The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

The (β1, β2)-skew Brownian motion: an explicitrepresentation of its transition densities and its

exact simulation

Sara Mazzonetto(Universitat Potsdam and Universite Lille 1)

joint work with Sylvie Roelly and David Dereudre

Les Houches, April 21st 2016

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Index

1 The generalized RS scheme

2 The (β1, β2)-skew BM

3 Sketch of the proofs

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Generalized rejection sampling method

Rejection sampling methodAssume to know how to sample the r.v. Y with density g(x).Then one can sample the r.v. X with density h(x) if:

(i) ∃M > 0 such that h(x) ≤ Mg(x) for all x ∈ R;

(ii) f (x) := 1M

h(x)g(x) can be evaluated.

X(d)= (Y |U < f (Y )) i.e. an exact simulation is possible.

Theorem

Replacing (ii) by

(ii’) there exists a sequence of functions (fn)n converging pointwise to fat a decreasing rate (δn)n.Moreover for each x ∈ R fn(x), δn(x) can be evaluated,

then X(d)= (Y |∃n;U < fn(Y )− δn(Y )).

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Toy example

Suppose p is an unknown parameter such that

∃(pn)n, (δn)n s.t. δnn→∞→ 0 decreasing, and |p − pn| < δn.

Then it is possible to simulate exactly a Bernoulli of parameter p sinceX := 1{∃n; |U−pn|>δn, U<pn} ∼ Bp .

0 1pu pn

pn − δn pn + δn

0 1p u′pn

pn − δn pn + δn

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Heuristics on the (β1, β2)-skew BM

Let β1, β2 ∈ [−1, 1]

1+β1

21−β1

2

0

1+β2

21−β2

2

1

Example: y 7→ p(0.3,0.7)(1, 0.5, y) is the density of X1 where (Xt)t is the(0.3, 0.7)−SBM starting at 0.5.

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Heuristics on the (β1, β2)-skew BM

Let β1, β2 ∈ [−1, 1]

1+β1

21−β1

2

0

1+β2

21−β2

2

1

Example: y 7→ p(0.3,0.7)(1, 0.5, y) is the density of X1 where (Xt)t is the(0.3, 0.7)−SBM starting at 0.5.

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

The (β1, β2)-SBM with drift

The process is defined for β1, β2 ∈ [−1, 1], µ ∈ R, z1, z2 ∈ R,

through the SDE, by{dXt = dWt + µdt + β1dL

z1t (X ) + β2dL

z2t (X );

Lzit (X ) =∫ t

0I{Xs=zi}dL

zis (X )

the infinitesimal generator is Lf = 12 ∆f +µ∇f +

∑j=1,2 βj〈δzj ,∇f 〉.

Lemma (Divergence form operator)L = 1

2h(x)∇ (h(x)∇)

Dom(L) ={f ∈ H1

0 (h(x)dx); h∇f ∈ H1(h−1(x)dx)}

h(x) = e2µxk(x)

where k(x) =

14 (1− β1)(1− β2) x < z1,14 (1 + β1)(1− β2) z1 ≤ x < z2,14 (1 + β1)(1 + β2) x ≥ z2.

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The transition densities of the (β1, β2)-SBM

Proposition

p(β1,β2)(t, x , y) = p(0,0)(t, x , y) · v (β1,β2)(t, x , y)

where, if z = z2 − z1,

v (β1,β2)(t, x , y) =∞∑k=0

(−β1β2)k4∑

j=1

cj(y)e−(aj (x,y)+2zk)2

2t e−|x−y |aj (x,y)+2zk

t

c1(y) ≡ 1

c2(y) =(21[z1,+∞)(y)− 1

)β1

c3(y) =(21[z2,+∞)(y)− 1

)β2

c4(y) =(1− 21[z1,z2)(y)

)β1β2

a1(x , y) ≡ 0

a2(x , y) = |y − z1|+ |x − z1| − |y − x |a3(x , y) = |y − z2|+ |y − z2| − |y − x |a4(x , y) = 2 (z2 −max(x , y , z1))+ + 2 (min(x , y , z2)− z1)+

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Application of the GRS method

Lemma

There exists an upper bound for v (β1,β2)(t, x , y) uniform in x and y :

supx,y∈R

∣∣∣v (β1,β2)(t, x , y)∣∣∣ ≤ v t :=

(1 + |β1|)(1 + |β2|)1− |β1β2|e−

2z2

t

Lemma

The remainder of the truncated series is bounded uniformly in x , y :

|RNv (β1,β2)(t, x , y)| ≤ v t

(|β1β2|e−

2z2

t

)N+1

.

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Approximation Vs generalized rejection sampling method

In the simulation we fix t = 1 and barriers z1 = 0 and z2 = 1, moreoverwe will sample 50000 times with GRSM y 7→ p(−0.7,0.3)(1, 0.5, y)

The 50000 simulations through GRS method are EXACT and the densityis approximated with its truncation at the 10th term.

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Asymmetric cases with β1β2 > 0 and β1β2 < 0

y 7→ p(0.3,−0.7)(1, 0.5, y) y 7→ p(−0.8,−0.6)(1, 0.5, y)

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Reflected skew Brownian motion and only one barrier

y 7→ p(1,−0.5)(1, 0.5, y) y 7→ p(0,−0.5)(1, 0.5, y)

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Simulation of the process (0.7,−0.2)-SBM

Assume the initial condition X0 = 0.5

t

Xt

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

The transition density for the (β1, β2)-SBM with drift

Main result:Assume β1, β2 ∈ (−1, 1) and µ ∈ R.

p(β1,β2)µ (t, x , y) = p(0,0)

µ (t, x , y)v (β1,β2)µ (t, x , y)

where the function v(β1,β2)µ is given by a series of Fourier transforms. It is

a series involving two functions{J0(ω, βiµ

√t) := e−

ω2

2

√2πe

(βiµ√

t+ω)2

2 Φc(ω + βiµ√t) with i ∈ {1, 2}

J1(ω) := −e−ω2

2 ,

evaluated in ω ∈{ωj,k :=

aj (x,y)+2zk+|y−x|√t

, j = 1, 2, 3, 4, k ∈ N}

.

