Download Discrete time approximation of BSDEs B. Bouchard

Discrete time approximation of BSDEs
B. Bouchard
LPMA University Paris 6 and CREST
SOME MOTIVATIONS
BSDE and finance
• Hedging of the European option g(X1) (interest rate r)

Z t
Z t



Xt = X0 +
diag [Xs] b(Xs)ds +
diag [Xs] σ(Xs)dWs ,



0
0





 Yt
= g(X1) −
Z 1
t
(Ys − φ0s1)rds −
Z 1
t
φ0sσ(Xs)dWs
• Hedging of the European option g(X1) (interest rate rb for
borrowing, rl for lending)

Z t
Z t



Xt = X0 +
diag [Xs] b(Xs)ds +
diag [Xs] σ(Xs)dWs ,



0
0





 Yt
= g(X1) −
Z 1h
t
(Ys − φ0s1)+r l − (Ys − φ0s1)−rb ds −
i
Z 1
t
φ0sσ(Xs)dWs
American option hedging
• Hedging of the American option g(X) (interest rate r)

Z t
Z t



Xt = X0 +
diag [Xs] b(Xs)ds +
diag [Xs] σ(Xs)dWs ,



0
0




Z 1
Z 1
0 1)rds −
0 σ(X )dW + K − K

Y
=
g(X
)
−
(Y
−
φ
φ

s
s
s
t
t
1
1
s
s


t
t

Z 1




( Ys − g(Xs) )dKs = 0 ,
 Yt ≥ g(Xt) , t ≤ 1 and
0
⇒ Yt = esssup E
τ ≤1
h
e−rτ g(Xτ ) | Ft
i
Game option hedging
• Hedging of the Game option g(X) (interest rate r)



Xt













 Yt
= X0 +
Z t
0
= g(X1) −
diag [Xs] b(Xs)ds +
Z 1
t
(Ys − φ0s1)rds −

`(Xt) ≤ Yt ≤ h(Xt) , t ≤ 1


Z 1





0
=
( Ys − `(Xs) )dKs+ ,



Z01




 0
=
( Ys − h(Xs) )dKs− .
0
Z t
0
Z 1
t
diag [Xs] σ(Xs)dWs ,
φ0sσ(Xs)dWs + K1 − Kt
⇒
Yt =
ess inf esssup E
θ≤1
τ ≤1
h
e−r(τ ∧θ)
n
(g(Xτ )1τ =T + `(Xτ )1τ <T ) 1τ ≤θ + h(Xθ )1τ >θ
o
| F
Default options
• Free of arbitrage pricing rule:
h
u1(t, x) := Et,x g1(XT )1τ >T + g0(XT )1τ ≤T
i
with default time: τ ∼ E(%).
• Decomposition :
"
u1(t, x) := Et,x g1(XT )e−(T −t)% +
u0(t, x) := Et,x [g0(XT )]
Z T
t
u0(s, Xs)%e−%(s−t)ds
#
Default options
∂
1 2 2 2
u0 + x σ D u0 = 0 , u0(T, ·) = g0 = [5 − ·]+ ∧ 0.5
∂t
2
1 2 2 2
∂
u1 + x σ D u1 + %(u0 − u1) = 0 , u1(T, ·) = g1 = [5 − ·]+
∂t
2
• BSDE:
dXt = XtσdWt
Mt = µ((0, t]) [2]
and
−dYt =
%1Mt=1 − λ Utdt − Utµ̄(dt) − Zt · dWt
YT = g1(XT )1MT =1 + g0(XT )1MT =0
BSDE and finance...
• g-expectations
• Optimal investment: Non Lipschitz coefficients !
Semilinear PDEs I: no non-local term

Z t
Z t



Xt = X0 +
b(Xs)ds +
σ(Xs)dWs ,



0
0





 Yt
= g(X1) +
Z 1
t
f (Xs, Ys, Zs)ds −
Z 1
t
ZsdWs
⇒ Yt = u(t, Xt) with u solution (in a weak sens) of
1
0 = ut + b0Du + Tr[σσ 0D2u] + f (·, u, Du0σ)
2
g = u(1, ·) .
⇒ Zt = (Du0σ)(t, Xt)
Semilinear PDEs II: with integral term

Z t
Z t
Z tZ



Xt = X0 +
b(Xs)ds +
σ(Xs)dWs +
β(Xs−, e)µ̄(de, ds) ,



0
0
0 E





 Yt
= g(X1) +
Z 1
t
f (Θs) ds −
with Θ := (X, Y, Γ, Z) and Γ :=
Z 1
t
Z
E
ZsdWs −
Z 1Z
t
E
Us(e)µ̄(de, ds)
ρ(e)U (e)λ(de).
⇒ Yt = u(t, Xt) with u solution (in a weak sens) of
0 = Lu + f ·, u, Du0σ, I[u]
g = u(1, ·) ,
where
1
0
Lu := ut + b Du + Tr[σσ 0D2u] + {u(·, · + β(·, e)) − u − Du0β(·, e)} λ(de)
2
E
Z
I[u] :=
{u(·, · + β(·, e)) − u} ρ(e) λ(de) .
Z
E
Semilinear PDEs II: with integral term

