithm

X

← x

, for i = 0, . . . , n − 1;

for j = 1, 2, . . . , max

τ

do

Randomize afresh {U

, . . . , U

} in ˜ Q

; X

= ϕ

(X

, U

), for i = 0, . . . , n − 1;

Compute the permutation π

(sort the states);

end for

Estimate µ by the average ¯ Y

= ˆ µ

.

Theorem: The average ¯ Y

is an unbiased estimator of µ.

Can estimate Var[ ¯ Y

] by the empirical variance of m indep. realizations.

7

Array-RQMC algorithm

X

← x

, for i = 0, . . . , n − 1;

for j = 1, 2, . . . , max

τ

do

Randomize afresh {U

, . . . , U

} in ˜ Q

; X

= ϕ

(X

, U

), for i = 0, . . . , n − 1;

Compute the permutation π

(sort the states);

end for

Estimate µ by the average ¯ Y

= ˆ µ

.

Theorem: The average ¯ Y

is an unbiased estimator of µ.

Can estimate Var[ ¯ Y

] by the empirical variance of m indep. realizations.

8

Mapping chains to points

Multivariate sort:

Sort the states (chains) by first coordinate, in n

packets of size n/n

. Sort each packet by second coordinate, in n

packets of size n/n

n

.

.. .

At the last level, sort each packet of size n

by the last coordinate.

Choice of n

, n

, ..., n

?

Generalization:

Define a sorting function v : X → [0, 1)

and apply the multivariate sort (in c dimensions) to the transformed points v(X

).

Choice of v: Two states mapped to nearby values of v should be

approximately equivalent.

8

Mapping chains to points

Multivariate sort:

Sort the states (chains) by first coordinate, in n

packets of size n/n

. Sort each packet by second coordinate, in n

packets of size n/n

n

.

.. .

At the last level, sort each packet of size n

by the last coordinate.

Choice of n

, n

, ..., n

?

Generalization:

Define a sorting function v : X → [0, 1)

and apply the multivariate sort (in c dimensions) to the transformed points v(X

).

Choice of v: Two states mapped to nearby values of v should be

approximately equivalent.

9

A (4,4) mapping

States of the chains

0.00.0 0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s

s s

s s

s

s

s s

s s s

s

s s

s

Sobol’ net in 2 dimensions with digital shift

0.00.0 0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0

1.0

s

s s

s s

s

s

s s

s s

s s

s

s

s

10

A (4,4) mapping

States of the chains

0.00.0 0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s s

s s

s s s

s s s

s s

s s s

s

Sobol’ net in 2 dimensions with digital shift

0.00.0 0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s s s

s

s s s

s

s s s s

s s

s

s

11

A (4,4) mapping

0.0 0.0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s s s

s

s s s

s

s

s s s

s s

s s

s s

s s

s s s

s

s s

s s

s s s

s

12

A (16,1) mapping, sorting along first coordinate

0.0 0.0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s

s s

s

s s s

s s

s s

s s

s s

s

s s

s

s s

s

s s

s s

s

s s s

s

s

13

A (8,2) mapping

0.0 0.0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s s

s s

s s

s s

s s

s s

s s

s s

s s

s s

s s

s s

s s

s s

s s

s

s

14

A (4,4) mapping

0.0 0.0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s s s

s

s s s

s

s

s s s

s s

s s

s s

s s

s s s

s

s s

s s

s s s

s

15

A (2,8) mapping

0.0 0.0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s

s s

s s s

s s

s s

s s

s

s s

s

s s

s s s

s s

s

s s s s

s s

s s

16

A (1,16) mapping, sorting along second coordinate

0.0 0.0

0.1 0.1

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

0.6 0.6

0.7 0.7

0.8 0.8

0.9 0.9

1.0 1.0

s s

s s s

s s

s s

s

s s

s s

s

s

s s

s s s s

s s

s s

s

s s s

s s

17

Dynamic programming for optimal stopping times

Suppose the stopping time τ is a decision determined by a stopping policy π = (ν

, ν

, . . . , ν

) where ν

: X → {stop now, wait}.

Suppose also that must stop at or before step T .

Dynamic programming equations: V

(x) = g

(x),

Q

(x) = E [V

(X

) | X

= x], (continuation value) V

(x) = max[g

(x), Q

(x)], (optimal value) ν

(x) =

( stop now if g

(x) ≥ Q

(x)

wait otherwise, (optimal decision)

for j = T − 1, . . . , 0 and all x ∈ X .

