Geometric projection of stochastic differential equations

Transcription

1 Geometric projection of stochastic differential equations John Armstrong (King s College London) Damiano Brigo (Imperial) August 9, 2018

2 Idea: Projection

3 Idea: Projection Projection gives a method of systematically reducing the dimension of an ODE Projection onto a linear subspace is the standard numerical method for solving PDEs Projecting onto a curved manifold may be more effective if we know the solution is close to this manifold e.g. perhaps the known soliton solutions to the KdV equation might give good approximations to solutions to a pertubed KdeV equation?

4 This talk Question: How should the notion of projection be extended to stochastic differential equations? Answer: There is a Stratonovich Projection which is best understood using Stratonovich calculus. There is an Extrinsic Ito Projection which is best understood... We will using Ito calculus. There is an Intrinsic Ito Projection which is best understood by using jet bundles. Define these various notions of projection and discuss their motivation and theoretical justifications Describe a geometric formulation of SDEs using 2-jets to understand the Intrinsic Ito projection Look at some numerical results when projection is applied to nonlinear filtering

5 Setup M is a submanifold of R r ψ : U R n is a chart for M φ = ψ 1 We have an SDE on R r dx t = a dt + α b α dw α t, X 0 and want to approximate this using an SDE on R n.

6 Definition: Stratonovich projection 1. Write the SDE in Stratonovich form dx t = a(x t ) dt + α b α (X t ) dw α t, X 0 2. Apply the projection operator Π to each coefficient to obtain an SDE on M dx t = Π Xt a(x t ) dt + α Π Xt b α (X t ) dw α t, ψ(x 0 )

7 Justifications What are the justifications for using the Stratonovich projection? It is clearly a well defined SDE. (Contrast with projecting Itô coefficients) It is clearly generalizes projection of ODEs - i.e. when b = 0 we get ODE projection. It gives good numerical results when applied to the filtering problem It generalizes the Galerkin method which can be interpreted as projection onto a linear subspace.

8 A justification for ODE projection Consider an ODE on R r dx dt = a(x ), X 0 Look for an ODE on R n of the form dx dt = a(x), ψ(x 0) such that φ(x t ) X t 2 is as small as possible.

9 A justification for ODE projection Compute Taylor expansion to see that leading term is minimized when: a(ψ(x 0 )) = ψ Π X0 A(x 0 ) Therefore ODE projection is the unique asymptotically optimal ODE approximating the original ODE at all points on M. (Linear projection operator gives solution to a quadratic optimization problem)

10 Repeat idea for SDEs Equation in larger space R r : dx = a(x, t) dt + b(x, t) dw t Equation in chart: dx = A(x, t) dt + B(x, t) dw t We have Itô Taylor series estimates (Kloeden and Platen): E( X t φ(x t ) ) = b 0 φ B 0 t + O(t) E(X t φ(x t )) = a 0 φ A ( B α,0 φ )B β,0 [W α, W β ] t + O(t 2 )

11 Extrinsic Ito Projection To minimize first estimate: φ B = Πb If we define B like this for whole chart, second estimate is minimized when: φ A = Πa 1 2 Π( B α φ )B β [W α, W β ] Given φ, define A and B using these equations This defines an SDE on the manifold We call this the Extrinsic Itô projection It is different from the Stratonovich projection

12 Discussion The Extinsic Itô Projection is optimal in the sense that it asymptotically minimizes two measures of the divergence of the approximation to the SDE from the true solution. Measure one is on the expectation of the absolute value. This determines the martingale part of our equation Measure two is on the absolute value of the expectation. This determines the bounded variation part of our equation The Extrinsic Itô Projection is greedy in that it finds the best approximation over short time horizons and hopes they will do well over long time horizons. Numerical test on the filtering problem indicate that it slightly outperforms the Stratonovich projection in practice over moderate time horizons. Over longer time horizons, it is random which performs better.

