Mean-square Stability Analysis of an Extended Euler-Maruyama Method for a System of Stochastic Differential Equations

Transcription

1 Mean-square Stability Analysis of an Extended Euler-Maruyama Method for a System of Stochastic Differential Equations Ram Sharan Adhikari Assistant Professor Of Mathematics Rogers State University Mathematical Finance, Probability, and Partial Differential Equation Conference Rutgers University, New Brunswick, May 17-19, 2017

2 Table of contents Introduction 1 Introduction

3 Introduction Consider the system of stochastic differential equations X(s) =x + R s 0 b (X(r)) dr + P R M s k=1 0 k (X(r) k dw k (r), X(0) =x 2 R d, (1.1) where b : R d! R d, k : R d! R, k 2 R d and W k (t) are independent, one dimensional Brownian motions. Assume b and k are such that a weak solution to (1.1) exists and is unique in probability law. We would like to construct an accurate approximations to solutions of (1.1) on fixed time intervals.

4 Introduction SDEs have wide range of applications: Population dynamics, protein kinetics and genetics, psychology and neuronal activity, Math finance, turbulent diffusion and radio-astronomy, Seismology and structural mechanics and so on. As with most ordinary differential equations we can not generally solve SDEs explicitly. Two ways of approximating solution to SDE: Strong approximation. Weak approximation.

5 Introduction We may be interested to approximate E(g(X T )), T > 0. For instance E e rt (X(T) K) +. Better approximation on the probability distribution of solutions is sufficient. Definition Weak convergence of order Ef(X(T)) Ef(Y(N)) apple Kh, N = T h.

6 The Numerical Method Literature review [Anderson and Mattingly, 2011],[Bruti-Liberati and Platen, 2008], [Gaines and Lyons, 1997],[Kloeden and Platen, 1992]. We propose a numerical method to approximate the solution of (1.1) with order of convergence one. Let := {t 0 = 0 < t 1 <... < t N = T} be a subdivision of [0, T]. Consider the same initial condition X(0) =Y 0 = x 0. (i) 1k, (i) : i 2 N, k 2{1, 2,...M} is a collection of mutually 2k independent normal random variables with mean zero and variance 1. Define [a] + = max{a, 0} = a _ 0 for all a 2 R.

7 The Numerical Method Fix 2 (0, 1) and define, 1 = 1 2 (1 ) and 2 = 1 1 = Discretization step h > 0. For i = 1, 2, 3,..., Step 1 : Y 1 = Y i 1 + b(y i 1 ) h + Step 2 : Y i = Y 1 + b(y i 1)(1 MX + k=1 MX k= (1 ) )h q 2 [ 1 k (Y 1 ) 2 2 k (Y i 1)] + k i 2k k (Y i 1 ) k i 1k p h, q (1 )h.

8 Motivation Behind The Algorithm Equation (1.1) is distributionally equivalent to Z t X(t) =X(0)+ + MX Z 1 k k=1 0 b (X(s)) ds 0 Z t 0 I [0, 2 k (X(s))(u) k (du ds). (2.1) k are independent space-time white noise processes. Must approximate k (A [0,h) ( 2 )) to approximate the diffusion k term in (2.1) over the interval [0, h). A [0,h) ( 2 k ) is the region under the curve 2 (X(t)) for 0 apple t apple h. k

9 Motivation Behind The Algorithm Figure: 1 A graphical illustration of Extended Euler-Maruyama Scheme for = 1 2 To approximate X(h), we focus on the double integral in (2.1) for a single k.

10 Motivation Behind The Algorithm Take = 1 2 for illustration. Approximate X( h 2 ) using the Euler-Maruyama method on the interval [0, h 2 ) Denote it by Y 1 To do so, we calculate k (Region 1), where Region 1 is the grey shaded region in the figure. k (Region 1) d = N(0, 2 k (X(0))h 2 ) d = k (X(0)) r h N(0, 1) 2 This is equivalent in distribution to step 1 of our algorithm and exactly carried by step 1 of our algorithm.