17

Dynamic programming for optimal stopping times

Suppose the stopping time τ is a decision determined by a stopping policy π = (ν

, ν

, . . . , ν

) where ν

: X → {stop now, wait}.

Suppose also that must stop at or before step T . Dynamic programming equations:

V

(x) = g

(x),

Q

(x) = E[V

(X

) | X

= x], (continuation value) V

(x) = max[g

(x), Q

(x)], (optimal value) ν

(x) =

( stop now if g

(x) ≥ Q

(x)

wait otherwise, (optimal decision)

for j = T − 1, . . . , 0 and all x ∈ X .

18

Hard to solve when the state space is large and multidimensional.

Can approximate Q

with a small set of basis functions.

{ψ

: X → R , 1 ≤ k ≤ m}:

Q ˜

(x) =

m

X

k=1

β

ψ

(x)

where β

= (β

, . . . , β

)

can be determined by least-squares regression, using an approximation W

of Q

(x

) at a set of points x

.

We solve

β min

X

i=1

Q ˜

(x

) − W

.

A set of representative states x

at each step j can be generated by

Monte Carlo, or RQMC, or array-RQMC.

19

Regression-based least-squares Monte Carlo

Tsistiklis and Van Roy (2000) (TvR);

Simulate n indep. trajectories of the chain {X

, j = 0, . . . , T }, and let X

be the state for trajectory i at step j ;

W

← g

(X

), i = 1, . . . , n;

for j = T − 1, . . . , 0 do

Compute the vector β

that minimizes

n

X

i=1 m

X

k=1

β

ψ

(X

) − W

!

.

W

← max[g

(X

), Q ˜

(X

)] , i = 1, . . . , n;

end for

return Q ˆ

(x

) = (W

+ · · · + W

)/n as an estimate of Q

(x

);

Longstaff and Schwartz (2001) (LSM): Define W

instead by W

=

( g

(X

) if g

(X

) ≥ Q ˜

(X

);

W

otherwise .

19

Regression-based least-squares Monte Carlo

Tsistiklis and Van Roy (2000) (TvR);

Simulate n indep. trajectories of the chain {X

, j = 0, . . . , T }, and let X

be the state for trajectory i at step j ;

W

← g

(X

), i = 1, . . . , n;

for j = T − 1, . . . , 0 do

Compute the vector β

that minimizes

n

X

i=1 m

X

k=1

β

ψ

(X

) − W

!

.

W

← max[g

(X

), Q ˜

(X

)] , i = 1, . . . , n;

end for

return Q ˆ

(x

) = (W

+ · · · + W

)/n as an estimate of Q

(x

);

Longstaff and Schwartz (2001) (LSM): Define W

instead by W

=

( g

(X

) if g

(X

) ≥ Q ˜

(X

);

W

otherwise .

20

Example: a simple put option

Asset price obeys GBM {S (t), t ≥ 0} with drift (interest rate) µ = 0.05, volatility σ = 0.08, initial value S (0) = 100.

For American version, exercise dates are t

= j/16 for j = 1, . . . , 16.

Payoff at t

: g

(S(t

)) = e

max(0, K − S (t

)), where K = 101.

European version: Can exercise only at t

= 1.

One-dimensional state X

= S (t

). Sorting for array-RQMC is simple. Basis functions for regression-based MC:

polynomials ψ

(x) = (x − 101)

for k = 1, . . . , 5.

For RQMC and array-RQMC, we use Sobol’ nets with a linear scrambling and a random digital shift, for all the results reported here.

Results are very similar for randomly-shifted lattice rule + baker’s

transformation.

20

Example: a simple put option

Asset price obeys GBM {S (t), t ≥ 0} with drift (interest rate) µ = 0.05, volatility σ = 0.08, initial value S (0) = 100.

For American version, exercise dates are t

= j/16 for j = 1, . . . , 16.

Payoff at t

: g

(S(t

)) = e

max(0, K − S (t

)), where K = 101.

European version: Can exercise only at t

= 1.

One-dimensional state X

= S (t

). Sorting for array-RQMC is simple.

Basis functions for regression-based MC:

polynomials ψ

(x) = (x − 101)

for k = 1, . . . , 5.