Z t
Z t
Z tZ



Xt = X0 +
b(Xs)ds +
σ(Xs)dWs +
β(Xs−, e)µ̄(de, ds) ,



0
0
0 E





 Yt
= g(X1) +
Z 1
t
f (Θs) ds −
with Θ := (X, Y, Γ, Z) and Γ :=
Z 1
t
Z
E
ZsdWs −
Z 1Z
t
E
Us(e)µ̄(de, ds)
ρ(e)U (e)λ(de).
⇒ Yt = u(t, Xt) with u solution (in a weak sens) of
0 = Lu + f ·, u, Du0σ, I[u]
g = u(1, ·) ,
⇒ Zt = (Du0σ)(t, Xt) and Ut(e) = u(t, Xt− + β(Xt−, e)) − u(t, Xt−)
Semilinear PDEs III: partially coupled systems
Pardoux, Pradeilles and Rao (97), Sow and Pardoux (04).
• System of κ PDE’s (i = 0, . . . , κ − 1)
1
i
0
i
0 = ut + biDu + Tr[σiσi0 D2ui] + fi(·, u, (Dui)0σi)
2
gi = ui(1, ·) .
• Define for i = 0, . . . , κ − 1


f˜(i, x, y, γ, z) = fi x, (. . . , y + γ κ−2, y + γ κ−1, |{z}
y , y + γ 1, y + γ 2, . . .), z 
i
• Set E = {1, . . . , κ − 1}, λ(de) = λ
Mt =
Pκ−1
k=1 δk (e) and
Z tZ
0
E
eµ(de, ds) [κ]
Semilinear PDEs III: partially coupled systems
Pardoux, Pradeilles and Rao (97), Sow and Pardoux (04).
• System of κ PDE’s (i = 0, . . . , κ − 1)
1
i
0
i
0 = ut + biDu + Tr[σiσi0 D2ui] + fi(·, u, (Dui)0σi)
2
gi = ui(1, ·) .
⇒ uMt (t, Xt) = Yt where
dXt = bMt (Xt)dt + σMt (Xt)dWt
−dYt = f˜(Mt, Xt, Yt, Ut, Zt)dt − λ
κ−1
X
k=1
Y1 = gM1 (X1)
U (k)tdt − ZtdWt −
Z
E
Ut(e)µ̄(de, dt)
Semilinear PDEs IV: free boundary problems

Z t
Z t



Xt = X0 +
b(Xs)ds +
σ(Xs)dWs ,



0
0




Z 1
Z 1

Yt = g(X1) +
f (Xs, Ys, Zs) ds −
ZsdWs + K1 − Kt



t
t

Z 1




( Ys − h(Xs) )dKs = 0 ,
 Yt ≥ h(Xt) , t ≤ 1 and
0
⇒ Yt = u(t, Xt) where u solves
0 = min −Lu − f (·, u, Du0σ) , u − h
n
g = u(1, ·)
o
DISCRETE TIME APPROXIMATION
PDE and Tree methods: Some references
Solve the PDE and they use MC simulation
• Ma, Protter and Yong (1994). Solving forward-backward stochastic
differential equations explicitly - a four step scheme, PTRF.
• Douglas, Ma and Protter (1996). Numerical Methods for ForwardBackward Stochastic Differential Equations, AAP.
PDE and Tree methods: Some references
Approximate the Brownian motion by a walk
• Coquet, Mackevičius and Mémin (1998). Stability in D of martingales
and backward equations under discretization of filtration, SPA.
• Briand, Delyon and Mémin (2001). Donsker-type theorem for BSDE’s,
ECP.
• Ma, Protter, San Martin and Torres (2002). Numerical Method for
Backward Stochastic Differential Equations, AAP.
• Antonelli and Kohatsu-Higa (2000). Filtration stability of backward
SDE’s, SPA.
Aim
⇒ Use a pure probabilistic approach to obtain the required regularity
under weaker conditions.
Discrete approximation approach: Some references
• Zhang J. (01). Some fine properties of backward stochastic differential equations. PhD thesis.
• Zhang J. (04). A numerical scheme for BSDEs. AAP 14.
• Bally V. and G. Pagès (03). Error analysis of the quantization algorithm for obstacle problems. SPA 106.
• Bouchard B. and N. Touzi (04). Discrete-Time Approximation and
Monte-Carlo Simulation of Backward Stochastic Differential Equations.
SPA, 111.
• Bouchard B. and R. Elie (05). Discrete time approximation of decoupled Forward-Backward SDE with jumps. Preprint.
• Ma J. and J. Zhang (05). Representations and regularities for solutions to BSDEs with reflections. SPA 115.
• Bouchard B. and J.-F. Chassagneux (06). Discrete time approximation for continuously and discretely reflected BSDEs. Preprint.
OUTLINE
1. Discretization
a. BSDEs
b. BSDEs with jumps
c. BSDEs with jumps and partially non-Lipschitz driver
d. Reflected BSDEs
2. Numerical implementation and simulation
a. Regression based algorithm
b. Malliavin calculus approach
c. Quantization method
BSDEs

