Numerical Methods for Stochastic Differential Equations presented by Kevin Burrage and Pamela Burrage, Math

1. Numerical Methods forStochastic DifferentialEquationspresented by  Kevin Burrage and Pamela Burrage, Maths Dept, Univ Queensland,Brisbane, Australiakb@maths.uq.edu.aupmb@maths.uq.edu.au Abstract More accurate modelling of physical systems involved the inclusion of random elements, and thus the theory of stochastic differential equations has been developed. As few SDEs can be solved analytically, methods must be developed for obtaining accurate numerical approximations efficiently. PowerPoint Eqn Font Sizes (Slides Sizes, then Default values) Full Subscript Sub-sub Symbol Sub 24pt 18pt 12pt 24pt 18 12pt 7pt 5pt 18pt 12 Abstract More accurate modelling of physical systems involved the inclusion of random elements, and thus the theory of stochastic differential equations has been developed. As few SDEs can be solved analytically, methods must be developed for obtaining accurate numerical approximations efficiently. PowerPoint Eqn Font Sizes (Slides Sizes, then Default values) Full Subscript Sub-sub Symbol Sub 24pt 18pt 12pt 24pt 18 12pt 7pt 5pt 18pt 12

2. Structure of Course Theme 1: Introduction to SDEs Theme 2: High strong order methods for SDEs Theme 3: Convergence results by B-Series Theme 4: Stability issues and implicit methods Theme 5: Numerical solution of SPDEs -- a hydrological example Theme 6: Implementation issues

3. Introduction to Stochastic Differential Equations K. Burrage and P.M. Burrage Department of Mathematics University of Queensland Brisbane 4072, Australia kb@maths.uq.edu.au pmb@maths.uq.edu.au Fields Institute, Toronto, October 2001

4. Contents Applications Theory -- stochastic integrals Theory -- SDEs Modelling -- different noise processes Taylor expansions Expectation of random variables

5. Stochastic O.D.E.s model physical system with random elements water catchment in a reservoir population dynamics stock market fluctuations many SDEs cannot be solved analytically main approaches for numerical solution are compute many sample paths, and determine the accuracy of a trajectory compute approximation to the probability distribution of the solution, and determine various statistical measures availability of supercomputing resources will have significant impact Stochastic D.E.s are used to model physical systems with random elements - by introducing randomness, a more accurate model is obtained. FOR EXAMPLE water catchment /seepage/evaporation/precipitation So an SDE is formed by introducing a random function in the representation of the ODE. Indeed there are many practical problems where the probability distribution or moments of an Ito process cannot be solved analytically, hence the requirement for a numerical approximation of their expected values. Because many sample paths must be computed, the use of supercomputers will have a significant impact on the time required (e.g.) for solving an SDE numerically.Stochastic D.E.s are used to model physical systems with random elements - by introducing randomness, a more accurate model is obtained. FOR EXAMPLE water catchment /seepage/evaporation/precipitation So an SDE is formed by introducing a random function in the representation of the ODE. Indeed there are many practical problems where the probability distribution or moments of an Ito process cannot be solved analytically, hence the requirement for a numerical approximation of their expected values. Because many sample paths must be computed, the use of supercomputers will have a significant impact on the time required (e.g.) for solving an SDE numerically.

6. An Application - Polymeric Flows �Stochastic Processes in Polymeric Fluids�� by H.C. ttinger I. Polymer Flows Non-Newtonian, memory effects, wide range of time scales, visco-elastic effects eliminate fast processes (local motion) through stochastic noise study on long time scales (visco-elastic, memory) polymer molecule as coarse-grained bead-spring models.

7. II. Modelling and Computation Process

8. III. Kinetic Theory pdfs (Fokker-Planck): solve highly nonlinear p.d.e.s OR stochastic d.e.s of motion for polymer molecules trajectories easier to refine models OR minimisation of the expectation of a functional-stochastic control theory - Hamilton Bellman Jacobi theory p.d.e.s

9. Mathematical Finance Problem System of SDEs models the stock price and both the instantaneous volatility and weighted average volatility of the stock. Initial value

10. Quadrants 1 and 2 show the fixed stepsize solution, with quadrants 3 and 4 giving the variable stepsize solution. Quadrants 1 and 3: Quadrants 2 and 4: Variable stepsize results - 171 steps attempted, 130 successful (fixed: 256 steps). Problem is non-commutative:

