%% 12. Random functions and random ODEs % % \setcounter{page}{141} % %% % % One of the most fascinating themes in the % mathematical sciences, whose importance is growing every year, % is randomness. In this chapter we say a word % about how randomness plays into the subject of ODE\kern .5pt s. % %% % % To begin the discussion, here are two examples of random % functions produced by the Chebfun {\tt randnfun} command. % ODEformats, rng(1), lambda1 = 1; dom = [0 10]; f1 = randnfun(lambda1,dom); subplot(2,1,1), plot(f1,'k',LW,1), ylim([-5 5]) title(['Fig.~12.1.~~Random functions with length scales ' ... '$\lambda = 1$ and 0.1'],FS,11); lambda2 = 0.1; f2 = randnfun(lambda2,dom); subplot(2,1,2), plot(f2,'k',LW,0.7), ylim([-5 5]) %% % \vskip 1.02 em %% % % \noindent % At the end of the chapter we will explain the precise mathematical % definition, but for the moment, the main thing to note % is that these are smooth functions defined on a prescribed % interval and with a prescribed length scale $\lambda$. % The first function, with $\lambda = 1$, has a typical distance % on the order of $1$ between maxima, whereas the second, % with $\lambda = 0.1$, wiggles ten times as fast. % The vertical scales are the same, and in fact, % at each fixed point $t$, each function produced by % {\tt randnfun} takes values corresponding to samples % from the standard normal distribution $N(0,1)$, with % mean 0 and variance 1. % %% % % Like so many commands in Chebfun, {\tt randnfun} % provides a continuous analogue of a familiar discrete % object. In Matlab, {\tt randn(n,1)} generates % an $n$-vector of random entries from $N(0,1)$. % Similarly \verb|randnfun(lambda,[a,b])| produces % a smooth random function of typical wavelength $\lambda$ % on the interval $[\kern .3pt a,b\kern .5pt ]$. % %% % % As our first random ODE problem, let us consider the simplest % ODE IVP of all, % $$y'(t) = f(t), \quad y(0) = 0, \eqno (12.1)$$ % whose solution is just the indefinite integral of $f$, % $$y(t) = \int_0^t f(s)\kern .7pt ds. \eqno (12.2)$$ % If we take $f$ to be the two functions plotted above, % we get these results. We call a curve like these a {\bf smooth random % walk.} % subplot(2,1,1), y1 = cumsum(f1); plot(y1,LW,1,CO,ivp), ylim([-4 4]) title(['Fig.~12.2.~~Their indefinite integrals: ' ... 'smooth random walks'],FS,11); subplot(2,1,2), y2 = cumsum(f2); plot(y2,LW,1,CO,ivp), ylim([-1 1]) %% % \vskip 1.02 em %% % % \noindent % Note that the first curve has a larger amplitude % than the second. The reason for this is a familiar matter % of statistics associated with cancellation of % random signs. These indefinite integrals % are essentially the average value of the integrand (times % 10, when we reach $t=10$), and according to the law of large % numbers, this average converges to $0$ as the number of % samples approaches $\infty$, which in our context means % as $\lambda$ approaches zero. Moreover, the convergence % will be in proportion to $\lambda^{1/2}$. So in fact, % our second curve should be expected to % be on the order of $\sqrt{10}$ times smaller than the first. % To eliminate this dependence on $\lambda$ we can renormalize % $f$ by dividing it % by $\lambda^{1/2}$, and in Chebfun, this (approximately) is what % is done if {\tt randnfun} is % called with the {\tt 'norm'} flag. From now on we will always % use {\tt 'norm'}. % %% % Here are three smooth random walks with $\lambda = 0.1$. lambda = 0.1; for k = 1:3 f = randnfun(lambda,dom,'norm'); y = cumsum(f); subplot(1,3,k), plot(y,LW,.6,CO,ivp), ylim([-12 12]) if k==2, title(['Fig.~12.3.~~Smooth random walks ' ... 'with $\lambda = 0.1$'],FS,11), end end %% % \vskip 1.02 em %% % % \noindent % Here are three smooth random walks with $\lambda = 0.01$. % lambda = 0.01; for k = 1:3 f = randnfun(lambda,dom,'norm'); y = cumsum(f); subplot(1,3,k), plot(y,LW,0.4,CO,ivp), ylim([-12 12]) if k==2, title(['Fig.