how to construct stochastic models ? theory and numerics wiedzy... · 2015-03-21 · how to...

How to construct stochastic models ? Theory andnumerics

Jacky CRESSON1,2

1LMAP/Université de Pau

2SYRTE/Observatoire de Paris

March 2015

joint work with Stefanie Sonner (TU Kaiserlautern), Benedicte Puig (Univ. Pau), Messoud Efendiev (TU

München and Helmholtz zentrum), Frederic Pierret (Observatoire de Paris), Sebastien Darses (Univ.

Aix-Marseille), Yasmina Khelou� (Univ. Pau)

Why we need stochastic systems ? some examplesI- Construction of stochastic models

The stochastisation problemStochastic di�erential equationsInvariance problemsFirst integralsInvariant manifoldsII- Validation of models : numerics

Position of the problemGhost equilibrium pointsStability/instability artefactsInvariance

Position of the problem

Why Stochastic models ?

In many cases, we have well established deterministic models

which are described by di�erential or a partial di�erential

equations.For some reasons, one has to take into account stochasticphenomenons or perturbations.

Stochastic : randomness depending on time.

Biology

I Hodgekin-Huxley model for neuronal dynamics 1 2

I Due to the stochasticity of stochastic behavior of neurons

and voltage-dependent ion channel

1. Fox, Stochastic versions of the Hodgkin-Huxley equations, Biophys. J. (1997).

2. Saarinen, Linne, Yli-Harja, SDE model for cerebellar granule cell excitability, Plos Comput. Biol.(2008).

Virology

I HIV dynamics 3 4

I stochastic e�ects arise by virtue of nature in the infection

process when virions bind to receptors and interact withuninfected cells

I randomness is also assumed in the transition of uninfected cellsinto latently infected or actively infected cells and described bythe transition probabilities p and 1− p, respectively 5.

3. Kamina, Makuch, Zhao, A stochastic modeling of early HIV-1 population dynamics, Math. Bios.(2001).

4. Tuckwell, Le Corfec, A stochastic model for early HIV-1 population dynamics, J. Theor. Biol.(1998).

5. An Evolutionary Role for HIV Latency in Enhancing Viral Transmission, Rouzine, Igor M. et al.Cell, Volume 160, Issue 5, 1002-1012.

Population dynamics

I Population dynamics 6

I Due to an environment that is subject to random

�uctuations.

I Tumour growth, chemotherapy and optimal therapy strategy 7

I The basic tumor growth kinetics is stochastically perturbedby the cytotoxic drug due to the chemotherapy

6. Braumann, Growth and extinction of pop. in randomly varying environments, Comput. Math.Appl. (2008).

7. Lisei, Julitz, A stochastic model for the growth of cancer tumors, Studia Univ. Babes-Bolay,(2008).

Physics

I Landau-Lipshitz model for ferromagnetism

I Due to stochastic behavior of the magnetic �eld

I Stochastic climate and weather modelling 8

I Due to unknown processes and multi scale phenomenon

8. C.L.E. Franzke and al., Stochastic climate theory and modelling,arxiv :1409.0423v1, 2014.

Physics

I Landau-Lipshitz model for ferromagnetismI Due to stochastic behavior of the magnetic �eld

I Stochastic climate and weather modelling 8

I Due to unknown processes and multi scale phenomenon

8. C.L.E. Franzke and al., Stochastic climate theory and modelling,arxiv :1409.0423v1, 2014.

Astronomy

I Two-body problem 9 10

I Due to the stochastic �uctuations of the zodiacal dust

around the sun

Figure: The dust sphere

9. N. Sharma and H. Parthasarathy, Dynamics of a stochastically perturbed two-body problem,Proc.R. Soc. A 2007 463, 979-1003

10. J.Cresson, F. Pierret, B. Puig, The Sharma-Parthasarathy stochastic two-body problem, Journalof Mathematical Physics, 56(3), 2015, 21.p

I Motion of a satellite around an oblate planet : J2 problem 11

I Due to the stochastic �uctuations of the shape of the

Earth.

Figure: Shape of the Earth

Figure: Earth's second degree zonal harmonic

11. E. Behar, J. Cresson, F. Pierret, Dynamics of a rotating �at ellipsoid with a stochastic oblateness,Arxiv 1410.0667, Oct. 2014, 14.p.

