Stability analysis of approximation schemes for … Stability and convergence Numerical Stability...
Transcript of Stability analysis of approximation schemes for … Stability and convergence Numerical Stability...
IntroductionStability and convergence
Numerical Stability
Stability analysis of approximation schemes for BSDEs
J-F Chassagneux (Imperial College London)joint work with .
A. Richou (Universite Bordeaux 1)
Advances in Financial Mathematics, Paris, 08-01-2014
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
IntroductionFrameworkBTZ scheme - (implicit Euler)
Stability and convergenceLinear Multi-step schemesL2-StabilityNumerical illustration
Numerical StabilityDefinitionSemi-explicit EulerImplicit Euler
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Outline
IntroductionFrameworkBTZ scheme - (implicit Euler)
Stability and convergenceLinear Multi-step schemesL2-StabilityNumerical illustration
Numerical StabilityDefinitionSemi-explicit EulerImplicit Euler
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Backward Sto. Diff. Eq. - Markovian Setting
I Decoupled Forward Backward SDE on [0,T ]:
Xt = X0 +
∫ t
0b(Xs)ds +
∫ t
0σ(Xs)dWs
Yt = g(XT ) +
∫ T
tf (Ys ,Zs)ds −
∫ T
tZsdWs
↪→ b, σ, f and g are Lipschitz functions.
I PDE representation: Yt = u(t,Xt), u solution to
−L(0)u = f (u, L(1)u) and u(T , .) = g
where L(0)u := ∂tu + LXu and L(1)u = ∂xu.σ.If smoothness,
Zt = L(1)[u](t,Xt) and f (Yt ,Zt) = −L(0)[u](t,Xt).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Backward Sto. Diff. Eq. - Markovian Setting
I Decoupled Forward Backward SDE on [0,T ]:
Xt = X0 +
∫ t
0b(Xs)ds +
∫ t
0σ(Xs)dWs
Yt = g(XT ) +
∫ T
tf (Ys ,Zs)ds −
∫ T
tZsdWs
↪→ b, σ, f and g are Lipschitz functions.I PDE representation: Yt = u(t,Xt), u solution to
−L(0)u = f (u, L(1)u) and u(T , .) = g
where L(0)u := ∂tu + LXu and L(1)u = ∂xu.σ.If smoothness,
Zt = L(1)[u](t,Xt) and f (Yt ,Zt) = −L(0)[u](t,Xt).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Non-linearity and finance
Example
The stock price is given by
Xt = X0 +
∫ t
0µXsds +
∫ t
0σXsdWs
The price of an european option with payoff g , assuming different ratefor borrowing (R) and lending (r) is given by
Yt = g(XT ) +
∫ T
t(−rYs +
µ− r
σZs + (R − r)[Ys −
Zs
σ]−)ds −
∫ T
tZsdWs .
↪→ These are the dynamics of the value of the optimal hedging portfolio.
I Non-linearity coming from f (y , z) = −ry + µ−rσ z + (R − r)[y − z
σ ]−
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs = Yti+1 +
∫ ti+1
tif (Ys ,Zs)ds (1)
I For the Y -part:
Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:
Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs ' Yti+1 + hf (Yti ,Zti ) (1)
I For the Y -part:
Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:
Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs ' Yti+1 + hf (Yti ,Zti ) (1)
I For the Y -part:
Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:
Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs ' Yti+1 + hf (Yti ,Zti ) (1)
I For the Y -part:Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]
↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:
Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs ' Yti+1 + hf (Yti ,Zti ) (1)
I For the Y -part:Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:
Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs ' Yti+1 + hf (Yti ,Zti ) (1)
I For the Y -part:Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:
Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs ' Yti+1 + hf (Yti ,Zti ) (1)
I For the Y -part:Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]
Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs ' Yti+1 + hf (Yti ,Zti ) (1)
I For the Y -part:Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Deriving the scheme
We are given an equidistant grid π = {0 = t0 < ... < ti < ... < tn = T},define h = T/n.