~12.4.~~Smooth random walks ' ... 'with $\lambda = 0.01$'],FS,11), end end %% % \vskip 1.02 em %% % % The sample paths we have shown in Figs.~12.2--12.4 are smooth, % but as $\lambda \to 0$, the smoothness % goes away. In this limit we get the precisely defined mathematical % notion of {\em Brownian motion}, where the sample % paths are continuous but not smooth. % A Brownian motion trajectory is also % called a {\em Wiener path}, and probabilists say that % a Wiener path is a sample from the {\em Wiener process}. % We can show the convergence as $\lambda\to 0$ by superimposing % three paths for successively smaller values of $\lambda$, % all based on the same random number seed set by % Matlab's {\tt rng} command.\footnote{In Matlab as in other programming % languages, successive calls to {\tt randn} give new random numbers, % but one can reinitialize the sequence for repeatability with the % command {\tt rng(k)}, where $k$ is a fixed integer. Chebfun's % {\tt randnfun} works the same way. This feature has been crucial for % us in writing this chapter, since we need reproducible random curves % if we are to comment on their particular features.} % clf for lambda = [1 1/4 1/16] rng(3), f = randnfun(lambda,dom,'norm'); y = cumsum(f); plot(y,LW,.4 + .7*lambda), ylim([-2.5 2.5]), hold on end title(['Fig.~12.5.~~Convergence to a Brownian path ' ... 'as $\lambda\to 0$'],FS,11), hold off %% % \vskip 1.02 em %% % % Here is a smooth random walk over a longer time scale, up to $t = 500$. % Note that the maximal amplitude is a bigger than % before, and yet, % the trajectory comes back repeatedly to zero --- whereupon, of course, % it starts over.'' In an infinitely long trajectory, % the path will cross zero infinitely often, and yet the % amplitudes will grow.\footnote{Statements like this hold % with probability 1'' or almost surely.'' In principle % a Brownian path could be any function at all, % and thus for example might remain bounded by~1 % forever, or even identically % zero, but the probability of such events will be zero. % With probability~1, a Brownian path is everywhere continuous % and nowhere differentiable.} % Probability theory is full of such paradoxes. % lambda = 0.1; rng(1) f = randnfun(lambda,[0 500],'norm'); y = cumsum(f); clf, plot(y,LW,0.3,CO,ivp),% ylim([-18 18]) title(['Fig.~12.6.~~Smooth random walk over a larger ' ... 'time interval'],FS,11) %% % \vskip 1.02 em %% % % Here are $n=10$ smooth random walks up to $t=100$, together % with their mean, shown as a thicker curve in black. % As the sample size $n$ approaches $\infty$, the mean will % approach the function $\varphi(t)$ that is the {\em expected} % value of $f(t)$ at each point $t$. For this simple example, % $\varphi(t) = 0$. % lambda = 0.1; F = randnfun(lambda,[0 100],10,'norm'); Y = cumsum(F); plot(Y,LW,0.3), ylim([-35 35]) title('Fig.~12.7.~~Ten smooth random walks and their mean',FS,11) hold on, plot(mean(Y,2),'k',LW,2), hold off %% % \vskip 1.02 em %% % % For a second random ODE problem, let us consider an % indefinite integral as before, but now a system of % equations in two variables % $y_1^{}$ and $y_2^{}$, % $$y_1'(t) = f_1(t), ~ y_2' = f_2(t), % \quad y_1^{}(0) = y_2^{}(0) = 0, \eqno (12.3)$$ % where $f_1^{}$ and $f_2^{}$ are independent random functions, % normalized again by division by $O(\lambda^{1/2})$. % The two variables are uncoupled, so in a sense there % is nothing new here. On the other hand, the trajectories % now take the interesting form of two-dimensional smooth % random walks, which in the limit $\lambda\to 0$ would % become 2D Brownian motion. Here are two sample paths with % $\lambda = 0.1$ on $[\kern .3pt 0,10]$. % clf, rng(2) for k = 1:2 f1 = randnfun(lambda,dom,'norm')/sqrt(2); y1 = cumsum(f1); f2 = randnfun(lambda,dom,'norm')/sqrt(2); y2 = cumsum(f2); subplot(1,2,k), plot(y1,y2,LW,0.5,CO,ivp) ax = get(gca,'pos'); ax(1) = ax(1)-(-1)^k*.03; set(gca,'pos',ax), axis(7*[-1 1 -1 1]), axis square set(gca,XT,-6:2:6,YT,-6:2:6) if k==1, title(['\kern -.4in Fig.