Z t
Z t



Xt = X0 +
b(Xs)ds +
σ(Xs)dWs ,



0
0





 Yt
= g(X1) +
Z 1
t
f (Xs, Ys, Zs)ds −
Z 1
t
ZsdWs
Construction
Step 1: Step-constant driver with π := {ti := i/n, i ≤ n}
Ȳtπi
Z t
1 π π π
i+1 π
π
Zt dWt ,
= Ȳti+1 + f Xti , Ȳti , Z̄ti −
n
t
i
where
Z t
i+1
ti
h
i
π
π
π
Zt dWt = Ȳti+1 − E Ȳti+1 | Fti
Construction
Step 1: Step-constant driver with π := {ti := i/n, i ≤ n}
Ȳtπi
Z t
1 π π π
i+1 π
π
Zt dWt ,
= Ȳti+1 + f Xti , Ȳti , Z̄ti −
n
t
i
where
Z t
i+1
ti
h
i
π
π
π
Zt dWt = Ȳti+1 − E Ȳti+1 | Fti
Step 2: Best L2(Ω × [ti, ti+1]) approximation of Z π Fti -meas. process
Z̄tπ
:= nE
"Z
ti+1
ti
#
Zsπ ds | Fti
= nE
h
Ȳtπi+1 (Wti+1 − Wti ) | Fti
i
Construction
Step 1: Step-constant driver with π := {ti := i/n, i ≤ n}
Ȳtπi
Z t
1 π π π
i+1 π
π
Zt dWt ,
= Ȳti+1 + f Xti , Ȳti , Z̄ti −
n
t
i
where
Z t
i+1
ti
h
i
π
π
π
Zt dWt = Ȳti+1 − E Ȳti+1 | Fti
Step 2: Best L2(Ω × [ti, ti+1]) approximation of Z π Fti -meas. process
Z̄tπ
:= nE
"Z
ti+1
ti
#
Zsπ ds | Fti
Step 3: Discrete scheme: Ȳtπi = E
= nE
h
h
Ȳtπi+1 (Wti+1 − Wti ) | Fti
i
i
1
Ȳtπi+1 | Fti + n f Xtπi , Ȳtπi , Z̄tπi .
Construction (an other interpretation)
Step 1: Euler scheme type approximation
Ȳtπi
1 π π π
π
∼ Ȳti+1 + f Xti , Ȳti , Z̄ti − Z̄tπi (Wti+1 − Wti ) ,
n
Construction (an other interpretation)
Step 1: Euler scheme type approximation
Ȳtπi
1 π π π
π
∼ Ȳti+1 + f Xti , Ȳti , Z̄ti − Z̄tπi (Wti+1 − Wti ) ,
n
Step 2: By taking expectation
Ȳtπi
= E
h
i
1 π π π
π
Ȳti+1 | Fti + f Xti , Ȳti , Z̄ti
n
Construction (an other interpretation)
Step 1: Euler scheme type approximation
Ȳtπi
1 π π π
π
∼ Ȳti+1 + f Xti , Ȳti , Z̄ti − Z̄tπi (Wti+1 − Wti ) ,
n
Step 2: By taking expectation
Ȳtπi
= E
h
i
1 π π π
π
Ȳti+1 | Fti + f Xti , Ȳti , Z̄ti
n
Step 3: By multiplying by (Wti+1 − Wti ) and taking expectation
Z̄tπi
= nE
h
Ȳ π (W
ti+1 − Wti ) | Fti
i
Discrete scheme
Ȳ1π
= g(X1π )
Z̄tπi
i
π
:= nE Ȳti+1 (Wti+1 − Wti ) | Fti
h
i
1 π π π
π
= E Ȳti+1 | Fti + f Xti , Ȳti , Z̄ti .
Ȳtπi
h
n
Implicit vs Explicit
• Implicit
Ȳtπi
= E
h
i
1 π π π
π
Ȳti+1 | Fti + f Xti , Ȳti , Z̄ti .
n
Implicit vs Explicit
• Implicit
Ȳtπi
= E
h
i
1 π π π
π
Ȳti+1 | Fti + f Xti , Ȳti , Z̄ti .
n
• Explicit
Ȳtπi
= E
h
i
i
1 h π π
π
π
Ȳti+1 | Fti + E f Xti , Ȳti+1 , Z̄ti | Fti .
n
Implicit vs Explicit
• Implicit
Ȳtπi
= E
h
i
1 π π π
π
Ȳti+1 | Fti + f Xti , Ȳti , Z̄ti .
n
• Explicit
Ȳtπi
= E
h
i
i
1 h π π
π
π
Ȳti+1 | Fti + E f Xti , Ȳti+1 , Z̄ti | Fti .
n
• It actually leads to the same approximation error: choose one or the
other depending on f .
First bound
Set


Z 1
Err2 := max E  sup |Ȳtπi − Yt|2 +
i
0
ti≤t≤ti+1
E
h
i
π
2
|Zt − Z̄t | dt
Theorem:
Z 1
Err2 ≤ C |π| +
0
h
i
E |Zt − Z̄t|2 dt
where
Z̄t := nE
=⇒ Regularity of Z ?
"Z
ti+1
ti
!
#
Zsds | Fti
Regularity of Z (simplification: f depends only on X )
• Malliavin calculus approach
Proposition:
solves
(Yt, Zt) admits a Malliavin derivative and (DsY, DsZ)
DsYt = ∇g(X1)DsX1 +
Z 1
t
∇f (Xr )DsXr dr −
Z 1
t
DsZr dWr
• Since
Yt = Y0 −
Z t
0
f (Xr )dr +
Z t
0
Zr dWr
we have
"
Zt = DtYt = E ∇g(X1)DsX1 +
"
= E ∇g(X1)∇X1 +
Z 1
t
Z 1
t
#
∇xf (Xr )DsXr dr | Ft
#
∇xf (Xr )∇Xr dr | Ft (∇Xt)−1σ(Xt)
Regularity of Z (simplification: f depends only on X )
• Zt = (Vt − αt)(∇Xt)−1σ(Xt) where
"
Vt = E ∇g(X1)∇X1 +
αt =
Z t
0
Z 1
0
#
∇xf (Xr )∇Xr dr | Ft
∇xf (Xr )∇Xr dr
• We have for some ∈ L2(Ω × [0, 1])
Vt = V0 +
Z t
0
σ̃sdWs
thus
n−1
X Z ti+1
i=0 ti
E
h
i
|Vt − Vti |2 dt ≤ C
n−1
X Z ti+1 Z t
i=0 ti
Z 1
≤ C |π|
0
ti
E
h
E
h
i
2
|σ̃s| dsdt
i
2
|σ̃s| ds .
Discrete time approximation error
Theorem: Assume that all the coefficients are Lipschitz continuous,
then
Err2 ≤ C |π| +
Z 1
0
h
i
!
E |Zt − Z̄t|2 dt ≤ C |π|
and
Z 1
0
E
h
i
|Zt − Z̄t|2 dt ≤
X
Z t
i+1
i≤n−1 ti
E
h
i
2
|Zt − Zti | dt ≤ C|π|
where
Z̄t := nE
"Z
ti+1
ti
#
Zsds | Fti
Discrete time approximation error
Theorem: Assume that all the coefficients are Lipschitz continuous,
then
Err2 ≤ C |π| +
Z 1
0
h
i
!
E |Zt − Z̄t|2 dt ≤ C |π|
and
Z 1
0
E
h
i
|Zt − Z̄t|2 dt ≤
X
Z t
i+1
i≤n−1 ti
E
h
i
2
|Zt − Zti | dt ≤ C|π|
where
Z̄t := nE
"Z
ti+1
ti
#
Zsds | Fti
Discrete time approximation error
• Gobet and Labart (2006). Error expansion for the discretization of
Backward Stochastic Differential Equations.
⇒ Under smoothness conditions and uniform ellipticity:
Ȳtπi − Yti = Du(ti, Xti )0(Xtπi − Xti ) + Oi(|π|) + O(|Xtπi − Xti |2)
√
and weak convergence of n(Y π − Y ) where Y π is a continuous-time
extension of Ȳ π .
• Similar result for Z̄ π .
Discrete-time scheme II:
BSDEs with jumps