The stochastisation problem

Give a sense todx

dt= f (x , t) + ”noise”

Problem : What do we mean by "noise" ?

I Construction of the stochastic model ?

I Find a suitable framework to deal with stochasticity and

dynamics so to de�ne the "noise".I In this framework, de�ne the suitable stochastic analogue of

classical constraints 12.I Explain how to preserve the constraints.

I Validation of the stochastic model ?

I Find a numerical methods adapted to stochastic systems andpreserving the speci�c constraints of the model.

12. Not all constraints are interesting. We must consider constraints which arepresent independently of the modelling


Give a sense todx





dynamics so to de�ne the "noise".

I In this framework, de�ne the suitable stochastic analogue ofclassical constraints 12.

I Explain how to preserve the constraints.





Give a sense todx






classical constraints 12.

I Explain how to preserve the constraints.





Give a sense todx











Give a sense todx







I Validation of the stochastic model ?I Find a numerical methods adapted to stochastic systems and

preserving the speci�c constraints of the model.


Stochastic framework

As we are dealing initially with di�erential or partial di�erentialequations, we want to consider a stochastic framework includingthese objects. A common framework is given by

Stochastic di�erential or partial di�erential equations 13

(SDEs or SPDEs)

13. For another strategy see : E. Nelson, Dynamical theories of Brownian mo-tion, Second edition, Princeton University Press, 2001 and J. Cresson, S. Darses,Stochastic embedding of dynamical systems, J. Math. Phys. 48, 072703 (2007)(54 pages)

Itô and Stratonovich formalismsWe consider systems of Itô's equations

dXt = f (t,Xt)dt + σ(t,Xt)dWt ,X |t=0 = X0,

(1)

where X = (X1, . . . ,Xd), f = (f1, . . . , fd) and σ = (σ1, . . . , σd)and dWt are "Brownian" increments.

This notation is formally de�ned by the integral relation

Xt = X0 +

∫ t

0

f (s,Xs) ds +

∫ t

0

σ(s,Xs) dWs ,

where the second integral is an Itô integral.

Chain rule in Itô stochastic calculus : Itô formula

df (t,Xt) =∂f

∂tdt + (∇T

Xf )dXt +

1

2(dXT

t )(∇2Xf )dXt , (2)

where ∇Xf = ∂f /∂X is the gradient of f w.r.t. X , ∇2Xf = ∇X∇T

Xf

is the Hessian matrix of f w.r.t. X, δ is the Kronecker symbol andthe following rules of computation are used : dtdt = 0,dtdWt,i = 0, dWt,idBt,j = δijdt.

Stratonovich stochastic di�erential equations

dXt = µ(t,Xt)dt + σ(t,Xt) ◦ dWt , (3)

Xt = x +

∫ t

0µ(s,Xs)ds +

∫ t

0σ(s,Xs) ◦ dWt , (4)

where the second integral is a Stratonovich integral.

Transformation formula

Solutions of the Stratonovich di�erential equation (3) correspondsto solutions of a modi�ed Itô equation :

dXt = µcor(t,Xt)dt + σ(t,Xt)dWt , (5)

where

µcor(t, x) =

[µ(t, x) +

1

2σ′(t, x)σ(t, x)

]. (6)

The correction term is also called the Wong-Zakai correctionterm.

µcor,i(t, x) = µi (t, x) +1

2

p∑j=1

n∑k=1

∂σi,j∂xk

σk,j , 1 ≤ i ≤ n. (7)

Advantage of the Stratonovich integral : the chain rule formulas isthe classical one !

Invariance

This is an ubiquitous property of many deterministic models : a setof variables remain in a �xed set under the dynamical evolution ofthe system.Examples :

I densities must belong to the interval [0, 1]

I the oblateness of a planet must remain in a given admissibleinterval

I a population remains positive

These invariance constraints are independent of the underlyingnature of the dynamics : deterministic or stochastic.

Invariance for deterministic systems

Stochastic invariance : a de�nition

De�nitionWe call the subset K ⊂ Rm invariant for the stochastic system(f , g) if for every initial data X0 ∈ K and initial time t0 ≥ 0 thecorresponding solution X (t), t ≥ t0, satis�es

P ({X (t) ∈ K , t ∈ [t0,∞[}) = 1,

i.e., the solution almost surely attains values within the set K .

Additive noise

{du = 0 dt + dWt

u(0) = u0,(8)

I The positivity is not preserved by the solutions of theperturbed stochastic system (8).