~12.8.~~2D smooth ' ... 'random walks to $t=10$'],FS,11,HA,'left'), end end %% % \vskip 1.02 em %% % % \noindent % Following our usual trick in 2D, we could equally well % have generated these images using a single complex random % function instead of two real ones: % for k = 1:2 f = randnfun(lambda,dom,'norm','complex'); y = cumsum(f); subplot(1,2,k), plot(y,LW,0.5,CO,ivp) ax = get(gca,'pos'); ax(1) = ax(1)-(-1)^k*.03; set(gca,'pos',ax) axis(7*[-1 1 -1 1]), axis square, set(gca,XT,-6:2:6,YT,-6:2:6) if k==1, title(['\kern -.9in Fig.~12.9.~~2D smooth random walks ' ... 'via complex arithmetic'],FS,11,HA,'left'), end end %% % \vskip 1.02 em %% % % \noindent % These trajectories look different, but only because we have % rolled the dice again. Beneath the superficial distinction of % complex scalars vs.\ real 2-vectors, these are % independent sample paths from the same distribution. % %% % % Our random ODE\kern .5pt s so far have been trivial, just indefinite integrals. % Let us explore some more substantial examples, which will give % an idea of some of the fascination of the field of % {\bf stochastic differential equations} % {\bf (SDE\kern .5pt s)}. In all of the next six figures, $f$ is % a smooth random function of some small fixed time scale and amplitude on % the interval $[\kern .3pt 0,5]$. % In each case several sample trajectories are plotted. % %% % % First we look at an equation featuring {\em additive noise,} % $$y' = y + f, \quad y(0) = 0. \eqno (12.4)$$ % Without $f$, the solution would be $y(t) = 0$, but the noise % term breaks this symmetry. At first, so long as $|y|$ is small, % trajectories look like random walks, with signs varying from % $+$ to $-$, but as $|y|$ gets larger % the exponential element overwhelms the random one, and a % path shoots off to $-\infty$ or $\infty$ with probability~1. % By symmetry, it is clear that both fates are equally likely. % clf, rng(0), lambda = 0.1; dom = [0 5]; L = chebop(dom); L.op = @(y) diff(y) - y; L.lbc = 0; for k = 1:6 f = randnfun(lambda,dom,'norm'); y = L\f; plot(y,LW,0.7), hold on end title('Fig.~12.11.~~Six solutions to (12.4): unstable',FS,11) ylim([-10 10]), hold off %% % \vskip 1.02 em %% % % \noindent % On a larger vertical scale the same curves look simply like exponentials. % title('Fig.~12.11.~~The same paths on a larger scale',FS,11) ylim([-100 100]), hold off %% % \vskip 1.02 em %% % Next, we reverse the sign in (12.4) and consider % $$y' = -y + f, \quad y(0) = 0. \eqno (12.5)$$ % Now the process is % stable, showing random oscillations about 0 that % remain bounded as $t$ increases. % L.op = @(y) diff(y) + y; L.lbc = 0; for k = 1:6 f = randnfun(lambda,dom,'norm'); y = L\f; plot(y,LW,0.5), hold on end title('Fig.~12.12.~~Six solutions to (12.5): stable',FS,11) ylim([-3 3]), hold off %% % \vskip 1.02 em %% % % Now let us change (12.4) into an equation with % {\em multiplicative noise}, % $$y' = fy , \quad y(0) = 1, \eqno (12.6)$$ % where $f$ is again random. % We find that the amplitudes of the solutions of this new equation vary widely. % dom = [0 5]; rng(1), L = chebop(dom); L.lbc = 1; for k = 1:6 f = randnfun(lambda,dom,'norm'); L.op = @(t,y) diff(y) - f*y; y = L\0; plot(y,LW,0.7), hold on end title(['Fig.~12.13.~~Solutions to (12.6): ' ... 