11 Motivation Behind The Algorithm Consider the whole region Region 5. k (Region 5) = d N 0, ( 2 k (X(0)) 2( 2 k (X(0)) 2 k (Y 1 ))h 2 d = N 0, (2 2 k (Y 1 ) 2 k (X(0)))h 2 Returning q to step 2 of our q algorithm: For i = 1, we have (2 2 k (Y 1 )) 2 k (X(0)) h 2 N(0, 1) in step 2 of our algorithm.

12 Functional Setting We define C k (R d ) in the following way: C k (R d )={f : R d! R s.t. D f(x) exists, bounded and continuous for all x 2 R d }, (3.1) where is such that i 2 Z + [{0} for each i = 1,...,d and applek. We define the norm kfk k = sup n D f(x) : x 2 R d, =( 1,..., d ), applek o. (3.2) B(R d ) is the family of real-valued, bounded and Borel measurable functions defined on R d.

13 Functional Setting Define the Markov semigroup P t : B(R d )!B(R d ) related to (1.1) by (P t f)(x) : def = E x f(x(t)), where X(0) =x (3.3) The Markov semigroup P h : B(R d )!B(R d ) by. (P h f)(y) : def = E y f(y 1 ), where Y 0 = y (3.4)

14 Assumptions Introduction (A1) The coefficients of (1.1) satisfy the Lipschitz and linear growth conditions: MX b(x) b(y) + k (x) k (y) appleapple x y, b(x) + k=1 MX k (x) appleapple(1 + x ), k=1 (3.5) for all k = 1,...,M and x, y 2 R d, where apple is a positive constant.

15 Assumptions Introduction (A2) For each k = 1,...,M, we have inf x2r d{ k (x)} > 0. In addition, there exists a positive constant 2 (0, 1] so that for any x, 2 R d we have 2 apple T a(x) apple 1 2, (3.6) where T denotes the transpose of and a(x) := P M k=1 (A3) For all multi-index with apple4, we have D b(x) + 2 k (x) k T k. MX D k (x) applek(1 + x p ), for all x 2 R d, (3.7) k=1 where K and p are positive numbers.

16 The Generators Introduction (Af)(x) =f 0 [b(x)](x)+ 1 2 (B 1 f)(x) =f 0 [b(x 0 )](x)+ 1 2 MX k=1 MX k=1 2 k (x)f 00 [ k, k ](x), (3.8) k (x 0 ) 2 f 00 [ k, k ](x), (3.9) (Bf)(x) =f 0 [b(x 0 )](x) (3.10) + 1 MX [ k (Y 1 ) 2 2 k (x 0)] + f 00 [ k, k ](x). k=1 Where f 0 [ ](z) is the derivative of f in the direction evaluated at the point z.

17 Main Theorems Introduction Theorem (Local Approximation ) Assume (A1) (A3). Then there exist a constant K so that P h P h 4!0 apple Kh 2 for all h > 0 sufficiently small. Theorem (Global Approximation ) Assume that b 2 C 4 and 8k, k 2 C 4 with inf x k (x) > 0. Then for any T > 0 there exists a constant C(T) such that sup P nh 0applenhappleT P n h 4!0 apple C(T)h

18 Main Theorems Introduction The proof of the Local Approximation Theorem depends on the following lemma Lemma Assume (A1) (A4). Then for all h > 0 sufficiently small and f 2 C 4, we have E[f(Y 1 )+(Bf)(Y 1 )(1 )h] =f(x 0)+(Af)(x 0 )h + O(h 2 ).

19 Example 1 Geometric Brownian Motion For one dimensional case we have the Black-Scholes model. dx(t) = X(t)dt + µx(t)dw(t), X(0) =x 2 R (4.1) This Stochastic differential equation (4.1) is used to model asset price in financial mathematics. The solution to this SDE is X(t) =X(0)e ( 1 2 µ2 )t+µw(t) (4.2) The expected value of the solution is E[X(t)] = E[X(0)]e t (4.3)

20 Example 1 Geometric Brownian Motion We test the weak convergence of our method over [0, 1] for = 2, µ = 0.1, and X(0) =1. We used step sizes h p = 2 p 10, for 1 apple p apple 5 to sample over discretized brownian paths of the equation (4.1). We then compute the error and the outcome of the numerical experiment is shown in Figure 2 where we have plotted the weak error against h on log-log scale.