I Start with: Yti +∫ ti+1
tiZsdWs ' Yti+1 + hf (Yti ,Zti ) (1)
I For the Y -part:Take conditional expectation, Yti ' Eti
[Yti+1 + hf (Yti ,Zti )
]↪→ Yi := Eti[Yi+1 + hf (Yi ,Zi )]
I For the Z -part:Multiply (1) by ∆Wi := Wti+1 −Wti , take conditional expectation:
Eti
[∫ ti+1
tiZsds
]' Eti
[∆WiYti+1
]Say Eti
[∫ ti+1
tiZsds
]' hZti , =⇒ hZti ' Eti[∆WiYi+1]
↪→ Zi := Eti[HiYi+1] with Hi := h−1∆Wi .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Objectives
I The Scheme: given the terminal condition Yn = g(Xn), thetransition from step i + 1 to i is
Yi := Eti[Yi+1 + hf (Yi ,Zi )]
Zi := Eti[HiYi+1]
↪→ The curse of computing the conditional expectations.I Goals:
(i) Prove the convergence of the scheme (using a first notion of stability)(ii) Understand the behaviour of the scheme in practice (using a second
notion of stability)
I Remark: for this scheme, convergence has been proved by Zhangand Bouchard - Touzi (2004) and Gobet-Labart (2007).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Objectives
I The Scheme: given the terminal condition Yn = g(Xn), thetransition from step i + 1 to i is
Yi := Eti[Yi+1 + hf (Yi ,Zi )]
Zi := Eti[HiYi+1]
↪→ The curse of computing the conditional expectations.
I Goals:
(i) Prove the convergence of the scheme (using a first notion of stability)(ii) Understand the behaviour of the scheme in practice (using a second
notion of stability)
I Remark: for this scheme, convergence has been proved by Zhangand Bouchard - Touzi (2004) and Gobet-Labart (2007).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Objectives
I The Scheme: given the terminal condition Yn = g(Xn), thetransition from step i + 1 to i is
Yi := Eti[Yi+1 + hf (Yi ,Zi )]
Zi := Eti[HiYi+1]
↪→ The curse of computing the conditional expectations.I Goals:
(i) Prove the convergence of the scheme (using a first notion of stability)(ii) Understand the behaviour of the scheme in practice (using a second
notion of stability)
I Remark: for this scheme, convergence has been proved by Zhangand Bouchard - Touzi (2004) and Gobet-Labart (2007).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Objectives
I The Scheme: given the terminal condition Yn = g(Xn), thetransition from step i + 1 to i is
Yi := Eti[Yi+1 + hf (Yi ,Zi )]
Zi := Eti[HiYi+1]
↪→ The curse of computing the conditional expectations.I Goals:
(i) Prove the convergence of the scheme (using a first notion of stability)
(ii) Understand the behaviour of the scheme in practice (using a secondnotion of stability)
I Remark: for this scheme, convergence has been proved by Zhangand Bouchard - Touzi (2004) and Gobet-Labart (2007).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Objectives
I The Scheme: given the terminal condition Yn = g(Xn), thetransition from step i + 1 to i is
Yi := Eti[Yi+1 + hf (Yi ,Zi )]
Zi := Eti[HiYi+1]
↪→ The curse of computing the conditional expectations.I Goals:
(i) Prove the convergence of the scheme (using a first notion of stability)(ii) Understand the behaviour of the scheme in practice (using a second
notion of stability)
I Remark: for this scheme, convergence has been proved by Zhangand Bouchard - Touzi (2004) and Gobet-Labart (2007).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
FrameworkBTZ scheme - (implicit Euler)
Objectives
I The Scheme: given the terminal condition Yn = g(Xn), thetransition from step i + 1 to i is
Yi := Eti[Yi+1 + hf (Yi ,Zi )]
Zi := Eti[HiYi+1]
↪→ The curse of computing the conditional expectations.I Goals:
(i) Prove the convergence of the scheme (using a first notion of stability)(ii) Understand the behaviour of the scheme in practice (using a second
notion of stability)
I Remark: for this scheme, convergence has been proved by Zhangand Bouchard - Touzi (2004) and Gobet-Labart (2007).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Outline
IntroductionFrameworkBTZ scheme - (implicit Euler)
Stability and convergenceLinear Multi-step schemesL2-StabilityNumerical illustration
Numerical StabilityDefinitionSemi-explicit EulerImplicit Euler
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Designing more acurate scheme, Y -part
I We want to find coefficients (aj), (bj):
Yti = Eti
r∑j=1
ajYti+j + hr∑
j=0
bj f (Yti+j ,Zti+j )
+ εYi
with the truncation error εYi = Oi (hm).