Z t
Z tZ
Z t



Xt = X0 +
b(Xs)ds +
σ(Xs)dWs +
β(Xs−, e)µ̄(de, ds) ,



0
0
0 E





 Yt
= g(X1) +
Z 1
t
f (Θs) ds −
with Θ := (X, Y, Γ, Z) with Γ :=
Z 1
t
Z
E
ZsdWs −
Z 1Z
t
E
ρ(e)U (e)λ(de).
Us(e)µ̄(de, ds)
Construction
Step 1: Step-constant driver with π := {ti = i/n, i ≤ n}
Ȳtπi
Z t
Z t
Z
1 π π π π
i+1 π
i+1
π
Zt dWt −
= Ȳti+1 + f Xti , Ȳti , Γ̄ti , Z̄ti −
Utπ µ̄(de, dt) ,
n
ti
ti
E
where
Z t
i+1
ti
Ztπ dWt +
Z t
Z
i+1
ti
h
i
π
π
π
Ut µ̄(de, dt) = Ȳti+1 − E Ȳti+1 | Fti
E
Construction
Step 1: Step-constant driver with π := {ti = i/n, i ≤ n}
Ȳtπi
Z t
Z t
Z
1 π π π π
i+1 π
i+1
π
Zt dWt −
= Ȳti+1 + f Xti , Ȳti , Γ̄ti , Z̄ti −
Utπ µ̄(de, dt) ,
n
ti
ti
E
where
Z t
i+1
ti
Ztπ dWt +
Z t
Z
i+1
ti
h
i
π
π
π
Ut µ̄(de, dt) = Ȳti+1 − E Ȳti+1 | Fti
E
R
2
π
π
Step 2: Best L (Ω×[ti, ti+1]) approximation of Z and Γ = E U π (e)ρ(e)λ(de)
by Fti -meas. process
Z̄tπ
Γ̄π
t
"Z
ti+1
#
h
i
π
π
:= nE
Zs ds | Fti = nE Ȳti+1 (Wti+1 − Wti ) | Fti
ti
"Z
#
Z
ti+1
π
:= nE
Γπ
ρ(e)µ̄(de, (ti, ti+1]) | Fti
s ds | Fti = nE Ȳti+1
ti
E
Construction
Step 1: Step-constant driver with π := {ti = i/n, i ≤ n}
Ȳtπi
Z t
Z t
Z
1 π π π π
i+1 π
i+1
π
Zt dWt −
= Ȳti+1 + f Xti , Ȳti , Γ̄ti , Z̄ti −
Utπ µ̄(de, dt) ,
n
ti
ti
E
where
Z t
i+1
ti
Ztπ dWt +
Z t
Z
i+1
ti
h
i
π
π
π
Ut µ̄(de, dt) = Ȳti+1 − E Ȳti+1 | Fti
E
R
2
π
π
Step 2: Best L (Ω×[ti, ti+1]) approximation of Z and Γ = E U π (e)ρ(e)λ(de)
by Fti -meas. process
Z̄tπ
Γ̄π
t
"Z
ti+1
#
h
i
π
π
:= nE
Zs ds | Fti = nE Ȳti+1 (Wti+1 − Wti ) | Fti
ti
"Z
#
Z
ti+1
π
:= nE
Γπ
ρ(e)µ̄(de, (ti, ti+1]) | Fti
s ds | Fti = nE Ȳti+1
ti
E
Step 3: Discrete scheme: Ȳtπi =
E
h
i
1
π
π
π
π
π
Ȳti+1 | Fti + n f Xti , Ȳti , Γ̄ti , Z̄ti .
First bound
Set