I u(t) = u(0) +Wt .

Multiplicative noise

We consider a linear, multiplicative noise of the form{du = 0 dt + αu ◦ dWt

u(0) = u0,(9)

where the constant α ∈ R.

I The stochastic problem (9) preserves positivity.

I

u(ω, t) = u0e−(α

2

2t−αWt(ω))

Invariance for stochastic systems 14

TheoremLet I ⊂ {1, . . . ,m} be a non-empty subset. Then, the set

K+ := {x ∈ Rm : xi ≥ 0, i ∈ I}

is invariant for the stochastic system (f , σ) if and only if

fi (t, x) ≥ 0 for x ∈ K+ such that xi = 0,

σi ,j(t, x) = 0 for x ∈ K+ such that xi = 0, j = 1, . . . , r ,

for all t ≥ 0 and i ∈ I .This result is valid independently of Itô's or Stratonovich'sinterpretations.

14. Cresson, Puig, Sonner, Validating stochastic models : invariance criteria forsystems of SDEs and the selection of a stochastic Hodgkin-Huxley type model,Int. J. Biomath. Biostat. (2013).

Selection of models : example of HIV-1 stochastic dynamics

The model studied by H.C. Tuckwell and E. Le Corfec is thefollowing system of SDEs

dT = (λ− µT − k1TV )dt −√k1TVdW1,

dL = (k1pTV − µL− αL)dt +√k1pTVdW2,

dI = (k1(1− p)TV + αL− aI )dt +√k1(1− p)TVdW3,

dV = (cI − γV − k2TV )dt −√k2TVdW4.

(10)

The constants a, c , k1, k2, α, γ, λ and µ are positive and p ∈ (0, 1).The model variables T , L, I and V represent cell densities, whichare necessarily non-negative.The positivity of solutions is preserved by the deterministic model.

What about the stochastic version ?

Selection of models 15

Proposition

The stochastic model for HIV-1 population dynamics (10) has thefollowing properties :

(i) The solutions of the underlying deterministic system remainnon-negative.

(ii) For any constant k1 6= 0 the stochastic model does notpreserve the positivity of solutions, independent of Itô's orStratonovich's interpretation of SDEs.

As a consequence, this model is not viable !

15. Cresson, Puig, Sonner, Stochastic models in biology and the invarianceproblem, submitted, 2015.

Simulations

-50

0

50

100

150

200

250

0 5 10 15 20 25 30 35 40

one sam ple path of T, uninfected CD4^ + T cells (for 1m m ^ 3)

t im e in days

T(t

)

-50

0

50

100

150

200

0 5 10 15 20 25 30 35 40

one sam ple path of I, act ively infected CD4^ + T cells (for 1m m ^ 3)

t im e in days

I(t)

Figure: Variable T and I

-5

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40

one sam ple path of L, latent ly infected CD4^ + T cells (for 1m m ^ 3)

t im e in days

L(t)

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

0 5 10 15 20 25 30 35 40

one sam ple path of V, plasm a virions

t im e in days

V(t

)

Figure: Variable L and V

The authors of this model was aware that it does not work !In ( 16 p.454, Section 3) :" In the simulations we found it expeditious to insert re�ectingbarriers at very small uninfected cells and virions numbers [. . .].This prevented the random �uctuation terms from taking thesevariables to unphysical negative values [. . .]. �

However, the use of re�ecting barriers lead to di�culties. Inparticular :

I Can we rely on the numerical simulations to validate the modelbehaviour and deduce biological interpretations ?

I What is the relation between the simulations and the originalmodel ? Could we formulate a modi�ed stochastic model thatre�ects the absorbing barriers used in the simulations ?