I Observing that:
Eti
[Yti+j
]=
m∑k=0
cYk (ti )|jh|k + O(hm+1)
Eti
[f (Yti+j ,Zti+j )
]= −
m∑k=1
cYk (ti )|jh|k + O(hm)
we end up matching coefficients to cancel terms...
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Designing more acurate scheme, Y -part
I We want to find coefficients (aj), (bj):
Yti = Eti
r∑j=1
ajYti+j + hr∑
j=0
bj f (Yti+j ,Zti+j )
+ εYi
with the truncation error εYi = Oi (hm).I Observing that:
Eti
[Yti+j
]=
m∑k=0
cYk (ti )|jh|k + O(hm+1)
Eti
[f (Yti+j ,Zti+j )
]= −
m∑k=1
cYk (ti )|jh|k + O(hm)
we end up matching coefficients to cancel terms...
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Designing more acurate scheme, Z -part
I We want to find coefficients (αj), (βj):
Zti = Eti
r∑j=1
αjHYi ,jYti+j + h
r∑j=1
βjHfi ,j f (Yti+j ,Zti+j )
+ εZi .
such that εZi = Oi (hm)
I The following expansions hold true:
Eti
[HYi ,jYti+j
]=
m−1∑k=0
cZk (ti )|jh|k + O(hm)
Eti
[H fi ,j f (Yti+j ,Zti+j )
]= −
m−1∑k=1
cZk (ti )|jh|k + O(hm−2)
where Hi ,j = 1jh
∫ ti+j
tiϕ( s−tihj )dWs for some ’special’ functions ϕ.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Designing more acurate scheme, Z -part
I We want to find coefficients (αj), (βj):
Zti = Eti
r∑j=1
αjHYi ,jYti+j + h
r∑j=1
βjHfi ,j f (Yti+j ,Zti+j )
+ εZi .
such that εZi = Oi (hm)I The following expansions hold true:
Eti
[HYi ,jYti+j
]=
m−1∑k=0
cZk (ti )|jh|k + O(hm)
Eti
[H fi ,j f (Yti+j ,Zti+j )
]= −
m−1∑k=1
cZk (ti )|jh|k + O(hm−2)
where Hi ,j = 1jh
∫ ti+j
tiϕ( s−tihj )dWs for some ’special’ functions ϕ.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Linear multi-step scheme
Definition(i) To initialise the scheme with r steps, r ≥ 1, we are given r terminalconditions (Yn−j ,Zn−j), Ftn−j -measurable square integrable randomvariables, 0 ≤ j ≤ r − 1.(ii) For i ≤ n − r , the computation of (Yi ,Zi ) involves r steps and isgiven by
Yi = Eti
[∑rj=1 ajYi+j + h
∑rj=0 bj f (Yi+j ,Zi+j)
]Zi = Eti
[∑rj=1 αjH
Yi ,jYi+j + h
∑rj=1 βjH
fi ,j f (Yi+j ,Zi+j)
]where aj , bj , αj , βj are real numbers.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
A pertubation approach
We consider a pertubed scheme: Yi = Eti
[∑rj=1 aj Yi+j + h
∑rj=0 bj f (Yi+j , Zi+j)
]+ ζYi
Zi = Eti
[∑rj=1 αjH
yi ,j Yi+j + h
∑rj=1 βjH
fi ,j fi ,j f (Yi+j , Zi+j)
]+ ζZi
set δYk := Yk − Yk , δZk := Zk − Zk .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Definition: L2-Stability
DefinitionThe multi-step scheme is said to be L2-stable if
max0≤i≤n−r
E[|δYi |2
]+
n−r∑i=0
hE[|δZi |2
]≤ C
(max
0≤j≤r−1E[|δYn−j |2 + h|δZn−j |2
]+ h
n−r∑i=0
E[
1
h2|ζYi |2 + |ζZi |2
])for all sequences ζYi ,ζZi of L2(Fti )-random variable and terminal values(Yn−j ,Zn−j), (Yn−j , Zn−j) belonging to L2(Ftn−j ), 0 ≤ j ≤ r − 1.
