Numerical Simulation of Stochastic Differential Equations: Lecture 2, Part 1. Recap: SDE. Euler Maruyama. Lecture 2, Part 1: Euler Maruyama

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1 Numerical Simulation of Stochastic Differential Equations: Lecture, Part 1 Des Higham Department of Mathematics Universit of Strathclde Lecture, Part 1: Euler Maruama Definition of Euler Maruama Method Weak Convergence Strong Convergence Linear Stabilit p.1/5 Recap: SDE Given functions f and g, the stochastic process X(t) is a soluton of the SDE dx(t) = f(x(t))dt + g(x(t))dw(t) if X(t) solves the integral equation X(t) X() = t f(x(s)) ds + t g(x(s)) dw(s) Discretize the interval [, T ]: let = T/N and = n Compute X n X( ) Initial value X is given Eact solution: X(+1 ) = X( ) + Euler Maruama: Euler Maruama f(x(s)) ds + X n+1 = X n + f(x n ) + W n g(x n ) where W n = W(+1 ) W( ) (Left endpoint Riemann sums) In MATLAB, W n becomes sqrt(dt)*randn g(x(s)) dw(s) p.3/5

2 ) = µ and g() = σ, µ =, σ =.1, X() = 1 Solution: X(t) = X()e (µ 1 σ )t+σw(t) Disc. Brownian path with δt =, E-M with = δt: 5 Convergence? X n and X( ) are random variables at each In what sense does X n X( ) as? X 3 There are man, non-equivalent, definitions of convergenc for sequences of random variables t X N X(T ) =.9 Reducing to = δt gives X N X(T ) =.1 Reducing to = δt gives X N X(T ) =. The two most common and useful concepts in numerical SDEs are Weak convergence: error of the mean Strong convergence: mean of the error p.5/5 Weak Convergence Weak convergence: capture the average behaviour Given a function Φ, the weak error is e weak := sup E [Φ(X n )] E [Φ(X( ))] T Φ from e.g. set of polnomials of degree at most k Converges weakl if e weak, as Weak order p if e weak K p, for all < In practice we estimate E[Φ(X n )] b Monte Carlo simulation over man paths 1/ M sampling error f() = µ and g() = σ, µ =, σ =.1, X() = Solution has E[X(t)] = e µt Measure weak endpoint error a M e µt over M = 1 5 discretized Brownian paths. Tr = 5,, 7,, 9 E[X(T)] Sample average of X N t Least squares fit: power is 1.11 (Confidence intervals smaller than graphics smbols) Suggests weak order p = 1 p.7/5

3 Weak Euler Maruama X n+1 = X n + f(x n ) + W n g(x n ) ( where P Wn = ) ( = 1 = P Wn = ) E.g. use sqrt(dt)*sign(randn) or sqrt(dt)*sign(rand-.5) 1 Weak Euler Maruama Generall, EM and weak EM have weak order p = 1 on appropriate SDEs for Φ( ) with polnomial growth Can prove via Fenman-Kac formula that relates SDEs to PDEs E[X(T)] Sample average of X N Least squares fit: power is t p.9/5 Strong Convergence Strong convergence: follow paths accuratel Strong error is e strong := sup E [ X n X( ) ] T Converges strongl if e strong, as Strong order p if e strong K p, for all < f() = µ and g() = σ, µ =, σ = 1, X() = Solution: X(t) = X()e (µ 1 σ )t+σw(t) M = 5, disc. Brownian paths over [, 1] with δt = 11 For each path appl EM with = δt, δt, δt, 1δt, 3δt, δ Record E [ X N X(1) ] for each δt Sample average of X N X(T) t Least squares fit: power is.51 p.11/5

4 Strong Convergence Strong Convergence Generall EM has strong order p = 1 on appropriate SDEs Can prove using Ito s Lemma, Ito isometr and Gronwall Note: strong convergence weak convergence, but this doesn t recover the optimal weak order Euler Maruama has Markov inequalit sas E [ X n X( ) ] K 1 P ( X > a) E[ X ], a for an a > ( ) Taking a = 1 gives P X n X( ) 1 K 1, i.e. ( ) P X n X( ) < 1 1 K 1 Along an path error is small with high prob. p.13/5 Higher Strong Order If g() is constant, then EM has strong order p = 1 More generall, strong order p = 1 is achieved b the Milstein method X n+1 = X n + f(x n ) + W n g(x n ) + 1 g(x n)g (X n ) ( W n ) (More complicated for SDE sstems.) Even Higher Strong Order: Warning! Numerical methods for stochastic differential equations Joshua Wilkie Phsical Review E, Claims to derive arbitraril high (strong?) order methods, with a Runge Kutta approach. But using onl Brownian increments, W n, rather than more general integrals like dw 1 (s)dw (t) there is an order barrier of p = 1 (Rümelin, 19). p.15/5

5 Beond Convergence... Stochastic Theta Method Numerical methods approimate the continuous b the discrete: X n X( ), with +1 =: Convergence: How small is X n X( ) at some finite? Stabilit (Dnamics): Does lim n X n look like lim t X(t)? X n+1 = X n + (1 θ)f(x n ) + θf(x n+1 ) + g(x n ) W n where we recall that W n = W(+1 ) W( ), so W n = V n, with V n Normal(, 1) i.i.d. X n X( ) in the SDE (Itô) dx(t) = f(x(t))dt + g(x(t))dw(t), X() = X Stud stabilit b appling the method to a class of test problems, where information about X(t) is known. Hope to show good behavior either for all >, or at least for sufficientl small. p.17/5 Stochastic Test Equation dx(t) = µx(t)dt + σx(t)dw(t) (Asset model in math-finance) Mean-square stabilit lim t E(X(t) ) = µ + σ < Mean-square stabilit Saito & Mitsui, SIAM J Num Anal 199 θ < 1 : SDE stable method stable iff < µ + σ µ (1 θ) STM gives X n+1 = (a + bv n )X n, with a := 1 + (1 θ)µ, b := σ 1 θµ 1 θµ θ = 1 : SDE stable method stable > < θ 1: SDE stable method stable > 1 p.19/5

6 Stabilit Regions Stochastic Test Equation Let := µ and := σ SDE stable < Method stable < (θ 1) 1 theta = 1 theta =.5 Asmptotic stabilit dx(t) = µx(t)dt + σx(t)dw(t) lim X(t) =, with prob. 1 µ t σ < theta = theta = 1 Recall that STM gives X n+1 = (a + bv n )X n, with a := 1 + (1 θ)µ, b := σ 1 θµ 1 θµ p.1/5 Asmptotic Stabilit: lim n X n =, w.p. 1 X n = ( n 1 i= a + bv i ) X SLLN: lim n X n = E(log a + bv i ) < Can be epressed in terms of Meijer s G-function Difficult to deal with analticall No simple epression for stabilit region boundar Asmptotic Stabilit for Backward Euler (θ = 1 theta = p.3/5

7 Man open equestions regarding asmptotic stabilit E.g. is there an A-stable method? Generalizations to nonlinear SDEs are also possible p.5/5