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Stochastic Calculus 1 Contents 3 Stochastic Calculus 1 3.1 References...................................... .............. 1 3.2 GaussianWhiteNoise .............................. ............... 2 3.3 StochasticIntegration . .................. 3 3.3.1 Langevinequationindifferentialform . ................... 3 3.3.2 Defining the stochastic integral . ................. 3 3.3.3 Summary of properties of the Itˆostochastic integral ....................... 5 3.3.4 Fokker-Planckequation. ............... 7 3.4 Stochastic Differential Equations . ...................... 7 3.4.1 Itˆochangeofvariablesformula . ................. 8 3.4.2 Solvabilitybychangeofvariables . .................. 9 3.4.3 MulticomponentSDE... .... ... .... .... .... ... .... ............ 9 3.4.4 SDEs with general α expressed as ItˆoSDEs (α =0)........................ 10 3.4.5 ChangeofvariablesintheStratonovichcase . .................... 11 3.5 Applications.................................... ............... 12 3.5.1 Ornstein-Uhlenbeckredux . ............... 12 3.5.2 Time-dependence ............................... ............ 13 3.5.3 Colorednoise .................................. ........... 13 3.5.4 Remarksaboutfinancialmarkets . ............... 15 i ii CONTENTS Chapter 3 Stochastic Calculus 3.1 References – C. Gardiner, Stochastic Methods (4th edition, Springer-Verlag, 2010) Very clear and complete text on stochastic methods, with many applications. – Z. Schuss, Theory and Applications of Stochastic Processes (Springer-Verlag, 2010) In-depth discussion of continuous path stochastic processes and connections to partial differential equations. – R. Mahnke, J. Kaupuzs,ˇ and I. Lubashevsky, Physics of Stochastic Processes (Wiley, 2009) Introductory sections are sometimes overly formal, but a good selection of topics. – H. Riecke, Introduction to Stochastic Processes and Stochastic Differential Equations (unpublished, 2010) Good set of lecture notes, often following Gardiner. Available online at: http://people.esam.northwestern.edu/~riecke/Vorlesungen/442/Notes/notes 442.pdf – J. L. McCauley, Dynamics of Markets (2nd edition, Cambridge, 2009) A physics-friendly discussion of stochastic market dynamics. Crisp and readable. Despite this being the second edition, there are alas a great many typographical errors. 1 2 CHAPTER 3. STOCHASTIC CALCULUS 3.2 Gaussian White Noise Consider a generalized Langevin equation of the form du = f(u,t)+ g(u,t) η(t) , (3.1) dt where η(t) is a Gaussian random function with zero mean and η(t) η(t′) = φ(t t′) . (3.2) − The spectral function of the noise is given by the Fourier transform, ∞ ˆ −iωs 1 2 φ(ω)= ds φ(s) e = lim ηˆT (ω) , (3.3) T →∞ T Z −∞ D E using the notation of 2.6.3. When φ(s) = Γδ(s) , we have φˆ(ω) = Γ , i.e. independent of frequency. This is the § case of Gaussian white noise. When φˆ(ω) has a nontrivial dependence on frequency, the noise is said to be colored. Gaussian white noise has an infinite variance φ(0), which leads to problems. In particular, the derivative u˙ strictly speaking does not exist because the function η(t) is not continuous. As an example of the sort of problem this presents, consider the differential equation u˙(t) = η(t) u(t). Let’s integrate this over a time period ∆t from tj to tj+1, where tj = j ∆t. We then have u(tj+1)= 1+ η(tj )∆t u(tj ). Thus, we find u(t )= 1+ η(t ) ∆t 1+ η(t ) ∆t u(t ) . (3.4) N N−1 ··· 0 0 Now let’s compute the average u(t ) . Since η(t ) is uncorrelated with η(t ) for all k = j, we can take the average N j k 6 of each of the terms individually, and since η(t ) has zero mean, we conclude that u(t ) = u(t ). On average, j N 0 there is no drift. Now let’s take a continuum limit of the above result, which is to say ∆t 0 with N∆t finite. Setting t = 0 and → 0 tN = t, we have t u(t)= u(0) exp ds η(s) , (3.5) Z0 and for Gaussian η(s) we have t t u(t) = u(0) exp 1 ds ds′ η(s) η(s′) = u(0) eΓ t/2 . (3.6) 2 Z Z 0 0 In the continuum expression, we find there isnoise-induced drift. The continuum limit of our discrete calculation has failed to match the continuum results. Clearly we have a problem that we must resolve. The origin of the problem is the aforementioned infinite variance of η(t). This means that the Langevin equation 3.1 is not well- defined, and in order to get a definite answer we must provide a prescription regarding how it is to be integrated1. 