13 Geodesic projection map Let π denote the smooth map defined on a tubular neighbourhood

14 An alternative justification for ODE projection Consider an ODE on R r dx dt = a(x ), X 0 Look for an ODE on R n of the form dx dt = a(x), ψ(x 0) such that φ(x t ) X t 2 d(x t, ψ π(x t )) is as small as possible. d is induced Riemannian distance.

15 Intrinsic Itô projection Repeating the ideas used to derive the Extrinsic Itô projection: Definition The Intrinsic Itô projection is the best approximation to π(x t ) in the sense that it asymptotically minimizes both: E(d(x t, ψ π((x t ))) d(e(x t ), E(ψ π(x t )))

16 Discussion All three projections are distinct. Which is better? Lemma (Factorizable SDEs) Suppose that S is an SDE for X on R r such that π(x ) solves an SDE S on M then the Stratonovich and intrinic Itô projections are both equal to S. However, the extrinsic projection may be different. Example The SDE S on R 2 dxt = σyt dwt dy t = σx t dw t In polar coordinates, solutions satisfy: dθ = 1 2 σ2 sin(4θ) dt + σ cos(2θ) dw t

17 Understanding the Intrinsic Itô projection Definition The Intrinsic Itô projection is the best approximation to π(x t ) in the sense that it asymptotically minimizes both: E(d(x t, ψ π((x t ))) d(e(x t ), E(ψ π(x t ))) For applications, one must calculate this in local coordinates, but the resulting expression is complex One can understand this projection more intuitively, and express the answer more elegantly, using the language of 2-jets.

18 Euler Scheme All being well in the limit the Euler scheme δx t = a(x ) δt + b(x ) δw t converges to a solution of the SDE dx t = a(x ) dt + b(x ) dw t d, δ, + imply vector space structure This is highly coordinate dependent

19 Curved Scheme Let γ x be a choice of curve at each point x of M. γ x (0) = x Consider the scheme X t+δt = γ Xt (δw t ) X 0

20 Concrete example γ E (x 1,x 2 ) (s) = (x 1, x 2 ) + s( x 2, x 1 ) + 3s 2 (x 1, x 2 ) First order term is rotational vector Second order term is axial vector

21 Simulation: Large time step γ E (x 1,x 2 ) (s) = (x 1, x 2 ) + s( x 2, x 1 ) + 3s 2 (x 1, x 2 )

22 Simulation: Smaller time step γ E (x 1,x 2 ) (s) = (x 1, x 2 ) + s( x 2, x 1 ) + 3s 2 (x 1, x 2 )

23 Simulation: Even smaller γ E (x 1,x 2 ) (s) = (x 1, x 2 ) + s( x 2, x 1 ) + 3s 2 (x 1, x 2 )

24 Simulation: Convergence γ E (x 1,x 2 ) (s) = (x 1, x 2 ) + s( x 2, x 1 ) + 3s 2 (x 1, x 2 )

25 Formal argument Write: γ x (s) = x + γ x(0)s γ x (0)s 2 + O(s 3 ) Then: X t+δt = γ t (δw t ) Rearranging: = X t + γ X t (0)δW t γ X t (0)(δW t ) 2 + O ( (δw t ) 3) δx t = X t+δt X t = γ X t (0)δW t γ X t (0)(δW t ) 2 + O ( (δw t ) 3) Taking the limit: where dx t = b(x )dw t + a(x )(dw t ) 2 + O ( (dw t ) 3) = b(x )dw t + a(x )dt b(x ) = γ X (0) a(x ) = γ X (0)/2

26 Comments The curved scheme depends only on the 2-jet of the curve SDEs driven by 1-d Brownian motion are determined by 2-jets of curves The first derivative determines the volatility term The second derivative determines the drift term ODEs correspond to 1-jets of curves SDEs correspond to 2-jets of curves Rigorous proof of convergence of quadratic scheme can be proved using standard results on Euler scheme dx t = a(x )dt + b(x )dw t = a(x ) ( d(w 2 t ) 2W t d(w t ) ) + b(x )dw t a(x ) ( δ(w 2 t ) 2W t δ(w t ) ) + b(x )δw t = a(x ) ( (δw t ) 2) + b(x )δw t For general curved schemes some analysis needed.