Z 1
Err2 := max E  sup |Ȳtπi − Yt|2 +
i
0
ti≤t≤ti+1
E
h
i
π
2
π
2
|Zt − Z̄t | + |Γt − Γ̄t | dt
Theorem:
Err2 ≤ C |π| +
Z 1
0
h
i
E |Zt − Z̄t|2 dt +
Z 1
0
h
i
!
E |Γt − Γ̄t|2 dt
where
Z̄t := nE
"Z
ti+1
ti
#
Zsds | Fti
=⇒ Regularity of Z and Γ
and
Γ̄t := nE
"Z
ti+1
ti
#
Γsds | Fti
Discrete time approximation error
• Assumption : For each e ∈ E, the map x ∈ Rd 7→ β(x, e) admits a
Jacobian matrix ∇β(x, e) such that the function
(x, ξ) ∈ Rd × Rd 7→ a(x, ξ; e) := ξ 0(∇β(x, e) + Id)ξ
satisfies one of the following condition uniformly in (x, ξ) ∈ Rd × Rd
a(x, ξ; e) ≥ |ξ|2K −1
or
a(x, ξ; e) ≤ −|ξ|2K −1 .
Proposition: Under the above condition
n−1
X Z ti+1
i=0 ti
E
h
i
2
|Zt − Zti | dt ≤ C |π|
Theorem: Under the above condition
Err2 ≤ C |π| .
• The same results hold if σ, b, β(·, e), f and g are Cb1 functions with
K-Lipschitz continuous derivatives, uniformly in e ∈ E.
Discrete time approximation error
• Assumption : For each e ∈ E, the map x ∈ Rd 7→ β(x, e) admits a
Jacobian matrix ∇β(x, e) such that the function
(x, ξ) ∈ Rd × Rd 7→ a(x, ξ; e) := ξ 0(∇β(x, e) + Id)ξ
satisfies one of the following condition uniformly in (x, ξ) ∈ Rd × Rd
a(x, ξ; e) ≥ |ξ|2K −1
or
a(x, ξ; e) ≤ −|ξ|2K −1 .
Proposition: Under the above condition
n−1
X Z ti+1
i=0 ti
E
h
i
2
|Zt − Zti | dt ≤ C |π|
Theorem: Under the above condition
Err2 ≤ C |π| .
• The same results hold with h1−ε without condition. ...................................
.........................................................................;
Discrete time approximation error: extensions
• b, σ, β, and f can depend on t if 1/2-Hölder in t
Discrete time approximation error: extensions
• b, σ, β, and f can depend on t if 1/2-Hölder in t
• Consider the system
Mt = M0 +
Xt = X0 +
Z t
Z 0t
0
bM (Mr )dr +
Z tZ
b(Mr , Xr )dr +
Yt = g(M1, X1) +
Z 1
t
0 E
Z t
0
βM (Mr−, e)µ̄(de, dr)
σ(Mr , Xr )dWr +
f (Mr , Θr ) dr −
Z 1
t
Z tZ
Zr dWr −
0 E
Z 1Z
t
β(Mr−, Xr−, e)µ̄(de, dr)
E
Ur (e)µ̄(de, dr)
Discrete time approximation error: extensions
• b, σ, β, and f can depend on t if 1/2-Hölder in t
• Consider the system
Mt = M0 +
Xt = X0 +
Z t
Z 0t
0
bM (Mr )dr +
Z tZ
b(Mr , Xr )dr +
Yt = g(M1, X1) +
Z 1
t
0 E
Z t
0
βM (Mr−, e)µ̄(de, dr)
σ(Mr , Xr )dWr +
f (Mr , Θr ) dr −
Z 1
t
Z tZ
Zr dWr −
0 E
Z 1Z
t
β(Mr−, Xr−, e)µ̄(de, dr)
E
Ur (e)µ̄(de, dr)
⇒ M independent of X and its Malliavin derivative equals zero.
Discrete time approximation error: extensions
• b, σ, β, and f can depend on t if 1/2-Hölder in t
• Consider the system
Mt = M0 +
Xt = X0 +
Z t
Z 0t
0
bM (Mr )dr +
Z tZ
b(Mr , Xr )dr +
Yt = g(M1, X1) +
Z 1
t
0 E
Z t
0
βM (Mr−, e)µ̄(de, dr)
σ(Mr , Xr )dWr +
f (Mr , Θr ) dr −
Z 1
t
Z tZ
Zr dWr −
0 E
Z 1Z
t
β(Mr−, Xr−, e)µ̄(de, dr)
E
Ur (e)µ̄(de, dr)
⇒ M independent of X and its Malliavin derivative equals zero.
⇒ Inversibility condition on ∇β or smoothness of b(m, ·), σ(m, ·), β(m, ·),
f (m, ·) and g(m, ·) (for each m) sufficient.
Discrete-time scheme III
systems of semilinear PDEs
Mt =
Z tZ
0
eµ(de, ds) [κ]
E
dXt = b(Mt, Xt)dt + σ(Mt, Xt)dWt
−dYt = f (Mt, Xt, Yt, Zt, Γt)dt − ZtdWt −
Y1 = g(M1, X1)
Where f is not Lipschitz in M !
Z
E
Ut(e)µ̄(de, dt)
Construction
R·
• M = 0 eµ(de, dt) [κ] can be simulated perfectly.
• By adding the jump times of M in the Euler scheme X π of X one has

E

sup
t∈[ti,ti+1 ]
|Xt − Xtπi |2 ≤ C |ti+1 − ti|
even if b, σ and β are not Lipschitz in m.
Construction
R·
• M = 0 eµ(de, dt) [κ] can be simulated perfectly.
• By adding the jump times of M in the Euler scheme X π of X one has