i.e. the overall error is the sum of the error done at each step +initialization error. C will depend ’dramatically’ on T and Lip(f ).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Main result
Recall the scheme:Yi = Eti
[∑rj=1 ajYi+j + h
∑rj=0 bj f (Yi+j ,Zi+j)
]Zi = Eti
[∑rj=1 αjH
Yi ,jYi+j + h
∑rj=1 βjH
fi ,j f (Yi+j ,Zi+j)
]TheoremAssume that f is Lipschitz continuous and the following holds
(H) The coefficients (aj) are non-negative,∑r
j=1 aj = 1 and for1 ≤ j ≤ r , aj = 0 =⇒ αj = 0,
then, the multi-step scheme is L2-stable, for h small enough.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Conclusion
I Stability + control of truncation error =⇒ convergence
I Possibly high order rate of convergence, if smoothness
I Scheme initialisation: use one-step method: Runge-Kutta method!
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Example 1
Example
In this example, we work with d = 1. The decoupled FBSDE is given by
Xt = 1 +
∫ t
0µXsds +
∫ t
0σXsdWs
Yt =X1
1 + X1+
∫ 1
tZs(σYs −
µ
σ)ds −
∫ 1
tZsdWs , with (µ, σ) = (0.05, 0.4) .
A direct application of Ito formula yields that (Yt ,Zt) = ( Xt1+Xt
, σXt(1+Xt)2 ).
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Graph Example 1
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Example 2
Example
In this example, we work with d = 3 and X = W . We consider thefollowing BSDE
Yt =ω(1,W1)
1 + ω(1,W1)+
∫ 1
t(Z 1
s + Z 2s + Z 3
s )(5
6− Ys)ds −
∫ 1
tZsdWs .
where ω(t, x) = exp (x1 + x2 + x3 + t). Applying Ito formula, we verifythat the solution is given by
Yt =ω(t,Wt)
1 + ω(t,Wt)and Z l
t =ω(t,Wt)
(1 + ω(t,Wt))2, l ∈ {1, 2, 3} , t ∈ [0, 1] .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
Linear Multi-step schemesL2-StabilityNumerical illustration
Graph Example 2
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Outline
IntroductionFrameworkBTZ scheme - (implicit Euler)
Stability and convergenceLinear Multi-step schemesL2-StabilityNumerical illustration
Numerical StabilityDefinitionSemi-explicit EulerImplicit Euler
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Some questions to answer
I Many possible schemes, some better than other? (with same orderof convergence)
I L2-stability, convergence: asymptotic result when h is small... Whatif f has a big Lipschitz constant? Stiff problems.
I What if T is big?
I Will the scheme always return a reasonable value? Are there someconstraints on h?
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Framework
We will discuss the two following schemes:
I BTZ-scheme ’Implicit Euler’:
Yi := Eti[Yi+1 + hf (Yi ,Zi )]
Zi := Eti[HiYi+1]
I ’Semi explicit’ Euler:
Yi := Eti[Yi+1 + hf (Yi+1,Zi )]
Zi := Eti[HiYi+1]
I Eti[Hi ] = 0, Eti
[|Hi |2
]≤ Λ, for some given Λ > 0.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Assumptions
I f (0, 0) = 0 =⇒ (0, 0) is solution with terminal condition 0.
I f is Lipschitz continuous in Y (uniformly), i.e.
|f (y , z)− f (y ′, z)| ≤ LY |y − y ′| .
and
yf (y , 0) ≤ −lY |y |2
where LY , lY are non-negative real numbers.
I We also assume that f is Lipschitz continuous in z (uniformly) i.e.
|f (y , z)− f (y , z ′)| ≤ LZ |z − z ′| .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Assumptions
I f (0, 0) = 0 =⇒ (0, 0) is solution with terminal condition 0.
I f is Lipschitz continuous in Y (uniformly), i.e.
|f (y , z)− f (y ′, z)| ≤ LY |y − y ′| .
and
yf (y , 0) ≤ −lY |y |2
where LY , lY are non-negative real numbers.