16. Tuckwell, Le Corfec, A stochastic model for early HIV-1 population dynamics, J. Theor. Biol.(1998).

A viable model

Using our result, we can describe a class of viable models :

dT = (λ− µT − k1TV )dt −√k1TVdW1,

dL = (k1pTV − µL− αL)dt + h1(L)√k1pTVdW2,

dI = (k1(1− p)TV + αL− aI )dt + h2(I )√k1(1− p)TVdW3,

dV = (cI − γV − k2TV )dt −√k2TVdW4,

(11)

where h1 and h2 are two functions satisfying h1(0) = h2(0) = 0.The conditions on h1 and h2 imply that the solutions of ourmodi�ed model (11) remain non-negative for both, Itô's andStratonovich's interpretation of SDEs. One simple choice are thefunctions

h1(L) =√aL, h2(I ) =

√bI

where a and b are two positive constants. The intensities of thestochastic perturbations in the equations for L and I then dependon these variables and not only on the cell densities T and V .

simulations

0

50

100

150

200

0 5 10 15 20 25 30 35 40

10 sam ple paths of T, uninfected CD4^ + T cells (for 1m m ^ 3)

t im e in days

T(t

)

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35 40

10 sam ple paths of I, act ively infected CD4^ + T cells (for 1m m ^ 3)

t im e in days

I(t)

Figure: Variable T and I

0

10

20

30

40

50

0 5 10 15 20 25 30 35 40

10 sam ple paths of L, latent ly infected CD4^ + T cells (for 1m m ^ 3)

t im e in days

L(t)

0

1000

2000

3000

4000

5000

6000

7000

0 5 10 15 20 25 30 35 40

10 sam ple paths of V, plasm a virions

t im e in days

V(t

)

Figure: Variable L and V

Invariance : make your own stochastic perturbation

Classical theory for precession and nutation : movement of thegeographic pole

due to the non-spherical form of the Earth but...he real movementof the pole is :

Stochastic J2 problem : produce a stochastic deformation of aplanetEasy with the invariance characterisation ! We de�ne a "tyo-model"with an ad-hoc stochastic viable perturbation...

Strange coincidence with the real behaviour !

Conservation laws for stochastic systems

The classical de�nition :

De�nitionA function I : Rn 7→ R is called a �rst integral if for all solutions xtof the equation we have I (xt) = I (x0) for all t. If I is su�ciently

smooth we deduce dI (xt)dt = 0.

What is the tochastic analogue of a �rst integral ?Two possibilities 17

De�nition (Strong �rst integral)

A function I : Rn → R is a strong �rst integral of (1) if for allsolutions Xt of (1), the stochastic process I (Xt) is a constantprocess, i.e. I (Xt) = I (X0) a.s. (almost surely) or d(I (Xt)) = 0.

Very strong property ! Most of the time not satis�ed !

17. Thieullen M. and Zambrini J.C, Probability and quantum symmetries I.The theorem of Noether in Schrödinger's euclidean mechanics, Ann. Inst. HenriPoincaré, Physique Théorique, Vol. 67, 3, p. 297-338 (1997).

A weaker version

A weaker property is :

De�nition (Weak stochastic �rst integral)

A function I : Rn → R is a weak stochastic �rst integral of (1) iffor all solutions Xt of (1), the stochastic process I (Xt) satis�esE(I (Xt)) = E(I (X0)) where E denotes the expectation.

Many deterministic systems are preserving �rst integrals in theweak sense under stochastisation !

Example : the stochastic two-body problem

Classical conserved quantities : angular momentum and energyde�ned by

M = mr2w , H =1

2m(v2 + r2w2)− k

r.

dM(Xt) = mrσφdBφt , dH(Xt) = mrvσrdB

rt+mrwσφdB

φt +

m

2

[σ2r r

2 + σ2φ]dt.

LemmaThe angular momentum is a weak �rst integral of the stochastictwo-body problem.

Can be used to test a numerical simulation !

stochastic two-body problem

Using a stochastic Runge-Kutta methode developed by Kasdin andal 18.

Rapid divergence between the perturbed and unperturbed case !Is the simulation accurate ?

18. N.J. Kasdin, L.J. Stankievech, On simulating randomly driven dynamicsystems, The journal of Astronautical Sciences, Vol. 57,nos.1 and 2 (2009), pp.289-311.

Simulations two-body problem

Good preservation of the weak �rst integral !

Figure: Left : E(M(Xt)). Right : E(H(Xt)).

Invariant manifolds and conservation laws

De�nitionA deterministic system leaves a given manifold of Rd , denoted byM, invariant if for all initial condition x0 ∈ M, the maximalsolution xt(x0) starting in x0 when t = 0 remains in M, i.e. satis�edxt(x0) ∈ M for all t ∈ R+.

Let M be de�ned by

M = {x ∈ Rn, F (x) = 0},

where F : Rd → R is C 1.

If F is a �rst integral then for all xo ∈ M, F (xt) = F (x0) = 0 andthe manifold M is invariant.