'smooth geometric Brownian motion'],FS,11) ylim([0 120]), set(gca,XT,0:5), hold off %% % \vskip 1.02 em %% % % \noindent % The greatly differing amplitudes may seem surprising at first, % but in fact, (12.6) is nothing more than the exponential of % (12.1). We can verify this by rewriting (12.6) as $y'/y = f$, that is, % $$(\log y)' = f , \quad \log y(0) = 0. \eqno (12.7)$$ % So for any given $f$, the solution $y$ of (12.6) is the % exponential of the solution $y$ of (12.1). % %% % % Equations (12.5) and (12.6) are first-order linear equations, of type % {\sf FLAShI} and {\sf FLaSHI}, respectively. Of course, equations involving % $a \kern .5pt y'' + b\kern .5pt y' + c\kern .5pt y$ as in (7.10) % in which the coefficients $a,b,c$ all vary with $t$ can % also be considered, as can nonlinear equations. % Let us consider a nonlinear example with a % bistable flavor. Without the random term $f$, the equation % $$y' = y - y^3 +f \eqno (12.8)$$ % would have stable fixed points $y=\pm 1$. % Taking 20 trajectories from the initial value $y=0$, and putting % the amplitude scale of $f$ at $0.2$, we % find that about half end up oscillating about each of these values. % By symmetry, the positive and negative % behaviors must be must be equally likely. (These % fates are not permanent. Since Gaussians take % arbitrarily large values, though rarely, % further sign flips will happen with probability~1 for sufficiently % large values of $t$.) % N = chebop(dom); rng(0) N.lbc = 0; N.op = @(t,y) diff(y) - y + y^3; for k = 1:20 f = 0.2*randnfun(lambda,dom,'norm'); y = N\f; plot(y,LW,0.4), hold on end title(['Fig.~12.14.~~Random switching in the nonlinear ' ... 'equation (12.8)'],FS,11), hold off %% % \vskip 1.02 em %% % On the other hand, suppose we bias the switch slightly % by taking the initial value $y(0) = 0.20$. Both positive and % negative fates are again possible, but among twenty test % trajectories, just two now go negative. N.lbc = 0.2; for k = 1:20 f = 0.2*randnfun(lambda,dom,'norm'); y = N\f; plot(y,LW,0.4), hold on end title(['Fig.~12.15.~~Random switching with a ' ... 'positive initial condition'],FS,11), hold off %% % \vskip 1.02 em %% % % We promised at the beginning % to explain the definition of the smooth % random functions delivered by {\tt randnfun}. % The essential idea here is the use of finite Fourier series % with normally distributed random coefficients all of % equal variance. We start from the notion of a % periodic function on the interval $[\kern .3pt 0,L]$, defined % by a Fourier series % $$f(t) = a_0^{} + \sqrt 2 \kern 2pt % \sum_{k=1}^m a_k^{} \cos\left({2\pi k\kern .5pt t\over L}\right) + % b_k^{} \sin\kern -1pt \left({2\pi k\kern .5pt t\over L}\right), % \eqno (12.9)$$ % where each $a_k^{}$ and $b_k^{}$ is an independent sample from % the $N(0,1/(2m+1))$ % distribution, i.e., with mean 0 and variance $1/(2m+1)$. % The space scale $\lambda$ is fixed by setting % $m$ to be the integer closest to $L/\lambda$. % In the normalized'' mode as specified in Chebfun by the % \verb|'norm'| flag, we have the same formula but % with $a_k^{}$ and $b_k^{}$ coming from a distribution whose variance % does not diminish as $m\to\infty$ for fixed $L$. Such a series almost % surely does not % converge as $m\to\infty$, but its integrals almost surely do, % such as the indefinite integral % $$\int^t \kern -3pt f(s)\kern .5pt ds = a_0^{}\kern .7pt t % + {L\over \sqrt 2\kern .5pt \pi }\kern 1pt % \sum_{k=1}^m k^{-1}\kern -2pt \left[ a_k^{} % \sin\kern -1pt \left({2\pi k\kern .5pt t\over L}\right) - % b_k^{} \cos\kern -1pt \left({2\pi k\kern .5pt t\over L} % \right)\right]. \eqno (12.10)$$ % Random infinite series of the form (12.10) % go back to Paley, Wiener, and Zygmund in 1933 and 1934, % and both (12.9) and (12.10) could be % called {\em finite Fourier--Wiener series.