27 Itô s lemma Given a family of curves γ x we will write: X t j 2 (γ x (dw t )) if X t is the limit of our scheme. If X t j 2 (γ x (dw t )) and f : X Y then: f (X ) t j 2 (f γ x (dw t )) Itô s lemma is simply composition of functions.

28 Usual formulation Is equivalent to: X t j 2 (γ x (dw t )) dx t = a(x )dt + b(x )dw t, a(x ) = 1 2 γ X (0), b(x ) = γ X (0) We calculate the first two derivatives of f γ X : (f γ X ) (t) = (f γ X ) (t) = n i=1 n j=1 i=1 + f (γ X (t)) dγ X x i dt n i=1 n 2 f (γ X (t)) dγi X dγ j X x i x j dt dt f (γ X (t)) d2 γ X x i dt 2 So f (X t ) j 2 (f γ x (dw t )) is equivalent to standard Itô s formula

29 Example γ E (x 1,x 2 ) (s) = (x 1, x 2 ) + s( x 2, x 1 ) + 3s 2 (x 1, x 2 ) Clearly polar coordinates might be a good idea. So consider the transformation φ : R 2 /{0} [ π, π] R by: φ(exp(s) cos(θ), exp(s) sin(θ)) = (θ, s),

30 The process j 2 (φ γ E ) plotted using image manipulation software The process j 2 (φ γ E ) plotted by applying Itô s lemma d(θ, s) = ( 0, 7 ) dt + (1, 0) dw t. 2

31 Drawing SDEs The following diagram commutes: SDE for X Itô s lemma SDE for f (X ) Draw Draw Picture of SDE for X in R n f (Picture of SDE for X) f

32 Intrinsic Itô Projection: 2-jet formulation If original SDE is: X t j 2 (γ x (dw t )) then intrinsic Itô projection is: x t j 2 (π γ x (dw t ))

33 Local coordinate formulation Calculate Taylor series for π to second order to compute: dx = A dt + B α dw α t, x 0 where: and: B i α = (π ) i β bβ α A i = (π ) i αa α + ( 1 2 φ γ 2 x α x β (π ) a γ(π ) α δ (π ) β ɛ + 2 φ ɛ x α x β (π ) β δ haα 2 φ γ x α x β (π ) β ɛ (π ) η γ(π ) ζ δ h ηζh aα b δ κb ɛ ι[w κ, W ι ] t. )

34 Numerical example The linear filtering problem has solutions given by Gaussian distributions Maybe approximately linear filtering problems can be well approximated by Gaussian distributions? Heuristic algorithms: Extended Kalman Filter Itô Assumed Density Filter Stratonovich Assumed Density Filter Stratonovich Projection Filter Algortihms based on optimization arguments: Extrinsic Itô Projection Filter Intrinsic Itô Projection Filter

35 Relative performance (Hellinger Residuals) All projections performed w.r.t. the Hellinger metric Intrinsic Ito Projection Extrinsic Ito Projection Stratonovich Projection Ito ADF Extended Kalman Filter Time

36 Summary - projection methods Extrinsic Ito Intrinsic Ito Stratonovich Optimal? Yes Yes Factorizable SDE Surprising Expected Expected Aesthetics Elegant Practice Best short term Best medium term Acceptable Note that our notion of optimal is based on expectation of squared residuals Other risk measures could be used

37 Summary - 2 jets 2-jets allow you to draw pictures of SDEs They provide an intuitive and elegant reformulation of Itô s lemma They provide an alternative route to coordinate free stochastic differential geometry to operator opproaches