E

sup
t∈[ti,ti+1 ]
|Xt − Xtπi |2 ≤ C |ti+1 − ti|
even if b, σ and β are not Lipschitz in m.
• The approximation scheme for Y is defined as follows:
Z̄tπ
Γ̄π
t
Ȳtπ
h
i
π
:= n E Ȳti+1 ∆Wi+1 | Fti
Z
:= n E Ȳtπi+1
ρ(e)µ̄(de, (ti, ti+1]) | Fti
E
"Z
#
h
i
ti+1
:= E Ȳtπi+1 | Fti + E
f Ms, Xtπi , Ȳtπi+1 , Z̄tπi , Γ̄π
ti ds | Fti
ti
for t ∈ [ti, ti+1), with the terminal condition Ȳ1π = g(M1, X1π ).
Discrete time approximation error
• We assume that, for each m, b(m, ·), σ(m, ·), β(m, ·), f (m, ·), g(m, ·)
satisfy the previous Lipschitz continuity assumption.
Discrete time approximation error
• We assume that, for each m, b(m, ·), σ(m, ·), β(m, ·), f (m, ·), g(m, ·)
satisfy the previous Lipschitz continuity assumption.
Proposition:
Let (Ft)t≤T and (F̄t)t≤T be defined, for t ∈ [ti, ti+1], by
"
Ft := f (Mt, Xti , Yti , Z̄ti , Γ̄ti ) , F̄t := E n
Z t
i+1
ti
f Ms, Xti , Yti , Z̄ti , Γ̄ti ds | Fti
#
Discrete time approximation error
• We assume that, for each m, b(m, ·), σ(m, ·), β(m, ·), f (m, ·), g(m, ·)
satisfy the previous Lipschitz continuity assumption.
Proposition:
Let (Ft)t≤T and (F̄t)t≤T be defined, for t ∈ [ti, ti+1], by
"
Ft := f (Mt, Xti , Yti , Z̄ti , Γ̄ti ) , F̄t := E n
Z t
i+1
ti
f Ms, Xti , Yti , Z̄ti , Γ̄ti ds | Fti
Then,
Err2 ≤ C |π| +
Z 1
0
h
i
!
E |Zt − Z̄t|2 + |Γt − Γ̄t|2 + |Ft − F̄t|2 dt
.
#
Discrete time approximation error
• We assume that, for each m, b(m, ·), σ(m, ·), β(m, ·), f (m, ·), g(m, ·)
satisfy the previous Lipschitz continuity assumption.
Proposition:
Let (Ft)t≤T and (F̄t)t≤T be defined, for t ∈ [ti, ti+1], by
"
Ft := f (Mt, Xti , Yti , Z̄ti , Γ̄ti ) , F̄t := E n
Z t
i+1
ti
f Ms, Xti , Yti , Z̄ti , Γ̄ti ds | Fti
Then,
Err2 ≤ C |π| +
Proposition:
Z 1
0
h
i
!
E |Zt − Z̄t|2 + |Γt − Γ̄t|2 + |Ft − F̄t|2 dt
Under the above conditions
Err2 ≤ C |π| +
Z 1
0
h
i
!
E |Ft − F̄t|2 dt −→ 0 .
.
#
Discrete-time scheme IV
Reflected BSDEs and free boundary problems

Z t
Z t



Xt = X0 +
b(Xs)ds +
σ(Xs)dWs ,



0
0




Z 1
Z 1

Yt = g(X1) +
f (Xs, Ys, Zs) ds −
ZsdWs + K1 − Kt



t
t

Z 1




( Ys − h(Xs) )dKs = 0 ,
 Yt ≥ h(Xt) , t ≤ 1 and
0
Previous works
• f independent of Z: Bally, Pages and Printemps (02,...), B. and
Touzi (04).
• f depends on Z: Bally (with randomization of the time horizon), Ma
and Zhang.
Construction
• Approximation scheme
Z̄tπi
Ỹtπi
Ȳtπi
h
i
π
= n Ei Ȳti+1 (Wti+1 − Wti )
h
i
π
= Ei Ȳti+1 + n−1 f (Xtπi , Ỹtπi , Z̄tπi )
π
π
= R ti , Xti , Ỹti , i ≤ N − 1 ,
with the terminal condition
ȲTπ = g(XTπ ) .
where for < = {rj , 0 ≤ j ≤ κ} ⊃ π
R(t, x, y) := y + [h(x) − y]+1{t∈<\{0,T }} , (t, x, y) ∈ [0, T ] × Rd+1 .
Construction
• Approximation scheme
Z̄tπi
Ỹtπi
Ȳtπi
h
i
π
= n Ei Ȳti+1 (Wti+1 − Wti )
h
i
π
= Ei Ȳti+1 + n−1 f (Xtπi , Ỹtπi , Z̄tπi )
π
π
= R ti , Xti , Ỹti , i ≤ N − 1 ,
with the terminal condition
ȲTπ = g(XTπ ) .
where for < = {rj , 0 ≤ j ≤ κ} ⊃ π
R(t, x, y) := y + [h(x) − y]+1{t∈<\{0,T }} , (t, x, y) ∈ [0, T ] × Rd+1 .
• f independent of Z (Bally and Pages): When < = π, error on Y
controled in
1
2
|π|
and by |π| when h is semi-convex.
Regularity of Z:
• Ma and Zhang approach: Integration by parts ⇒ uniform ellipticity
condition on σ
"
Zt = E g(X1)N1t +
Z 1
t
f (Θs)Nstds +
Z 1
t
#
NstdKs | Ft
where
Nrt := (r − t)−1
Z r
t
σ(Xs)−1∇XsdWs (∇Xt)−1σ(Xt)
Regularity of Z:
• Ma and Zhang approach: Integration by parts ⇒ uniform ellipticity
condition on σ
Theorem: (Ma and Zhang) If σ is uniformly elliptic, σ, b ∈ Cb1 and h
∈ Cb2, then
n−1
X Z ti+1
i=0 ti
E
h
i
1
2
|Zt − Zti | dt ≤ C |π| 2 .
If < = π, then
"
E
#
max|Yti − Ȳtπi |2
i≤n


1
π
2
π
2


2
+ max E
sup |Yt − Ȳti+1 |
+ kZ − ZkH2 ≤ C |π| ,
i≤n
t∈[ti,ti+1 ]
Regularity of Z: Discretely reflected case
• Discretely reflected BSDE (BC06 with simplication f = f (x))
Ỹtb = Yrbj+1 +
Z r
j+1
t
Ytb = R t , Xt , Ỹtb
f (Xs)ds −
Z r
j+1
t
ZsbdWs ,
on each [rj , rj+1] , j ≤ κ − 1 .
with Y1b = g(X1) and Θb = (X, Ỹ b, Z b).
Proposition: There is a version of Z b such that for each j ≤ κ − 1 and
t ∈ [rj , rj+1):
Ztb =
E ∇g(X1)∇X11{τj =1} + ∇h(Xτj )∇Xτj 1{τj <1} | Ft (∇Xt)−1σ(Xt)
+ E
h
i
Z τ
j
t
∇xf (Xu)∇Xudu | Ft (∇Xt)−1σ(Xt) .
where τj := inf{t ∈ < | t ≥ rj+1, h(Xt) > Ỹtb} ∧ 1 and s ∈ [rj , rj+1).
Regularity of Z: Discretely reflected case
1 and f ≡ 0. For simplicity:
• Main idea to conclude: Assume h = g is CL
Λ ≡ (∇Xt)−1σ(Xt) ≡ 1.
j
Ztb = Vt :=
h
E ∇h(Xτj )∇Xτj | Ft
i
is a martingale, thus, with ij s.t. tij = rj ,
n−1
X
i=0
"Z
E
ti+1
ti
#
|Ztb − Ztbi |2dt
=
i
−1
κ−1
X j+1
X Z ti+1
j=0 k=ij
≤ |π|
ti
−1
i
κ−1
X
X j+1
E
h
E
h
i
j
j 2
|Vt − Vt | dt
k
j
j
|Vt
|2 − |Vt |2
k+1
k
i
j=0 k=ij
i
h
κ−1
2
0
2
= |π| E |Vrκ | − |Vr0 |