11. Stochastic ODEs - theory a) Stochastic initial value problems 1 Wiener process f - drift coefficient g - diffusion coefficient (rapidly varying) in integral form 2nd integral is a stochastic integral �formally��, Gaussian white noise is

12. is a Wiener process, with independent increments, nowhere differentiable is Normal b) Multi-Wiener process case independent Wiener processes alternative representation

13. c) Stochastic Integrals the natural approximating sums converge in the mean-square sense to different values depending on mean-square convergence  W(t) is nowhere differentiable, so need to determine how to interpret the 2nd integral. A 1st approach is to approximate by sums. This converges ... Ito or Stratonovich? If the randomness can be approximated by a relatively smooth process, so that the usual rules of calculus apply, then the Stratonovich interpretation could be used. Having chosen which interpretation to use, it is always possible to convert to the other interpretation, and so use whichever is more advantageous at the time. Example:W(t) is nowhere differentiable, so need to determine how to interpret the 2nd integral. A 1st approach is to approximate by sums. This converges ... Ito or Stratonovich? If the randomness can be approximated by a relatively smooth process, so that the usual rules of calculus apply, then the Stratonovich interpretation could be used. Having chosen which interpretation to use, it is always possible to convert to the other interpretation, and so use whichever is more advantageous at the time. Example:

14. Example: The approximation converges to stochastic integral: Stratonovich stochastic integral:

15. d) integration w.r.t. Wiener process ( , Gikhman) non-anticipating - history of up to is independent of future evolution of after independent r.v.s representation

16. can use more general stochastic processes (martingales) as integrators -- integrals inherit martingale property from integrators. e) or Stratonovich Stratonovich - usual rules of calculus Stratonovich - limiting process in which sequences of r.v.s converge to W. P. - Wong Zakai theorem - integrals inherit martingale properties - if parameters perturbed by noise

17. f) Stochastic D.E.s then ! solution with real-valued r.v. Additive noise (B=constant), =Stratonovich Linear problem - no simple solution if do not commute for all j.

18. Fokker Planck for pdf of (5) The Stratonovich SDE and the SDE have the same solution.

19. g) Modelling with SDEs Logistic Population Model perturbed by a noise process can be solution of an SDE (i) White Noise

20. (ii) Coloured Noise - Ornstein Uhlenbeck Solution: (iii) Multiplicative Noise -

21. -Taylor Expansion Stochastic Calculus requires its own chain rule Apply with a=f (and g) in turn: expand Y(t) in a stochastic Taylor Series Consider the integral equation representation of Yt. Then, for a continuous function �a�, ... Ito�s formula is applied successively to Consider the integral equation representation of Yt. Then, for a continuous function �a�, ... Ito�s formula is applied successively to

22. Taylor Series looks like Each elementary differential corrresponds to a tree. Each tree has an associated weight, in this case an Ito integral. The Stratonovich Taylor Expansion Stratonovich integrals Trees - The expansion of the Taylor Series gets markedly more complex, with terms in various combinations of derivatives. By considering rooted trees of up to p nodes, each possible derivative term is covered. Indeed, each labelled tree corresponds to one elementary differential. For stochastic trees, need bi-coloured nodes. Up to 2nd order trees, with Y0 = Y(t0)), the Ito-Taylor series is:Trees - The expansion of the Taylor Series gets markedly more complex, with terms in various combinations of derivatives. By considering rooted trees of up to p nodes, each possible derivative term is covered. Indeed, each labelled tree corresponds to one elementary differential. For stochastic trees, need bi-coloured nodes. Up to 2nd order trees, with Y0 = Y(t0)), the Ito-Taylor series is:

23. Expectations in SDEs need to calculate expected value of products of stochastic integrals from Kloeden and Platen (1992):

24. Example 1

25. Example 2

26. Given expectation, need Stratonovich. Is a general recursive formula: Some general rules

27. Examples- use Maple to evaluate algebra:

28. References S.S. Artemiev and T.A. Averina (1997): Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations, VSP, Utrecht. P.M. Burrage (1999): Numerical methods for stochastic differential equations, Ph.D. Thesis, Univ. Queensland. T.C. Gard (1988): Introduction to Stochastic Differential Equations, Marcel Dekker, New York. P.E. Kloeden and E. Platen (1992): Numerical solution of stochastic differential equations, Springer-Verlag. P. Levy (1948): Processus stochastiques et mouvement Brownian, Monographies des Probabilites, Gauthier-Villars, Paris. H.C. Ottinger (1996): Stochastic processes in Polymeric Fluids, Springer.