1We will see that Eqn. 3.4 corresponds to the Itˆo prescription and Eqn. 3.5 to the Stratonovich prescription. 3.3. STOCHASTIC INTEGRATION 3 3.3 Stochastic Integration 3.3.1 Langevin equation in differential form We can make sense of Eqn. 3.1 by writing it in differential form, du = f(u,t) dt + g(u,t) dW (t) , (3.7) where t W (t)= ds η(s) . (3.8) Z0 This is because W (t) is described by a Wiener process, for which the sample paths are continuous with probability unity. We shall henceforth take Γ 1, in which case W (t) is Gaussianly distributed with W (t) =0 and ≡ h i W (t) W (t′) = min(t,t′) . (3.9) The solution to Eqn. 3.7 is formally t t u(t)= u(0) + ds f u(s),s + dW (s) g u(s),s . (3.10) Z Z 0 0 Note that Eqn. 3.9 implies d dW (t) dW (t′) W (t) W (t′) = Θ(t t′) = η(t) η(t′) = δ(t t′) . (3.11) dt′ − ⇒ dt dt′ − 3.3.2 Defining the stochastic integral Let F (t) be an arbitrary function of time, and let t be a discretization of the interval [0,t] with j 0,...,N . { j } ∈ { } The simplest example to consider is tj = j ∆t where ∆t = t/N. Consider the quantity N−1 S (α)= (1 α) F (t )+ α F (t ) W (t ) W (t ) , (3.12) N − j j+1 j+1 − j j=0 X h ih i where α [0, 1]. Note that the first term in brackets on the RHS can be approximated as ∈ F (τ )=(1 α) F (t )+ α F (t ) , (3.13) j − j j+1 where τ (1 α) t + αt [t ,t ]. To abbreviate notation, we will write F (t )= F , W (t )= W , etc. We j ≡ − j j+1 ∈ j j+1 j j j j may take t 0 and W 0. The quantities ∆W W W are independently and Gaussianly distributed with 0 ≡ 0 ≡ j ≡ j+1 − j zero mean for each j. This means ∆Wj =0 and 2 ∆Wj ∆Wk = (∆Wj ) δjk = ∆tj δjk , (3.14) where ∆t t t . Wick’s theorem then tells us j ≡ j+1 − j ∆Wj ∆Wk ∆Wl ∆Wm = ∆Wj ∆Wk ∆Wl ∆Wm + ∆Wj ∆Wl ∆Wk ∆Wm + ∆Wj ∆Wm ∆Wk ∆Wl = ∆tj ∆tl δjk δ lm + ∆tj ∆tk δjl δkm + ∆tj ∆ tk δjm δkl . (3.15) 4 CHAPTER 3. STOCHASTIC CALCULUS 2 4 2 EXERCISE: Show that WN = t and WN =3t . The expression in Eqn. 3.12 would converge to the integral t S = dW (s) F (s) (3.16) Z0 independent of α were it not for the fact that ∆W /∆t has infinite variance in the limit N . Instead, we will j j → ∞ find that S (α) in general depends on the value of α. For example, the Itˆointegral is defined as the N limit N → ∞ of S (α) with α =0, whereas the Stratonovich integral is defined as the N limit of S (α) with α = 1 . N → ∞ N 2 We now define the stochastic integral t N−1 dW (s) F (s) ms lim (1 α) F (tj )+ α F (tj+1) W (tj+1) W (tj ) , (3.17) α ≡ −N→∞ − − Z j=0 0 X h ih i where ms-lim stands for mean square limit. We say that a sequence SN converges to S in the mean square if lim (S S)2 =0. Consider, for example, the sequence S = N−1(∆W )2. We now take averages, using N→∞ N − N j=0 j (∆W )2 = t t ∆t . Clearly S = S = t. We also have j j+1 − j ≡ j h N i P N−1 N−1 2t2 S2 = (∆W )2 (∆W )2 = (N 2 +2N)(∆t)2 = t2 + , (3.18) N j k N j=0 k=0 X X where we have used Eqn. 3.15. Thus, (S S)2 =2t2/N 0 in the N limit. So S converges to t in the N − → → ∞ N mean square. Next, consider the case where F (t)= W (t). We find N−1 N−1 S (α)= (1 α) W (t )+ α W (t ) W (t ) W (t ) = W + α ∆W ∆W N − j j+1 j+1 − j j j j j=0 j=0 X h ih i X (3.19) N−1 N−1 2 2 2 = 1 W + ∆W W 2 + (2α 1) ∆W = 1 W 2 + (α 1 ) ∆W . 2 j j − j − j 2 N − 2 j j=0 j=0 X h i X Taking the average, N−1 S (α) = 1 t + (α 1 ) (t t )= α t . (3.20) N 2 N − 2 j+1 − j j=0 X Does S converge to S = αt in the mean square? Let’s define Q N−1(∆W )2, which is the sequence we N h N i N ≡ j=0 j analyzed previously. Then S = 1 W 2 + (α 1 ) Q . We then have N 2 N − 2 N P S2 = 1 W 4 + (α 1 ) W 2 Q + (α 1 )2 Q2 , (3.21) N 4 N − 2 N N − 2 N with N−1 N−1 N−1 N−1 4 2 2 WN = ∆Wj ∆Wk ∆Wl ∆Wm =3N (∆t) j=0 k=0 l=0 m=0 X X X X N−1 N−1 N−1 2 2 2 2 WN QN = ∆Wj ∆Wk (∆Wl) = (N +2N)(∆t) (3.22) j=0 k=0 l=0 X X X N−1 N−1 2 2 2 2 2 QN = (∆Wj ) (∆Wk) = (N +2N)(∆t) .
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