κ−2
i
X h
j
2
j+1
2
+
E |Vrj+1 | − |Vrj+1 | 
j=1
Regularity of Z: Discretely reflected case
where
E
h
2
|Vrjj+1 |2 − |Vrj+1
|
j+1
i
h
i
j
j+1
≤ E ηrj+1 |Vrj+1 − Vrj+1 |
h
h
i
= E ηrj+1 |E ∇h(Xτj )∇Xτj − ∇h(Xτj+1 )∇Xτj+1 | Frj+1
≤ E η̂(τj+1 −
1
τj ) 2
so that
κ−2
X
j=1
h
2
E |Vrjj+1 |2 − |Vrj+1
|
j+1
i
≤
√
κ E η̂(τκ−1 −
1
τ1) 2
• Similarly if ∇h ∈ C 2 :
E
h
2
|Vrjj+1 |2 − |Vrj+1
|
j+1
i
h
≤ E η̂(τj+1 − τj )
i
so that
κ−2
X
j=1
E
h
2
|Vrjj+1 |2 − |Vrj+1
|
j+1
i
≤
E η̂(τκ−1 − τ1)
Regularity of Z: Discretely reflected case
1 with L-Lipschitz derivative, then
Theorem: If h ∈ CL
n−1
X
"Z
E
ti+1
ti
i=0
#
|Ztb − Ztbi |2dt
≤ C
√
κ |π| .
2 with L-Lipschitz derivatives + σ ∈ C 1 , then
If h ∈ CL
b
n−1
X
i=0
"Z
E
ti+1
ti
#
|Ztb − Ztbi |2dt
≤ C |π| .
Convergence speed: Discretely reflected case
• Assumptions: b, σ, g and f are Lipschitz-continuous.
i with L-Lipschitz derivatives of order up to i, with
• Use Hi : h ∈ CL
i = 1 or 2.
Theorem: Let H1 hold. Then,
"
E

#

1
π
b
2
π
b
2


max|Ȳti − Yti | + max E
sup |Ȳti+1 − Yt |
≤ C κ 2 |π|
i≤n
i≤n−1
t∈(ti,ti+1 ]
and
X
i
"Z
E
ti+1
ti
#
|Z̄tπi − Ztb|2dt
≤ C κ |π|
Convergence speed: Discretely reflected case
• Assumptions: b, σ, g and f are Lipschitz-continuous.
i with L-Lipschitz derivatives of order up to i, with
• Use Hi : h ∈ CL
i = 1 or 2.
Theorem: Let H2 hold + σ ∈ Cb1. Then,
"

#
E max|Ȳtπi − Ytbi |2 + max E 
i≤n
i≤n−1

|Ȳtπi+1 − Ytb|2 ≤ C |π|
sup
t∈(ti,ti+1 ]
and
X
i
"Z
E
ti+1
ti
#
|Z̄tπi − Ztb|2dt
≤ C κ |π|
Convergence speed: Discretely reflected case
• Assumptions: b, σ, g and f are Lipschitz-continuous.
i with L-Lipschitz derivatives of order up to i, with
• Use Hi : h ∈ CL
i = 1 or 2.
Theorem: Let H1 hold and X π = X on π. Then,
X
"Z
E
ti+1
ti
i
#
|Z̄tπi − Ztb|2dt
≤ C
1
κ2
|π|
Let H2 hold and X π = X on π. Then,
X
i
"Z
E
ti+1
ti
#
|Z̄tπi − Ztb|2dt
≤ C |π|
Convergence speed: Continuously reflected case
Theorem: Take < = π. Let H1 hold. Then,
"