I We also assume that f is Lipschitz continuous in z (uniformly) i.e.
|f (y , z)− f (y , z ′)| ≤ LZ |z − z ′| .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Assumptions
I f (0, 0) = 0 =⇒ (0, 0) is solution with terminal condition 0.
I f is Lipschitz continuous in Y (uniformly), i.e.
|f (y , z)− f (y ′, z)| ≤ LY |y − y ′| .
and
yf (y , 0) ≤ −lY |y |2
where LY , lY are non-negative real numbers.
I We also assume that f is Lipschitz continuous in z (uniformly) i.e.
|f (y , z)− f (y , z ′)| ≤ LZ |z − z ′| .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Remarks
I Things can go wrong already for ODEs: y ′ = f (y) withf (y) = −ay , a > 0.
Explicit Euler scheme satisfies: yn = (1− ah)ny0
if h > 2a and n is big, we get NaN!
h = 2a is a critical value.
I For this talk, numerical illustration done using binomial tree (or
trinomial tree) for brownian motion (W ).
↪→ The terminal condition is given by cos(WT ).
↪→ T will be big to get a sharp characterisation of thestability/unstability.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Remarks
I Things can go wrong already for ODEs: y ′ = f (y) withf (y) = −ay , a > 0.
Explicit Euler scheme satisfies: yn = (1− ah)ny0
if h > 2a and n is big, we get NaN!
h = 2a is a critical value.
I For this talk, numerical illustration done using binomial tree (or
trinomial tree) for brownian motion (W ).
↪→ The terminal condition is given by cos(WT ).
↪→ T will be big to get a sharp characterisation of thestability/unstability.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Some Definitions
Let ξ be a bounded terminal condition (random).
I Numerical stability: We say that the scheme is asymptoticallynumerically stable if there exists h∗ > 0, such that for all h ≤ h∗
|Y0| ≤ C ||ξ||∞,
where C is a constant that does not depend on T .
I A-stability: if, moreover, h∗ =∞, then we say that the scheme isA-stable.
↪→ In practice, we could expect 2 regimes for the scheme:
I h < h: scheme returns a ’reasonable’ value
I h > h: scheme is unstable
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Some Definitions
Let ξ be a bounded terminal condition (random).
I Numerical stability: We say that the scheme is asymptoticallynumerically stable if there exists h∗ > 0, such that for all h ≤ h∗
|Y0| ≤ C ||ξ||∞,
where C is a constant that does not depend on T .
I A-stability: if, moreover, h∗ =∞, then we say that the scheme isA-stable.
↪→ In practice, we could expect 2 regimes for the scheme:
I h < h: scheme returns a ’reasonable’ value
I h > h: scheme is unstable
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Strict monotony (lY > 0) and one dimensional Y
TheoremAssume that
1− h∗|LY |2
2lY− LZh∗max
i|Hi | ≥ 0 (1)
then the scheme is numerically asymptoticaly stable for the Y part.
↪→ Scheme based on binomial tree, Linear case: f (y , z) = −ay + bz .the condition reads
√h ≤
√(b
a
)2
+2
a− b
a> 0 .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Strict monotony (lY > 0) and one dimensional Y
TheoremAssume that
1− h∗|LY |2
2lY− LZh∗max
i|Hi | ≥ 0 (1)
then the scheme is numerically asymptoticaly stable for the Y part.
↪→ Scheme based on binomial tree, Linear case: f (y , z) = −ay + bz .the condition reads
√h ≤
√(b
a
)2
+2
a− b
a> 0 .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Strict monotony (lY > 0) and multidimensional Y
TheoremAssume that
1−
(LY√
h∗ + LZ√
Λ)2
2lY≥ 0 (2)
then the scheme is numerically asymptoticaly stable for the Y part.
↪→ Scheme based on binomial tree, Linear case: f (y , z) = −ay + bz .the condition reads
√h∗ =
√2
a− b
a.
More restrictive condition.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Strict monotony (lY > 0) and multidimensional Y
TheoremAssume that
1−
(LY√
h∗ + LZ√
Λ)2
2lY≥ 0 (2)
then the scheme is numerically asymptoticaly stable for the Y part.