What about the stochastic case ?

Invariant manifols : the Itô case

The multidimensional Itô formula gives

d [F (xt ] = ∇F (xt)f (t, xt)dt +∇F (xt)σ(t, xt)dWt

+∑i,j

∂2F

∂xi∂xj(xt)

k∑l=1

σi,l(t, xt)σl,j(t, xt)dt.

The gradient of F always being normal to the manifold

∇F (xt) · f (t, xt) = 0

d [F (xt ] = ∇F (xt)σ(t, xt)dWt +∑i,j

∂2F

∂xi∂xj(xt)

k∑l=1


Two kind of notions can be developed in the stochastic case :

I strong persistence of the invariant manifold : dF = 0.

I weak persistence : persistence in expectation.

weak persistence

We always have :

E

[∫ t

0∇F (xs) · σ(s, xs)dWs

]= 0.

The stochastic term has no in�uences !

but

E

∫ t

0

∑i ,j

∂2F

∂xi∂xj(xt)

k∑l=1

σi ,l(t, xt)σl ,j(t, xt)dt

6= 0

in general !

We have generically no weak-persistence !What about strong persistence ?

Strong invariance : Itô caseWe must have :

∇F (xt)σ(t, xt) = 0.

Which is the deterministic invariance condition.

We obtain

d [F (xt ] =∑i,j

∂2F

∂xi∂xj(xt)

k∑l=1


Simplest case : σi ,j = δji .

d [F (xt)] =n∑

i=1

∂2F

∂2xi[σi,i (t, xt ]

2dt.

Theorem (Invariance-Itô case)The manifold M is invariant by the �ow of the stochastic system(1) in the Itô sense if and only if

n∑i=1

∂2F

∂x2i[σi,i (t, xt)]

2dt = 0,∀(t, x) ∈ R+ ×M.

Strong constraints !

We can specialize this result in the case of the sphere.

Corollary

The sphere Sd−1 is invariant under the �ow of the stochasticsystem (1) if and only if the stochastic perturbation is null on thesphere i.e.

d∑i=1

[σi ,i (t, xt)]2 = 0.

The perturbation is identically null on the sphere !

Example of a stochastic Landau-Lifshitz model

We use the model de�ned by P. Etoré, S. Labbé and J. Lelong 19 :

dµt = [−µt∧b−αµt∧(µt∧b)]dt−ε[−µt∧dWt−αµt∧µt∧dWt ],

The vector b is the magnetic �eld and α > 0 is the magnitude ofdamping term.

Is this model viable ?

19. P. Etoré, S. Labbé and J. Lelong. Long time behaviour of a stochasticnano particle, Journal of Di�erential Equations, 257(6), 2014.

No ! By symmetries arguments and physical arguments, themagnetic �eld must remains of norm one, i.e. belongs to the

sphere independently of the nature of the modelling.

Figure: Behaviour of the magnetic �eld-Itô case

This stochastic model is not viable !

The Stratonovich case

Computations with the Stratonovich integral behave as the usualdi�erential calculus.

d [F (xt ] = ∇F (xt) · f (t, xt)dt +∇F (xt) · σ(t, xt)dWt .

Theorem (Stochastic invariance for manifolds)

The manifold M is invariant under the �ow of the stochasticsystem in the Stratonovich sense if and only if

f (t, x) ∈ TxM and σ(t, x) ∈ TxM, for all (t, x) ∈ R+ ×M.

Invariant sphere - the Stratonovich case

TheoremThe sphere S is invariant under the �ow of the stochastic system(3) if and only if

x · f (t, x) = x · σ(t, x) = 0,∀(t, x) ∈ R+ ×M.

The Stratonovich version of the stochastic Landau-Lifshitz exampleis viable !

Behaviour of the magnetic �eld - Stratonovich case

Stochactic Hamiltonian systems

Introduced by J-M. Bismut in his book Mécanique aléatoire,Stochastic Hamiltonian systems are formally de�ned as :

De�nitionA stochastic di�erential equation is called stochastic Hamiltoniansystem if we can �nd a �nite family of smooth functionsH = {Hr}r=0,...,m, Hr : R2n 7→ R, r = 0, . . . ,m such that

dP i = −∂H∂qi

dt −m∑r=1

∂Hr

∂qi(t,P,Q)◦dB r

t ,

dQ i =∂H

∂pidt +

m∑r=1

∂Hr

∂pi(t,P,Q)◦dB r

t .

