Some Tools From Stochastic Analysis

W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click on the logo!

J. Potthoff, Some Tools From Stochastic Analysis Introduction 1 Introduction Where to begin? 2 / 34

J. Potthoff, Some Tools From Stochastic Analysis A Bit of History 1 Introduction Where to begin? Robert Brown (1827) rapid oscillatory motion of pollen sustained in water is sex the driving force of this movement? first correct ideas about the origin of the motion: C. Wiener (1863), I. Carbonelle (1874), J. Delsaux (1877) 3 / 34

J. Potthoff, Some Tools From Stochastic Analysis A Bit of History Louis Bachelier (1900) studies random walks as models for the time development of stock prices in his dissertation Théorie de la Spéculation derives Brownian motion as a limit of random walks, derives the transition probabilities Albert Einstein (1905) independently derives the formula for the transition probabilities of Brownian motion, relation to the heat equation calculates the value of the diffusion coefficient measurement of Avogadro s number Einstein s formulae are at the basis for experiments by Jean Perrin for the determination of Avogadro s number Perrin receives the Nobel prize for this work in 1926 4 / 34

J. Potthoff, Some Tools From Stochastic Analysis A Bit of History Nobert Wiener (1923) constructs the first complete mathematical model of Brownian motion: Wiener space and Wiener process his construction is based on the theory of integration by Lebesgue, Borel and Daniell which just had been established derivation of first path properties Paul Lévy (1939) deep analysis of the path properties of Brownian motion: precise modulus of continuity, arcsine-law, running maximum,... construction and study of local time 5 / 34

J. Potthoff, Some Tools From Stochastic Analysis A Bit of History Construction of (continuous time) stochastic processes until 1946: Consider in a domain D R d a (uniformly elliptic) second order differential operator L of the form L = 1 2 d i,j=1 a ij (x) 2 x i x j + d b i (x) x i i=1 A.N. Kolmogorov and the associated heat equation ( t + L ) u(t, x) = 0, t > 0, x D consider its fundamental solution p(t; x, y) as the transition probability of a stochastic process W.S. Feller 6 / 34