↪→ Scheme based on binomial tree, Linear case: f (y , z) = −ay + bz .the condition reads
√h∗ =
√2
a− b
a.
More restrictive condition.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Numerical illustration f (y , z) = −ay + bz , b = 5
Yellow = unstable, correct Y0 value ' 0.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Strict monotony (lY > 0) and multidim Y
TheoremAssume that
1
Λ− |L
Z |2
2ly≥ 0 ,
then the scheme is A-stable.
↪→ Scheme based on binomial tree, Linear case: f (y , z) = −ay + bz .the condition reads
1− b2
2a≥ 0 .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Strict monotony (lY > 0) and multidim Y
TheoremAssume that
1
Λ− |L
Z |2
2ly≥ 0 ,
then the scheme is A-stable.
↪→ Scheme based on binomial tree, Linear case: f (y , z) = −ay + bz .the condition reads
1− b2
2a≥ 0 .
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
One dimensional Y , possibly lY = 0
TheoremAssume that that
1− LZh∗maxi|Hi | ≥ 0 (3)
then the scheme is asymptotically numerically stable.
↪→ Scheme based on binomial tree, Linear case: f (y , z) = −ay + bz .the condition reads
h∗ ≤ 1
b2.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
One dimensional Y , possibly lY = 0
TheoremAssume that that
1− LZh∗maxi|Hi | ≥ 0 (3)
then the scheme is asymptotically numerically stable.
↪→ Scheme based on binomial tree, Linear case: f (y , z) = −ay + bz .the condition reads
h∗ ≤ 1
b2.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Financial application
I Consider the model with two different interest rates and thefollowing ’plausible’ parameters:
r = 0.03, R = 0.06, σ = 0.4.
I then (for binomial tree), we get
- Implicit scheme: A-stable!- Semi explicit scheme: h ≥ 12.
No worries, at least for d = 1...
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Financial application
I Consider the model with two different interest rates and thefollowing ’plausible’ parameters:
r = 0.03, R = 0.06, σ = 0.4.
I then (for binomial tree), we get
- Implicit scheme: A-stable!- Semi explicit scheme: h ≥ 12.
No worries, at least for d = 1...
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Numerical illustration −ay + bz , b = 5
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Von Neumann Stability Analysis, f (y , z) = bz
I For k ∈ R, ξ = e ikWtn (binomial tree),
Yi = yieikWti
with
yi = λyi+1 with λ := E[(1 + b∆Wi )e ik∆Wi
]
I The scheme is stable iff |λ| ≤ 1 i.e. (after some computations)
h|b|2 ≤ 1 !
I Remark: observe that the dimension of b (so W ) impacts thestability of the scheme.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Von Neumann Stability Analysis, f (y , z) = bz
I For k ∈ R, ξ = e ikWtn (binomial tree),
Yi = yieikWti
with
yi = λyi+1 with λ := E[(1 + b∆Wi )e ik∆Wi
]I The scheme is stable iff |λ| ≤ 1 i.e. (after some computations)
h|b|2 ≤ 1 !
I Remark: observe that the dimension of b (so W ) impacts thestability of the scheme.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Von Neumann Stability Analysis, f (y , z) = bz
I For k ∈ R, ξ = e ikWtn (binomial tree),
Yi = yieikWti
with
yi = λyi+1 with λ := E[(1 + b∆Wi )e ik∆Wi
]I The scheme is stable iff |λ| ≤ 1 i.e. (after some computations)
h|b|2 ≤ 1 !
I Remark: observe that the dimension of b (so W ) impacts thestability of the scheme.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Numerical illustration f (z) = bz
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Numerical illustration f (z) = b|z |
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Numerical illustration f (z) = sin(bz)
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Conclusion
I Above results hold true for US options.
I Convergent method for BSDEs with unbounded terminal time?
I Numerical stability condition for other schemes?
I Numerical experiments to run with other methods to computeconditional expectations, in greater dimension.
J-F Chassagneux Stability of approx. for BSDEs
IntroductionStability and convergence
Numerical Stability
DefinitionSemi-explicit EulerImplicit Euler
Non-linearity is beautiful!
Thank you!
J-F Chassagneux Stability of approx. for BSDEs