#
E max|Ytπi − Yti |2 + max E 
i≤n
with α(π) =
i≤n−1
1
2
|π|

|Ytπi+1 − Yt|2 ≤ CL α(π) ,
sup
t∈(ti,ti+1 ]
under H1 and α(π) = |π| under H2.
Moreover, under H1,
X
"Z
E
i
ti+1
ti
#
|Z̄tπi − Zt|2dt
≤ CL
1
|π| 2
,
If H2 + σ ∈ Cb1 holds and X π = X on π, then
X
i
"Z
E
ti+1
ti
#
|Z̄tπi − Zt|2dt
≤ CL |π| .
Numerical methods
a. Regression based algorithm
b. Malliavin calculus approach
c. Quantization method
How to compute the conditional expectations ?
Ȳ1π
= g(X1π )
Z̄tπi
i
π
:= nE Ȳti+1 (Wti+1 − Wti ) | Fti
h
i
1 π π π
π
= E Ȳti+1 | Fti + f Xti , Ȳti , Z̄ti .
Ȳtπi
h
n
Regression based algorithms
• Simulate N path (X π,j )j≤N of X π
Regression based algorithms
• Simulate N path (X π,j )j≤N of X π
π,j
• Initialize: Ȳ1
π,j
= g(X1 ) for j ≤ N
Regression based algorithms
• Simulate N path (X π,j )j≤N of X π
π,j
• Initialize: Ȳ1
π,j
= g(X1 ) for j ≤ N
π,j
π,j
j
• Compute the non parametric regression: (Ȳt )j≤N and (Ȳt (Wt −
i+1
i+1
i+1
j
π,j
Wti ))j≤N on (Xti )j≤N and set
π,j
Ȳtπi+1 | Xtπi = Xti
i
Ê Ȳtπi+1 (Wti+1 − Wti ) | Xtπi = Xtπ,j
i
i
Ê
h
h
=
=
X
k≤κ
X
k≤κ
π,j
α̂k pk (Xti )
π,j
β̂k pk (Xti )
• Gives
π,j
Z̄ti
π,j
Ȳti
= n
=
X
k≤κ
X
k≤κ
π,j
β̂k pk (Xti )
1 π,j π,j π,j π,j
α̂k pk (Xti ) + f Xti , Ȳti , Z̄ti
.
n
Regression based algorithms
• Carrière (1996). Valuation of the Early-Exercise Price for Options
using Simulations and Nonparametric Regression, Insurance : mathematics and Economics
• Longstaff and Schwartz (2001). Valuing American Options By Simulation : A simple Least-Square Approach. Review of Financial Studies.
• Clément, Lamberton and Protter (2002). An analysis of a least
squares regression method for American option pricing, Finance and
Stochastics.
• Gobet, Lemor and Warin (2005). Rate of convergence of empirical regression method for solving generalized BSDE. preprint Ecole
Polytechnique.
Malliavin calculus approach
• Simulate N path (X π,j )j≤N of X π
Malliavin calculus approach
• Simulate N path (X π,j )j≤N of X π
π,j
• Initialize: Ȳ1
π,j
= g(X1 ) for j ≤ N
Malliavin calculus approach
• Simulate N path (X π,j )j≤N of X π
π,j
• Initialize: Ȳ1
π,j
= g(X1 ) for j ≤ N
• Estimate the conditional expectation using the representation
"
π 1
E Ȳtπi+1 | Xtπi = Xtπ,j
=
E
Ȳ
ti+1
i
h

∼ 
i


#
π,j
i
{Xtπ ≥Xt }
i
Φi /E 1
#
π,j
i
{Xtπ ≥Xt }
i

X
π,l
l
l


Ȳt 1 π,l
1
π,j Φi /
π,l
π,j Φi
i+1 {Xt ≥Xt }
{Xt ≥Xt }
i
i
i
i
l
l
X
"
Φi
Malliavin calculus approach
• Simulate N path (X π,j )j≤N of X π
π,j
• Initialize: Ȳ1
π,j
= g(X1 ) for j ≤ N
• Estimate the conditional expectation using the representation
"
π 1
E Ȳtπi+1 | Xtπi = Xtπ,j
=
E
Ȳ
t
i+1
i
h
i

∼ 


#
π,j
i
{Xtπ ≥Xt }
i
"
Φi /E 1
#
π,j
i
{Xtπ ≥Xt }
i
Φi

X
π,l
l
l


Ȳt 1 π,l
1
π,j Φi /
π,l
π,j Φi
i+1 {Xt ≥Xt }
{Xt ≥Xt }
i
i
i
i
l
l
X
• Set for j ≤ N
π,j
Z̄ti
π,j
Ȳti
i
π,j
π
π
:= nÊ Ȳti+1 (Wti+1 − Wti ) | Xti = Xti
i
h
1 π,j π,j π,j π,j
π
π
= Ê Ȳti+1 | Xti = Xti + f Xti , Ȳti ,ti
.
n
h
With a good algorithm, the complexity IS NOT in N 2 but N ln(N )d !
Malliavin calculus approach
• Simulate N path (X π,j )j≤N of X π
π,j
• Initialize: Ȳ1
π,j
= g(X1 ) for j ≤ N
• Estimate the conditional expectation using the representation
"
π 1
E Ȳtπi+1 | Xtπi = Xtπ,j
=
E
Ȳ
t
i+1
i
h
i

∼ 


#
π,j
i
{Xtπ ≥Xt }
i
"
Φi /E 1
#
π,j
i
{Xtπ ≥Xt }
i
Φi

X
π,l
l
l


Ȳt 1 π,l
1
π,j Φi /
π,l
π,j Φi
i+1 {Xt ≥Xt }
{Xt ≥Xt }
i
i
i
i
l
l
X
• Set for j ≤ N
π,j
Z̄ti
π,j
Ȳti
i
π,j
π
π
:= nÊ Ȳti+1 (Wti+1 − Wti ) | Xti = Xti
i
h
1 π,j π,j π,j π,j
π
π
= Ê Ȳti+1 | Xti = Xti + f Xti , Ȳti ,ti
.
n
h
⇒ With a good algorithm, the complexity IS NOT in N 2 but N ln(N )d !
Malliavin calculus approach
• Fournié, Lasry, Lebuchoux, and Lions (2001). Applications of Malliavin calculus to Monte Carlo methods in finance II, Finance and Stochastics.
• Bouchard, Ekeland and Touzi (2004). On the Malliavin approach to
Monte Carlo approximation of conditional expectations, Finance and
Stochastics.
• Bouchard and Touzi (2004). Discrete-Time Approximation and MonteCarlo Simulation of Backward Stochastic Differential Equations. Stochastic Processes and their Applications.
• Lions and Regnier (2001). Calcul du prix et des sensibilités d’une
option américaine par une méthode de Monte Carlo, preprint.
Quantization method
• Replace X π by a process X̂ π taking values in a sequence of grids
(Γti )ti such that
1. X̂tπi is the projection of Xtπi on Γti
2. Γti minimize the L2 distance between X̂tπi and Xtπi among all the
grids having the same size.
• Write the algorithm on this process after having computed the transition probabilities of X̂ π by MC methods: finite dimensional space,
everything becomes explicit.
⇒ Bally and Pages (2002). A quantization algorithm for solving discrete time multidimensional optimal stopping problems. Bernoulli.
⇒ and many others... see the web page of Pages.