For RQMC and array-RQMC, we use Sobol’ nets with a linear scrambling and a random digital shift, for all the results reported here.

Results are very similar for randomly-shifted lattice rule + baker’s

transformation.

20

Example: a simple put option

Asset price obeys GBM {S (t), t ≥ 0} with drift (interest rate) µ = 0.05, volatility σ = 0.08, initial value S (0) = 100.

For American version, exercise dates are t

= j/16 for j = 1, . . . , 16.

Payoff at t

: g

(S(t

)) = e

max(0, K − S (t

)), where K = 101.

European version: Can exercise only at t

= 1.

One-dimensional state X

= S (t

). Sorting for array-RQMC is simple.

Basis functions for regression-based MC:

polynomials ψ

(x) = (x − 101)

for k = 1, . . . , 5.

For RQMC and array-RQMC, we use Sobol’ nets with a linear scrambling and a random digital shift, for all the results reported here.

Results are very similar for randomly-shifted lattice rule + baker’s

transformation.

21

European version of put option.

log

n

8 10 12 14 16 18 20

log

Var[ˆ µ

]

-40 -30 -20 -10

n

array-RQMC

PCA

BB Seq

standard MC

21

European version of put option.

log

n

8 10 12 14 16 18 20

log

Var[ˆ µ

]

-40 -30 -20 -10

n

array-RQMC PCA

BB Seq

standard MC

22

Histogram of states at step 16

States for array-RQMC with n = 2

in blue and for MC in red.

Theoretical dist.: black dots.

S

90 100 110 120

frequency

0

200

400

600

23

Histogram after transformation to uniforms (applying the cdf).

States for array-RQMC with n = 2

in blue and for MC in red.

Theoretical dist. is uniform (black dots).

0 0.5 1

frequency

0

200

400

600

24

log

n 8 10 12 14 16 18 20

log

Var[ˆ µ

]

-25 -20 -15 -10 -5

n

TvR, array-RQMC

TvR, RQMC bridge

TvR, standard MC

LSM, array-RQMC

LSM, RQMC bridge

LSM, standard MC

American put option: estimation for a fixed policy.

log

n

8 10 12 14 16 18 20

log

Var[ˆ µ

]

-20 -15 -10 -5

array-RQMC

RQMC PCA

RQMC bridge

RQMC sequential

standard MC

American put option: out-of-sample value for policy obtained from

LSM.

log

n

6 8 10 12 14

E [out-of-sample value]

1.95 2.00 2.05 2.10 2.15

2.1690 array-RQMC

RQMC PCA

standard MC

American put option: out-of-sample value for policy obtained from

TvR.

log

n

6 8 10 12 14

E [out-of-sample value]

2.05 2.10

2.15 2.1514 array-RQMC

RQMC PCA

standard MC

28

Example: Asian Option

Given observation times t

, t

, . . . , t

, suppose

S (t

) = S (t

) exp[(r − σ

/2)(t

− t

) + σ(t

− t

)

Φ

(U

)], where U

∼ U[0, 1) and S(t

) = s

is fixed.

State is X

= (S(t

), S ¯

), where S ¯

=

P

i=1

S (t

).

Transition:

(S (t

), S ¯

) = ϕ(S(t

), S ¯

, U

) =

S (t

), (j − 1)¯ S

+ S (t

) j

.

Payoff at step j is max

0, S ¯

− K .

We use the two-dimensional sort at each step; we first sort in n

packets

based on S (t

), then sort the packets based on ¯ S

.

29

GBM with parameters: S (0) = 100, K = 100, r = 0.05, σ = 0.15, t

= j /52 for j = 0, . . . , s = 13.

Basis functions to approximate the continuation value: polynomials of the form g (S , S ¯ ) = (S − 100)

(¯ S − 100)

, for k , m = 0, . . . , 4 and km ≤ 4.

Also broken polynomials max(0, S − 100)

for k = 1, 2, and

max(0, S − 100)(¯ S − 100).