} % To generate a nonperiodic random function, % {\tt randnfun} first constructs a periodic one on a % larger interval, and then % restricts it to the interval prescribed. % %% % % Without fully describing any of the mathematics, let us at least % mention some of the terminology that appears when % our smooth random ODE\kern .5pt s are related to SDE\kern .5pt s via the limit % $\lambda \to 0$. % A random function $f$ is a % sample from a certain {\bf Gaussian process} dependent on % the parameter $\lambda$. Suppose we write an % ODE involving $f$ in the form % $$y'(t) = \mu(t,y(t)) + \sigma(t,y(t)) f(t) \eqno (12.11)$$ % for some functions $\mu$ and $\sigma$. As $\lambda\to 0$, this % ODE approaches an {\bf SDE} that would normally be written as % $$dX_t^{} = \mu(t,X_t^{})\kern .5pt dt + \sigma(t,X_t^{}) \circ dW_t. \eqno (12.12)$$ % The two terms on the right are sometimes labeled {\em drift} and % {\em diffusion} (or {\em volatility}), respectively. % If $\mu$ is of the form of a constant times % $X_t^{}$ and $\sigma$ is a constant, as in (12.4) and (12.5), the SDE is a % {\bf Langevin equation,} and its solution is the % {\bf Ornstein--Uhlenbeck process.} % If $\mu$ and $\sigma$ are both of the form of a constant times % $X_t^{}$, as % in (12.6), we have the SDE of {\bf geometric Brownian motion.} % The small circle in (12.12) indicates that this is an SDE % of {\bf Stratonovich} type. The alternative of an {\bf It\^o} SDE % has a different definition and the notation % $$dX_t^{} = \tilde\mu(t,X_t^{})\kern .5pt dt + % \sigma(t,X_t^{}) \kern .5pt dW_t. \eqno (12.13)$$ % We have changed $\mu$ to $\tilde \mu$ because although % (12.12) and (12.13) have different meanings, they define the % same stochastic process provided $\tilde\mu$ and $\mu$ are % related by % $$\tilde\mu(t,X_t^{}) = \mu(t,X_t^{}) + {1\over 2} \kern 1pt % \sigma(t,X_t^{}) \kern 1pt{\partial \sigma\over \partial x} % (t,X_t^{}) \eqno (12.14)$$ % Details of the usual formulations of It\^o and Stratonovich calculus can % be found in many books of stochastic analysis. Results about % the convergence of random ODE\kern .5pt s to SDE\kern .5pt s stem from two papers % by E. Wong and M. Zakai in 1966; see also Sussmann, On the gap % between deterministic and stochastic ordinary differential equations,'' % {\em The Annals of Probability,} 1978. % %% % % \begin{center} % \hrulefill\\[1pt] % {\sc Application: metastability, radioactivity, and tunneling}\\[-3pt] % \hrulefill % \end{center} % %% % % Many systems in physics, chemistry, biology, % and social sciences have what are known as {\em metastable % states,} which means, states that may appear stable for % a long time but then suddenly undergo a transition. % Examples include financial bubbles, supercooled liquids, % and radioactive nuclei. % Often the effect can be explained by noting that there is a stable % fixed point of a noise-free system, but when noise is present, % it eventually kicks the system out of the stable state. % %% % We can illustrate the effect with the IVP % $$y' = y^3 - y + \varepsilon f(t), \quad y(0) = 0, \eqno (12.15)$$ % where $f$ is a smooth random function and $\varepsilon$ is a % noise amplitude parameter. % (Note that the signs are opposite to those in (12.8).) % Here are three solutions for $t\in [\kern .3pt 0,100\kern .3pt ]$ with % $\varepsilon = 0.2$. % In the absence of the noise term, $y=0$ is a stable fixed point % and $y=\pm 1$ are unstable fixed points. % When noise is added, however, the stable state will % eventually be left behind. lambda = 1; rng(11) N = chebop(0,100); N.op = @(y) diff(y) - y^3 + y; N.lbc = 0; N.maxnorm = 10; ep = 0.25; f1000 = randnfun(lambda,[0 1000],'norm',3); f = f1000{0,100}; for k = 1:3 y = N\(ep*f(:,k)); plot(y,LW,0.7), grid on, axis([0 100 -2 2]), hold on, drawnow end title(['Fig.~12.16.~~Metastability for (12.15) with ' ... '$\varepsilon = ' num2str(ep) '$'],FS,11), hold off %% % \vskip 1.02em %% % % Note that each trajectory stays near the stable state for a while, % and then at some moment escapes. % We cannot predict the precise moment % of escape, though it would appear that for this example, it happens % on a time scale in the range 10--100.