J. Potthoff, Some Tools From Stochastic Analysis A Bit of History Kiyosi Itô (1946) introduces stochastic integrals and stochastic differential equations w.r.t. a given Brownian motion (i.e., Wiener process) this gives a new method of pathwise construction of a large class of stochastic processes 7 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations 2 Stochastic Differential Equations Well-known: In R d L = 1 ( 2 p(t; x, y) = (2π t) d/2 exp 1 2t x y 2) B = (B t, t 0) where B is a Brownian motion on some probability space (Ω, A, P). In particular, ) u(t, x) := E x (f (B t ), t > 0, x R d solves the Cauchy problem ( t + L ) u(t, x) = 0 u(0+, x) = f (x) 8 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Consider first the case of constant coefficients : a ij (x) = a ij, b i (x) = b i coordinate transformation gives the fundamental solution ( p(t; x, y) = (2π t) d/2 (det a) 1/2 exp 1 ( x + bt y, a 1 (x + bt y) )) 2t Simple observation: Set σ := a, i.e., σ T σ = a, and X t := x + b t + σ B t, t 0 (X t, t 0) has transition probabilities given by the above p(t; x, y). 9 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Ansatz for non-constant coefficients a, b: X t = x + k b(x t k ) (t k+1 t k ) + k σ (X t k ) (B tk+1 B tk ) 10 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Ansatz for non-constant coefficients a, b: X t = x + k b(x t k ) (t k+1 t k ) + k σ (X t k ) (B tk+1 B tk ) Suppose that we can take the limit t 0 X t = x + stochastic integral equation t 0 b(x s ) ds + t 0 σ (X s ) db s Mathematical problems: Definition of the integral along the Brownian path: not rectifiable! existence and uniqueness path properties probabilistic properties... 2.5 2 1.5 B(t) 1 0.5 0 1 2 3 4 5 t 10 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Kiyosi Itô (1946) gives a definition of the stochastic integral develops its calculus proves existence and uniqueness of stochastic integral equations of the above type Simple example for illustration: d = 1, σ 1, drift b given by 2 b(x) 1 3 2 1 1 2 3 x 1 2 11 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations 2 1 0 1 2 3 4 5 t 1 path of Brownian motion B path of X: solution of the stochastic integral equation X t = x + t 0 b(x s ) ds + B t 12 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations 15 10 5 0 50 100 150 200 250 300 t path of Brownian motion B path of X: solution of the stochastic integral equation X t = x + t 0 b(x s ) ds + B t 13 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations The Itô integral has somewhat peculiar rules of calculus for example, in the one-dimensional case for f C 2 (R) f (B t ) f (B s ) = t s t s f (B u ) db(u) f (B u ) db(u) + 1 2 t s f (B u ) ds Because B t t, one needs a Taylor expansion to second order! Shorthand: df (B t ) = f (B t ) db t + 1 2 f (B t ) dt Itô s formula 14 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Work out Itô s formula for f (X t ), f C 2 (R d ): df (X t ) = i = i ( D i f (X t ) dbt i + b i (X t ) D i f (X t ) i + 1 2 i,j D i f (X t ) db i t + Lf (X t) dt ( σ T σ ) ij (X t) ( D i D j f ) (X t )) dt 15 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Work out Itô s formula for f (X t ), f C 2 (R d ): df (X t ) = i ( D i f (X t ) dbt i + b i (X t ) D i f (X t ) i + 1 2 i,j ( σ T σ ) ij (X t) ( D i D j f ) (X t )) dt = i D i f (X t ) db i t + Lf (X t) dt During the 50 s and 60 s: development of the theory of Martingales (Doob) and Markov processes (Dynkin) gave full power to Itô s calculus J.L. Doob E.B. Dynkin 15 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Consider the solution X t, t 0, of the SDE, and set ) u(t, x) := E x (f (X t ), t 0, x R d Use that the fact that X t, t 0, is a Markov process to prove that the mappings T t : f u(t, ), t 0 form a semigroup Use Itô s formula to compute its generator: L u(t, x) solves the Cauchy problem ( ) t + L u(t, x) = 0, t > 0, x R d lim u(t, x) = f (x) t 0 16 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Remark 1 Consider (for simplicity) the case a I (identity matrix), i.e. L = 1 2 + b(x) Martingale theory: make the Girsanov transformation, ) ( t E x (f (X t ) = E x (f (W t ) exp b(w s ) dw s 1 0 2 where W t, t 0, is a Brownian motion. t 0 b(w s ) 2 ds )) Analogue of the formula in the book by R.P. Feynman and A.R. Hibbs for a Schrödinger particle in an external electro-magnetic field ( minimal coupling ). 17 / 34

J. Potthoff, Some Tools From Stochastic Analysis Stochastic Differential Equations Remark 2 We have seen: for reasonable f ) u(t, x) = E x (f (X t ) solves the heat equation of L with initial condition f.? What about the fundamental solution? For this we would have to let f converge to a Dirac distribution δ y : p(t; x, y) =? ) E x (δ y (X t ) theory of generalized random variables ( 1980 2000): Hida-Malliavin-Calculus T. Hida P. Malliavin 18 / 34

J. Potthoff, Some Tools From Stochastic Analysis Potentials 3 Potentials? What happens, if we add a zero order term V : L = 1 2 d i,j=1 a ij (x) 2 x i x j + d b i (x) i=1 x i + V (x) Throughout: V is bounded from below, say, V (x) 0, x R d Note: the constant functions are no longer in the kernel of L the fundamental solution p(t; x, y) of ( / t + L) is no longer a probability density w.r.t. y X t, t 0, stochastic process with transition probability given by p(t; x, y): the probability that at time t > 0 we find the particle anywhere in R d is strictly less than 1 19 / 34

