Numerical Simulation of Stochastic Differential Equations: Lecture 2, Part 2

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1 Numerical Simulation of Stochastic Differential Equations: Lecture 2, Part 2 Des Higham Department of Mathematics University of Strathclyde Montreal, Feb p.1/17

2 Lecture 2, Part 2: Mean Exit Times Statement of Problem Differential Equation Formulation SDE/Monte Carlo Algorithm Convergence Property Montreal, Feb p.2/17

3 dx(t) = f(x(t))dt + g(x(t))dw(t), X(0) = X 0 Suppose X 0 is a constant in (a, b) Define the random variable T exit to be T exit := inf{t : X(t) = a or X(t) = b} In words: T exit is first time solution leaves (a, b) b a Problem: find the mean exit time T mean exit := E [T exit ] Montreal, Feb p.3/17

4 Monte Carlo for Mean Exit Time Choose a stepsize, t Choose a number of paths, M for s = 1 to M Set t n = 0 and X n = X 0 While X n > a and X n < b Compute a N(0,1) sample ξ n Replace X n by X n + tf(x n ) + t ξ n g(x n ) Replace t n by t n + t end set Texit s = t n 1 t 2 end set a M = M 1 M s=1 T exit s set b 2 M = M 1 1 M s=1 (T exit s a M) 2 Montreal, Feb p.4/17

5 Errors Two sources of error Sampling error: sample mean expected value Numerical SDE error: EM SDE paths Third source of error Discrete time: {t i } is monitored, not continuous time b a Montreal, Feb p.5/17

6 Analysis Let X 0 = x. Then T mean exit = u(x), where 1 g(x)2 d2 u 2 dx 2 + f(x)du dx = 1, for a < x < b with b.c. s u(a) = u(b) = 0 We can solve (analytically or numerically) to get exact solution. Then investigate accuracy of SDE/Monte Carlo. SDE/Monte Carlo approach is attractive for high-dimensional problems, e.g. X R 32 complicated geometries, computing u(x) at a single point Montreal, Feb p.6/17

7 u(x) = σ2 µ f(x) = µx and g(x) = σx (log(x/a) 1 (x/a)1 2µ/σ2 1 (b/a) 1 2µ/σ2 log(b/a) ) E.g. µ = 0.1, σ = 0.2, a = 0.5 and b = 2: u(x) Initial data, x Montreal, Feb p.7/17

8 f(x) = µx and g(x) = σx Replace X n by X n + tµx n + t ξ n σx n can be improved: Replace X n by X n exp ( (µ 1σ2 ) t + ) t ξ 2 n σ E.g. µ = 0.1, σ = 0.2, a = 0.5, b = 2 and X 0 = 1 with M = and t = 10 2 : First exit time Montreal, Feb p.8/17

9 Accuracy? T mean exit = With M = and t = 10 2 we get a M = with a 95% conf. int. of [7.7561, ] exact answer well outside conf. int. Monte Carlo method overestimates mean exit time Explanation: error from checking paths only at discrete time points {t i } is dominating the statistical sampling error decrease t rather than increase M t = 10 3 : a M = conf. int. [7.6641, ] t = 10 4 : a M = conf. int. [7.6200, ] Convergence rate... Montreal, Feb p.9/17

10 M = and try t = 10 1, 10 2, 10 3, 10 4 µ = 0.5, σ = 0.2, a = 0.5, b = 2, X 0 = 1.5: Sample mean + conf. interval Error in sample mean t Least-squares fit: power = 0.45, (resid = 0.12) t Montreal, Feb p.10/17

11 Convergence Rate The rate O( t 1 2 ) has been widely reported Overall error is then O( t / M) take M t 1 Montreal, Feb p.11/17

12 M = 10 4, t = 10 4, 40 X 0 values µ = 0.1, σ = 0.2, a = 0.5, b = 2: u(x) x Montreal, Feb p.12/17

13 f(x) = λ(µ x) and g(x) = σ x Safe EM step is Replace X n by X n + tλ(µ X n ) + t ξ n σ X n Take λ = 1, µ = 0.5, σ = 0.3, a = 1 and b = 2, M = 10 3 and t = 10 3 : u(x) Exact solution from bvp4c.m x Montreal, Feb p.13/17

14 Double-Well Potential: V (x) = x 2 (x 2) V(x) x dx(t) = V (X(t))dt + σdw(t) Take a = 3 and b = 1 Measuring expected time to climb over the central hump Montreal, Feb p.14/17

15 1 σ2 d2 u 2 dx 2 V (x) du dx = Mean exit time σ = 4 σ = σ = Initial data, x Montreal, Feb p.15/17

16 Convergence Fix X 0 = 0 and σ = 4 M = and t = 10 2, 10 3, 10 4, 10 5 : Sample mean + conf. interval Error in sample mean t Least-squares fit: power = 0.46, (resid = 0.017) t Montreal, Feb p.16/17

17 Monte Carlo/SDE for Mean Exit Time Research problems: Develop a provably O( t) algorithm. Ideas: Adaptively reduce t near boundary. After each step, calculate probability that exit was missed. Then draw from a uniform (0, 1) random number generator in order to decide whether to record an exit. Use random t n from a suitable exponential distribution. Develop an efficient method for high-dimension/complicated region Montreal, Feb p.17/17