(12)

We recover the classical algebraic structure of Hamiltoniansystems. What more ?

I Liouville's property : Let (P,Q) ∈ R2n, we consider thestochastic di�erential equation

dP = f (t,P,Q)dt +m∑r=1

σr (t,P,Q) ◦ dB rt ,

dQ = g(t,P,Q)dt +m∑r=1

γr (t,P,Q) ◦ dB rt .

(13)

The phase �ow of (13) preserves the symplectic structure ifand only if it is a stochastic Hamiltonian system.

I Hamilton's principle : Solutions of a stochastic Hamiltoniansystem correspond to critical points of a stochastic functionalde�ned by

LH(X ) =

∫ t

0H0(s,Xs) +

m∑r=1

Hr (s,Xs) ◦ dB rt . (14)

How to know if a stochastic system is Hamiltonian or not 20 ?

TheoremA 2n-system of stochastic di�erential equations of the form

dP = f (t,P,Q)dt +m∑r=1

σr (t,P,Q) ◦ dB rt ,

dQ = g(t,P,Q)dt +m∑r=1

γr (t,P,Q) ◦ dB rt ,

(15)

possesses a stochastic Hamiltonian formulation if and only if thecoe�cients satisfy the following set of conditions

∂σir∂pα

+∂γαr∂qi

= 0,∂σir∂qα

=∂σαr∂qi

, α 6= i ,

∂γir∂pα

=∂γrα∂pi

, α 6= i ,(16)

for i , α = 1, ...n.

20. Milstein G.N., Repin Y.M., Tretyakov M.V., Symplectic integration of Ha-miltonian systems with additive noise, SIAM J Numer. Anal., 2002.

Application : Hamiltonian structure for the stochastictwo-body problem 21 ?

The Stratonovich form of the stochastic two-body problem is givenby

dr = prm dt,

dφ =pφmr2

dt,

dpr = (p2φmr3− k

r2)dt +mσr r ◦ dB r

t ,

dpφ = mσφr ◦ dBφt .

(17)

We check that the stochastic two-body problem does notpossess a stochastic Hamiltonian formulation.

21. N. Sharma and H. Parthasarathy, Dynamics of a stochastically perturbedtwo-body problem, Proc. R. Soc. A 2007 463, 979-1003.

Validation of a stochastic model

How to validate a given stochastic model ?

simulations → comparison with the expected behaviour

simulations require an adapted numerical scheme → why ?

Numerical problems

Numerical problems related to simulations are already present inthe deterministic case.For example, use a Runge-Kutta of order 4 method. We have thefollowing well-known problems :

I Creation of ghost equilibrium points

I Non-preservation of the positivity (more generally invariance)

I Non-preservation of the variational structure

I Non-preservation of conservation laws

Illustration

We consider the following classical population dynamics model

dx

dt= x

(b −

(bx + ay

c+x

)),

dy

dt= y

(x

c+x − d),

where a, b, c , d are real constants.For the following simulations, we use a = 2, b = 1, c = 0.5, d = 6.

Creation of ghost equilibrium points and positivity

Numerical simulations for the initial conditions x0 = 15, y0 = 0.1.

Figure: h = 0.2 and h = 0.1

But the correct behaviour is :

Figure: h = 0.01

Stability/instability artefactsSimulations with initial conditions x0 = 0.3, y0 = 7.5.

Figure: h = 0.01

Figure: h = 0.1

Figure: h = 0.2

Figure: h = 0.3

Figure: h = 0.4

Figure: h = 0.5

Consequence :

If you make simulation with an unadapted numerical method, youobtain unphysical results

No possibilities to test your model !

A possible answer : Construct adapted numerical scheme !Variational integrators for the preservation of variationalstructuresMickens non standard methods to preserve invariance and somedynamical properties

More generally topological numerical scheme to preserve speci�cdynamical properties

Balance between approximation order and dynamical accuracy

of the scheme.In the stochastic case : you need a very hight number of

simulations to compute expectations : for implementation this isbetter a low order of approximation but an accurate dynamical

scheme.First step in this direction done by F. Pierret 22

22. F. Pierret, A nonstandard Euler-Maruyama scheme, Arxiv 1411.2220, Nov.2014, 20.p.

Thank you for your attention !

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