J. Potthoff, Some Tools From Stochastic Analysis Potentials Bring in an isolated point N ( nirvana ), and an independent expontially distributed random variable T. Let (X t, t 0) be the solution of the SDE as before (i.e., for V = 0). Consider the increasing stochastic process µ : R + t µ t = path dependent (random) time scale t 0 V (X s ) ds R + Construct new paths (Y t, t 0) from the paths of (X t, t 0) as follows: draw a value T (ω) for T and a path (X t (ω), t 0) define a random time τ(ω) as τ(ω) := first time t s.t. µ t (ω) T (ω) set X t (ω), t < τ(ω) Y t (ω) := N, t τ(ω) 20 / 34

J. Potthoff, Some Tools From Stochastic Analysis Potentials Example: d = 1, L = 1 2 + V, V = double well potential as before 1 0.5 0 0.5 1 1.5 2 t τ 0.5 1 path of Brownian motion B path of µ t = t 0 V (B s) ds: time scale of the Brownian path in the potential V value of T 21 / 34

J. Potthoff, Some Tools From Stochastic Analysis Potentials Example: d = 1, L = 1 2 + V, V = double well potential as before 1 0.5 0 0.5 1 1.5 2 t τ 0.5 1 path of Brownian motion B sent to N at time τ(ω) = 1.035 path of µ t = t 0 V (B s) ds: time scale of the Brownian path in the potential V value of T 22 / 34

J. Potthoff, Some Tools From Stochastic Analysis Potentials Extend the initial function f to R d N by f (N) := 0, and set ) u(t, x) := E x (f (Y t ) u solves the initial value problem of the heat equation with potential V : easy to perform the integration w.r.t. T ( ) )) t E x (f (Y t ) = E x (f (X t ) exp V (X s ) ds 0 which is the Feynman-Kac-Formula. R.P. Feynman M. Kac 23 / 34

J. Potthoff, Some Tools From Stochastic Analysis Boundary Conditions 4 Boundary Conditions Consider the simplest, non-trivial situation: L = 1 2 on R + = [0, + ) 24 / 34

J. Potthoff, Some Tools From Stochastic Analysis Boundary Conditions 4 Boundary Conditions Consider the simplest, non-trivial situation: L = 1 2 on R + = [0, + ) i.e., we are looking for a representation ) u(t, x) = E x (f (X t ), t 0, x R + for solutions of the initial-boundary-value problem t u(t, x) = 1 2 u(t, x), t > 0, x > 0 2 x2 u(0+, x) = f (x) β u(t, 0) + γ u (t, 0+) = 0 24 / 34

J. Potthoff, Some Tools From Stochastic Analysis Dirichlet Boundary Condition (a) Dirichlet BC Require u(t, 0) = 0, t 0 Heuristically: Consider a Brownian particle moving on R +. When it hits x = 0 it freezes, and its movement stops it gets absorbed at x = 0 1.4 1.2 1 B(t) 0.8 0.6 0.4 0.2 0.2 0.5 1 1.5 2 t 25 / 34

J. Potthoff, Some Tools From Stochastic Analysis Dirichlet Boundary Condition Let X t, t 0, be Brownian motion on R + with absorption at x = 0: Set and τ := inf { t 0, B t = 0 } B t, t < τ X t := 0, t τ With a small trick based on the strong Markov property of Brownian motion you can calculate the transition probabilities p a (t; x, y), x > 0, y > 0, for X t : p a (t; x, y) = p(t; x, y) p(t; x, y) where p(t; x, y) is the standard heat kernel on R. Now easy exercise: ) u(t, x) := E x (f (X t ) is the solution to the Dirichlet initial-boundary-value problem. 27 / 34

J. Potthoff, Some Tools From Stochastic Analysis Neumann Boundary Condition (b) Neumann BC Boundary condition x u(t, x) = 0, t 0 x=0+ 1 0.8 B(t) 0.6 0.4 0.2 0 0.2 0.5 1 1.5 2 t 0.4 0.6 0.8 28 / 34