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Representation of the transition density

Proposition

The infinitesimal generator L is self adjoint in L2(h(x)dx), and thefollowing equality holds:

p(β1,β2)µ (t, x , y) =

1

2πi

∫Γ

eλtG (β1,β2)µ (x , y ;λ)dλ,

where Γ is a complex contour of σ(L) ⊆ (−∞, 0] and

for y ∈ R and λ ∈ C \ (−∞, 0], x 7→ G(β1,β2)µ (x , y ;λ), the Green’s

function for the resolvent, solves{(λ− L)u(x) = δ{y}(x), u ∈ C2(R \ {z1, z2}) ∩ C(R)

h(z+i )u′(z+

i ) = h(z−i )u′(z−i ), i = 1, 2.

This is known as the Titchmarsh-Kodaira-Yoshida method (µ = 0).

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

The Green function for the (β1, β2)-SBM with drift

Lemma

The Green functions, if w :=√

2λ+ µ2 ∈ {v ∈ C;<(v) > 0}, are

G (x , y ;w) =1

weµ(y−x)

∑4j=1 cj(µ, y ;w)e−w(aj (x,y)+|x−y |)

β1β2e−2wz(w2 − µ2) + (w + β1µ)(w + β2µ).

cj(µ, y ;w) = w2cj,0(y) + wµcj,1(y) + µ2cj,2(y), wherec1,0 = 1,

c2,0 =(21[z1,+∞)(y)− 1

)β1

c3,0 =(21[z2,+∞)(y)− 1

)β2

c4,0 =(1− 21[z1,z2)(y)

)β1β2

c1,1 = β1 + β2

c2,1 = −β1 − c4,0

c3,1 = −β2 + c4,0

c4,1 = 0

c1,2 = β1β2

c2,2 = β1c3,0

c3,2 = −β2c2,0

c4,2 = −c4,0.

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Ideas of the proof

If φ(λ) =√

2λ+ µ2, we recall the contour integral representation

p(β1,β2)µ (t, x , y) =

1

2πi

∫Γ

eλtG (β1,β2)µ (x , y ;φ(λ))dλ,

and we will use techniques of complex analysis:

R

iR

Γ

φ(Γ)

R

No poles

φ(Γ)

ρM

|µ|−βµ

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Towards a series of Fourier transforms

For simplicity β1µ > 0, β2µ > 0, hence there is no pole.The formula for vµ

(β1,β2)(t, x , y) is

√t√

2πe

(x−y)2

2t

∫Re−

u2

2 t

∑4j=1−cj(µ, y ; iu)e−iu(aj (x,y)+|x−y |)

β1β2e−i2zu(u2 + µ2) + (u − iβ1µ)(u − iβ2µ)du.

Remark

For all u, µ ∈ R and β1, β2 ∈ (−1, 1)

|β1β2| (u2 + µ2) ≤ |(u − iβ1µ)(u − iβ2µ)|

Therefore

1

1 + β1β2(u2+µ2)e−i2zu

(u−iβ1µ)(u−iβ2µ)

=∞∑k=0

(−β1β2(u2 + µ2)

(u − iβ1µ)(u − iβ2µ)

)k

e−i2zuk

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Density’s uniform bound

Under the assumption β1µ > 0, β2µ > 0,v(β1,β2)µ (t, x , y) = e

|x−y|22t∑∞

k=0

∑4j=1 Fj,k(ωj,k), ωj,k :=

aj (x,y)+2zk+|y−x|√t

Fj,k := (−β1β2)k F(w 7→ e−

w2

2 cj(y , µ√t; iw) · −(w2+µ2t)k

(w−iβ1µ√t)k+1(w−iβ2µ

√t)k+1

).

Lemma

There exists an upper bound for v(β1,β2)µ (t, x , y) uniform in x and y :

supx,y∈R

∣∣∣v (β1,β2)µ (t, x , y)

∣∣∣ ≤ C (β1, β2)

1− e−2z2

t

|RNv (β1,β2)µ (t, x , y)| ≤ C (β1, β2)

1− e−2z2

t

e−2z2

t (N+1).

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Remerciements

Merci pour votre attention!

The generalized RS scheme The (β1, β2)-skew BM Sketch of the proofs References

Bibliography

[1] David Dereudre, Sara Mazzonetto, and Sylvie Roelly. Exactsimulation of one-dimensional brownian diffusions with driftadmitting several jumps. Preprint.

[2] David Dereudre, Sara Mazzonetto, and Sylvie Roelly. An explicitrepresentation of the transition densities of the skew brownian motionwith drift and two semipermeable barriers. Monte Carlo MethodsAppl., 22(1):1–23, 2016.

[3] Pierre Etore and Miguel Martinez. Exact simulation for solutions ofone-dimensional stochastic differential equations with discontinuousdrift. ESAIM Probab. Stat., 18:686–702, 2014.

[4] Bernard Gaveau, Masami Okada, and Tatsuya Okada. Second orderdifferential operators and Dirichlet integrals with singular coefficients.I. Functional calculus of one-dimensional operators. Tohoku Math. J.(2), 39(4):465–504, 1987.

[5] Antoine Lejay, Lionel Lenotre, and Geraldine Pichot. One-dimensionalskew diffusions: explicit expressions of densities and resolvent kernel,2015, hal-01194187.