21 Example 1 Geometric Brownian Motion Figure: 2 Log-Log plots of error versus step-size.

22 Example 1 Geometric Brownian Motion Detailed calculation shows that E[(X(t)) 2 ]=E[X(0) 2 ]e (2 +µ2 )t (4.4) We test the weak convergence of our method over [0, 1] for = 2, µ = 0.1, and X(0) =1. We used step sizes h p = 2 p 10, for 1 apple p apple 5 to sample over discretized brownian paths of the equation (4.1). The result of the numerical experiment is shown in the Figure 3 where we have plotted the weak error against h on log-log scale.

23 1 Geometric Brownian Motion Figure: 3 Log-Log plots of error versus step-size. It is observed that the slope of the best fit line is empirically one.

24 Background For Roundoff and truncation errors produced during simulation is natural. The spreading of such errors is absolutely necessary to understand in simulation. The usefulness of a numerical method depends upon its ability to control the propogation of errors. Numerical schemes with better numerical stability are demanded.

25 Background For The solutions of a differential equation are continuous in their initial values, at least over a finite time interval. The concept of stability is an extension of this idea to an infinite time interval, [Kloeden and Platen, 1992]. In some sense, the stability of a numerical scheme refers to the conditions under which the impact of an error vanishes asymptotically over time, [Bruti-Liberati and Platen, 2008]. Consider scalar Itô equation dx(t) =b(t, X(t))dt + (t, X(t))dW(t), X(t 0 )=x 0 2 R d (5.1) With a steady solution X(t) 0, so b(t, 0) =0 and (t, 0) =0.

26 Background For We recall the definitions for pth-mean stability and asymptotically stable in pth-mean from [Kloeden and Platen, 1992]. Definition Steady solution X t 0 is called stable in pth-mean if for every >0 and t = (t 0, ) > 0 such that E[ X t 0,x 0 t p ] < for all t t 0 and x 0 <, asymptotically stable in pth- mean if it is stable in pth-mean and there exists a 0 = 0 (t 0 ) > 0 such that lim t!1 E[ X t 0,x 0 t p ]=0 for all x 0 < 0.

27 Background For We give the following probabilistic definitions for stochastic stability from [Kloeden and Platen, 1992] Definition The steady solution X(t) 0 is called stochastically stable if for any >0 and t 0 0 lim P sup X t 0,x 0 x 0!0 t = 0, t t 0 and stochastically asymptotically stable if, in addition, lim x 0!0 P lim t!1 X t 0,x 0 t!0 = 1.

28 Background For Now it is natural to ask the following questions: 1 Do the numerical solutions of SDEs preserve the stability properties of the original SDEs? 2 and if yes for what stepsizes h does the numerical method reproduce the characteristics of the test equation? The above two questions have received quite a lot of attention and in fact stability analysis of numerical methods is motivated by them. We try to answer the above two questions. We will focus on the linear test equation X(t) =X(0)+ Z t 0 X(s)ds + Z t 0 µx(s)dw(s), t 0. (5.2)

29 Mean-square Stability Analysis For Stochastic ODEs The SDE (5.2) can be written as dx(t) = (X(t))dt + µ(x(t))dw(t), t 2 [0, T] X(0) =x 2 R (5.3) We assume that and µ are real constants. Assuming that X(0), 0 with probability 1, solutions of (5.3) have the following properties: lim t!+1 E( X(t) 2 )=0, 2 + µ 2 < 0 (5.4) lim t!+1 X(t) = 0, with probability 1, 1 2 µ2 < 0. (5.5) Here E(.) denotes the expected value.