European version of Asian call option

log

n

8 10 12 14 16 18 20

log

Var[ˆ µ

]

-40 -30 -20 -10

n

array-RQMC, split sort RQMC PCA

RQMC bridge

RQMC sequential

standard MC

31

European version, sorting strategies for array-RQMC.

log

n

8 10 12 14 16 18 20

log

Var[ˆ µ

]

-40 -30 -20 -10

n

n

array-RQMC, n

= n

array-RQMC, n

= n

array-RQMC, split sort

array-RQMC, sort on ¯ S

array-RQMC, sort on S

32

American-style Asian option with a fixed policy.

log

n

8 10 12 14 16 18 20

log

Var[ˆ µ

]

-25 -20 -15 -10 -5

array-RQMC, split sort RQMC PCA

RQMC bridge

RQMC sequential

standard MC

33

Fixed policy, choices of array-RQMC sorting.

log

n

8 10 12 14 16 18 20

log

Var[ˆ µ

]

-20 -15 -10

array-RQMC, n

= n

array-RQMC, n

= n

array-RQMC, split sort

array-RQMC, sort on ¯ S

array-RQMC, sort S

Out-of-sample value of policy obtained from LSM.

log

n

8 10 12 14

E [out-of-sample value]

2.17 2.19 2.22 2.24 2.27 2.29

2.32 2.3204 array-RQMC, split sort

RQMC PCA

standard MC

35

Out-of-sample value of policy obtained from TvR.

log

n

8 10 12 14

E [out-of-sample value]

2.27 2.28 2.29

2.30 2.2997 array-RQMC, split sort

RQMC PCA

standard MC

36

Call on the maximum of 5 assets

Five indep. asset prices obeys a GBM with s

= 100, r = 0.05, σ = 0.2.

The assets pay a dividend at rate 0.10, which means that the effective risk-free rate can be taken as r

= 0.05 − 0.10 = −0.05.

Exercise dates are t

= j /3 for j = 1, . . . , 9.

State at t

is X

= (S

, . . . , S

).

Basis functions for regression: 19 polynomials in the S

− 100, where S

, . . . , S

are the asset prices sorted in increasing order.

For array-RQMC, we sort on the m largest asset prices.

At each step we generate the next value first for the maximum, then for

the second largest, and so on.

36

Call on the maximum of 5 assets

Five indep. asset prices obeys a GBM with s

= 100, r = 0.05, σ = 0.2.

The assets pay a dividend at rate 0.10, which means that the effective risk-free rate can be taken as r

= 0.05 − 0.10 = −0.05.

Exercise dates are t

= j /3 for j = 1, . . . , 9.

State at t

is X

= (S

, . . . , S

).

Basis functions for regression: 19 polynomials in the S

− 100, where S

, . . . , S

are the asset prices sorted in increasing order.

For array-RQMC, we sort on the m largest asset prices.

At each step we generate the next value first for the maximum, then for

the second largest, and so on.

European version

log

n

8 10 12 14 16 18

log

Var[ˆ µ

]

-25 -20 -15 -10 -5 0

n

array-RQMC, split sort 3 max RQMC PCA

RQMC bridge

RQMC sequential

standard MC

38

log

n

8 10 12 14 16 18

log

Var[ˆ µ

]

-25 -20 -15 -10 -5 0

n

n

array-RQMC, split sort 5 max

array-RQMC, split sort 4 max

array-RQMC, split sort 3 max

array-RQMC, split sort 2 max

array-RQMC, sort 1 max

American version, fixed policy

log

n

8 10 12 14 16 18

log

Var[ˆ µ

]

-10 -5 0

array-RQMC, split sort 3 max RQMC PCA

RQMC bridge

RQMC sequential

standard MC

Fixed policy.

log

n

8 10 12 14 16 18

log

Var[ˆ µ

]

-12.5 -10 -7.5 -5 -2.5

array-RQMC, split sort 5 max

array-RQMC, split sort 4 max

array-RQMC, split sort 3 max

array-RQMC split, sort 2 max

array-RQMC, sort 1 max

41

Out-of-sample value of policy obtained from LSM.

log

n

8 10 12 14

E[out-of-sample value]

24 25 26

26.116 array-RQMC, split sort 3 max

RQMC PCA

RQMC bridge

RQMC sequential

standard MC

Out-of-sample value of policy obtained from TvR.

log

n

8 10 12 14

E[out-of-sample value]

25.0 25.5 26.0 26.5

26.124 array-RQMC, split sort 3 max

RQMC PCA

RQMC bridge

RQMC sequential

standard MC

43

Conclusion

Empirical results are excellent for fixed number of steps.

More modest but still interesting for random stopping time.

Proving the observed convergence rates seems difficult; we need help!