\ \ To put it another % way, the {\em half-life} of the system is evidently in this range, % where the half-life is defined (as in the Application of % Chapter~2) as the expected time $t_{1/2}^{}$ by which % the probability of escape has risen to $1/2$. % A related notion is that of a {\em mean exit time.}\footnote{For such % definitions to be mathematically precise, they must be based on % a precise definition of when a particle has escaped. The definition % implicit in the {\tt N.maxnorm} setting of our Chebfun code is % that a particle escapes when $|y|$ reaches the value $10$.} % %% % % Intuitively speaking, a system will escape from a metastable % state when the random fluctuations, by chance, happen to % deviate by an exceptionally large amount from their usual % state. The mathematical theory of % {\em large deviations} is used to analyze such effects. % One phenomenon one finds in this subject is that a % small change in a parameter may have a large effect on the % lifetime. Here, for example, we reduce $\varepsilon$ from % $0.25$ to $0.22$ and find that none of the three trajectories escapes. % ep = 0.22; for k = 1:3 y = N\(ep*f(:,k)); plot(y,LW,0.7), grid on, axis([0 100 -2 2]), hold on, drawnow end hold off title(['Fig.~12.17.~~Reduction to $\varepsilon = ' ... num2str(ep) '$'],FS,11) %% % \vskip 1.02em %% % % \noindent % Eventually, trajectories will still escape, as we can % see if we show results over all of $t\in [\kern .3pt 0,1000\kern .3pt ]$. % N.domain = [0 1000]; for k = 1:3 y = N\(ep*f1000(:,k)); plot(y,LW,0.4), grid on, ylim([-2 2]), hold on, drawnow end title(['Fig.~12.18.~~A longer time interval with ' ... '$\varepsilon = ' num2str(ep) '$'],FS,11), hold off %% % \vskip 1.02em %% % % Thanks to the power of exponentials, half-lives of % radioactive isotopes have been estimated ranging % over more than 50 orders of magnitude, from less % than $10^{-24}$ seconds to more than $10^{22}$ years. % Three famous examples are uranium-238, with a half-life % of 4.5 billion years, uranium-235 at 700 million years, % and carbon-14 at 5700 years. In quantum physics the % process of decay from a metastable state is called % {\em tunneling.} % %% % % \smallskip % {\sc History.} % The approach to stochastic differential equations taken in this % chapter, via smooth random functions, is nonstandard. % After foundational works of Bachelier (1900), Einstein (1905 and 1906), % Smoluchowski (1906), Langevin (1906), and Perrin (1909), % it became usual at least among mathematicians % since the work of Wiener (1923) % to regard randomness as intrinsically nonsmooth, % involving independent, instantaneous noise increments injected at % each instant of time. An advantage of this point of view is % that it is mathematically beautiful and just right as an % idealization, even if the physical world does not % contain elements on all scales down to infinitesimal. % A disadvantage is that it is mathematically advanced, so % that any discussion of randomness is faced with technical challenges % of measure theory and functional analysis % (or an apology for their omission) from page~1. % Indeed, one cannot even write SDE\kern .5pt s % in the usual form $y'(t) = f(t,y)$, since $y'$ does not exist --- it % would represent white noise, which to be truly white must have % infinite amplitude. Therefore new % notations as in eqs.~(12.12) and (12.13) are used instead. % Along with new notation go % new theories of SDE\kern .5pt s above and beyond the usual theory of % deterministic ODE\kern .5pt s (It\^o, Stratonovich); and these in turn must be solved % by numerical methods above and beyond the usual ones (Euler--Murayama, % Milstein$,\dots$). % %% % % \smallskip % {\sc Our favorite reference.