J. Potthoff, Some Tools From Stochastic Analysis Neumann Boundary Condition (b) Neumann BC Boundary condition 1 0.8 x u(t, x) = 0, t 0 x=0+ 1 0.8 B(t) 0.6 0.4 X(t) 0.6 0.4 0.2 0.2 0 0.2 0.5 1 1.5 2 t 0 0.2 0.5 1 1.5 2 t 0.4 0.4 0.6 0.6 0.8 0.8 path of Brownian motion B path of reflected Brownian motion X = B 28 / 34

J. Potthoff, Some Tools From Stochastic Analysis Neumann Boundary Condition Define reflected Brownian motion as X t := B t, t 0 Easy to compute the transition probabilities p r (t; x, y) of X: p r (t; x, y) = p(t; x, y) + p(t; x, y) where p(t; x, y) is the standard heat kernel on R. ) u(t, x) = E x (f (X t ), t > 0, x 0 solves the Cauchy problem for the heat equation with Neumann boundary conditions. 29 / 34

J. Potthoff, Some Tools From Stochastic Analysis Neumann Boundary Condition Remarks Even though the construction of reflected Brownian motion seems quite trivial, many of the deeper results of P. Lévy are based on the analysis of this process reflected Brownian motion can also be constructed by a stochastic differential equation involving a singular term this is the key to the construction of reflected Itô processes in the case of non-trivial coefficients b, σ and for rather general domains in R d (non-smooth boundary) this has been completed only rather recently by Dupuis and Ishii (1994) 30 / 34

J. Potthoff, Some Tools From Stochastic Analysis Newton Boundary Condition (c) Newton BC Newton s radiation boundary condition for γ > 0: u(t, x) x = γ u(t, 0), t 0 x=0+ Heuristically: the one-dimensional domain [0 + ) is loosing heat by radiation at x = 0 if you start at t = 0 with a probability density as initial distribution, it is no longer normalized at time t > 0 Brownian particles are being sent to nirvana N 31 / 34

J. Potthoff, Some Tools From Stochastic Analysis Newton Boundary Condition Want Construct a process X t, t 0, from a reflected Brownian motion B t, t 0, which is sent to nirvana N at an exponentially distributed random time, measured on the random time scale given by the time spent at x = 0 Problem Fix t > 0. The set Z t R + of time spent by a Brownian motion B in x = 0 up to time t is a random Cantor set in particular, λ(z t ) = 0 P. Lévy constructs local time L t as a singular limit shows many probabilistic and path properties 32 / 34

J. Potthoff, Some Tools From Stochastic Analysis Newton Boundary Condition Construction of the process X: bring in an exponentially distributed random time variable T with rate γ for given Brownian path B t (ω), t 0, and value T (ω) set ζ(ω) := inf { t 0, L t (ω) > T (ω) } set B t (ω), t < ζ(ω) X t (ω) := N, t ζ(ω) 33 / 34

J. Potthoff, Some Tools From Stochastic Analysis Newton Boundary Condition As for the case of a potential, one can do the integration w.r.t. T : ) u(t, x) = E x (f (X t ) = E x ( f ( B t ) exp ( γ L t ) ) = E x (f ( B t ) exp ( γ t 0 δ 0 (B s ) ds Newton boundary condition u (t, 0) = γ u(t, 0) corresponds to the introduction of a Dirac potential at the origin. )) 34 / 34

W H I T E That s it Thanks!

W H I T E Brownian motion or Wiener process: A family B = (B t, t 0) of R d valued random variables on (Ω, A, P) such that: for P a.e. ω Ω the paths t B t (ω) are continuous the increments B t B s, 0 s < t, form an independent, stationary family for s, t > 0, B t B s is normally distributed with mean zero and variance t s One model is the canonical coordinate process on Wiener space; but there are many others. back