30 Mean-square Stability Analysis For Stochastic ODEs For simplicity, let S P denotes the sets of order pairs of real problem parameters for which problem is stable. Clearly, S P = n, µ 2 R : 2 + µ 2 < 0 o. Applying the extended Euler-Maruyama method to (5.2) produces the following iterative sequence: Y n = h r i apple A + B (n) Yn (1 )h C + D (n) + E (n) 2 + (n) Y n 1, (5.6) for n = 1, 2,...,where { (n) 1, (n), n = 1, 2,...} are mutually 2 independent normal random variables with mean zero and variance one.

31 Mean-square Stability Analysis For Stochastic ODEs A, B, C, D and E in (5.6) are given by A := 1 + h, B := µ p h, C := µ 2 ( h h) D := 2 1 µ 3 ( h + 1) p h, E := 1 µ 4 h > 0. (5.7) Apparently C is positive when 0. When <0, we can find 0 < h < 1 positive. 1 + p 2 / 1 such that C is

32 Mean-square Stability Analysis For Stochastic ODEs The sequence (5.6) is mean-square stable if lim n!1 E( Y n 2 )=0) [Higham, 2000b]. It follows from (5.6) that E[ Y n 2 ]=E[ Y n 1 2 ] A 2 + B 2 +(1 appleh (n) )he C + D + E (n) 2 i (5.8) Lemma For 0 < h < p 2 / 1, we have apple E [C + D (n) + E( (n) 1 1 )2 ] + apple C + E + o(h 2 ). (5.9)

33 Mean-square Stability Analysis For Stochastic ODEs Putting (5.9) into (5.8) and using the expressions for A,...,E in (5.7), detailed computations reveal that E[ Y n 2 ] < E[ Y n 1 2 ] A 2 + B 2 +(1 )h(c + E + o(h 2 )) " = E[ Y n 1 2 ] 1 +(2 + µ 2 )h [ µ 2 + µ 4 ]h 2 # + o(h 2 ). (5.10) Furthermore, for the expression inside the brackets of the right-hand side of (5.10), we notice that µ 2 + µ 4 = µ2 + 4 µ4 > 0. (5.11)

34 Mean-square Stability Analysis For Stochastic ODEs Next we compute the discriminant =(2 + µ 2 ) ( µ 2 + µ 4 ) = µ 4 < 0. (5.12) Therefore, it follows that for any h > 0, 1 +(2 + µ 2 )h [ µ 2 + µ 4 ]h 2 > 0. The condition for mean square stability of the Extended Euler-Maruyama(EEM) method for (5.2) is 1 +(2 + µ 2 )h [ µ 2 + µ 4 ]h 2 < 1. (5.13)

35 Mean-square Stability Analysis For Stochastic ODEs We can rewrite equation (5.13) as 0 < h < 2(2 + µ 2 ) µ 2 + µ 4. (5.14) Sufficient condition for mean square stability of the weak Simpson method for (5.2) is ( 2(2 + µ 2 ) 0 < h < min µ 2 + µ, p ) 2 / 1. (5.15)

36 Mean-square Stability Analysis For Stochastic ODEs We have the following theorem: Theorem The following assertions are true: (a) Given (, µ) 2 S P, the extended Euler-Maruyama method is mean-square stable if the discretization stepsize h satisfies (5.15). Therefore the mean square stability of the process (5.2) implies the mean square stability of the extended Euler-Maruyama method if the discretization stepsize h satisfies (5.15). (b) Conversely, if the extended Euler-Maruyama method with discretization stepsize h > 0 is mean-square stable for (5.2), then the parameters and µ of (5.2) satisfies (, µ) 2 S P. In other words, the mean square stability of the extended Euler-Maruyama method implies that of the underlying stochastic process (5.2).

37 Mean-square Stability Analysis For Stochastic ODEs We can visualize the stability region for 2(2 + µ 2 ) µ 2 + µ < p 2 / 1. Using (5.13), we have S M := n (x, y) 2 R 2 : 2x 2 + 2xy + y 2 + 4x + 2y < 0 o. (5.16) The plot of mean-square stability domain for the EEM method S M is shown in Figure 9.

38 Mean-square Stability Analysis For Stochastic ODEs Figure: 9 Real mean-square stability domain for weak EEM and EM method

39 Thank You!

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