} % Jean-Pierre Kahane (1926--2017) was an expert in Taylor and Fourier % series with random coefficients. % As our favorite reference, we would like to highlight % his review paper A century of interplay between % Taylor series, Fourier series, and Brownian motion,'' {\em Bulletin % of the London Mathematical Society} 29 (1997), pp.~257--279. % The opening pages tell the fascinating story of % how an infinite Taylor series with random coefficients from $N(0,1)$, % for example, defines an analytic function in the open complex unit disk % $|z|<1$ and hence a smooth function of $\theta$ for $z=re^{i\theta}$ for % any $r<1$ (see Exercise 12.2). % As $r\to 1$, such functions approach white noise. % \smallskip % %% % % \begin{displaymath} % \framebox[4.7in][c]{\parbox{4.5in}{\vspace{2pt}\sl % {\sc Summary of Chapter 12.} % Smooth random functions with specified length scale $\lambda$ can % be defined via finite Fourier series with random coefficients. % Integrals of such functions give smooth random walks, % and random ODE\kern .5pt s can incorporate such functions either as % forcing terms or as coefficients. As $\lambda\to 0$, % smooth random ODE\kern .7pt s approach stochastic differential equations % (SDE\kern .5pt s) of the Stratonovich variety. % \vspace{2pt}}} % \end{displaymath} % %% % \smallskip\small\parskip=1pt\parindent=0pt % {\em \underline{Exercise $12.1$}. Tracking a random signal.} % Let $f$ be the function on $[\kern .3pt 0,50\kern .3pt]$ defined % by {\tt rng(0)}, \verb|randnfun(1,[0,50])|, and consider the IVP % $y' = -a(y(t) - f(t)),$ $y(0) = 0$, where $a>0$ is a constant. % Plot $f$ together with the solutions $y$ for $a = 0.1$, $1$, and % $10$ and discuss the results. Intuitively speaking, what is % happening here? % \par % {\em \underline{Exercise $12.2$}. Random and lacunary Taylor series.} % {\em (a)} % Define $f(z) = \sum_{k=0}^n c_k^{} z^k$, where % $c_0^{}, \dots,c_n^{}$ are independent random samples % from $N(0,1)$, with $n$ chosen big enough so that it is equivalent % to $\infty$ to plotting accuracy. % For a particular choice of random coefficients, % plot $\hbox{Re} (f(z))$ as a function of $\theta$ for % $z = re^{i\theta}$ with $r = 0.5, 0.9, 0.99$. % {\em (\kern .7pt b)} Another way to generate an analytic % function in the unit disk with a natural boundary on the unit % circle is by means of a {\em lacunary series} (i.e., one with % long gaps), an idea going back to Weierstrass. Make the same % plots as in {\em (a)} but now with $c_j^{}=1$ when $j$ is a power % of 2 and $c_j^{} = 0$ otherwise. % \par % {\em \underline{Exercise $12.3$}. Unbounded variation of a Brownian path.} % White noise has unbounded 1-norm with probability 1; so % its integral, Brownian motion, has unbounded variation. % Make a log-log plot of the 1-norms of normalized smooth random functions on % $[-1,1]$ as a function of wavelength parameter $\lambda$ for % $\lambda = 1, 1/2, \dots, 1/256$. What rate of increase do you % see as a function of $\lambda\kern .7pt ?$ % \par % {\em \underline{Exercise $12.4$}. Cumulative maximum of a Brownian path.} % Plot four smooth random walks with $\lambda = -0.1$ on % $[\kern .3pt 0,50\kern .3pt ]$ together with their cumulative % maxima, which you can calculate with {\tt cummax(f)}. Describe qualitatively % what you see. (It is known that with probability 1, % the maximum grows in a certain sense at a rate proportional to % $(t\log\log t)^{1/2}$ as $t\to\infty$.) % \par % {\em \underline{Exercise $12.5$}. Roots of a Brownian path.} % Calculate smooth random walk functions on % $[\kern .3pt 0,50\kern .3pt]$ for $\lambda= 16,8,4,\dots,1/16$, % initializing the random number seed with {\tt rng(1)} in each case. % Plot each function and calculate its roots. Describe qualitatively % how the sets of roots behave as $\lambda\to 0$. Find a way to % show this graphically. %