Diffusion processes

Appendix J Diffusion processes This appendix provides a brief introduction to diffusion processes and to some of the mathematical techniques used in their description and analysis. More comprehensive and rigorous accounts are provided by Gardiner (1985), Gillespie (1992), Arnold (1974), and Karlin and Taylor (1981). A diffusion process is a particular kind of stochastic process. It is a continuous-time Markov process with continuous sample paths (and other properties described below). Markov processes Let U(t) for t ≥ t0 be a stochastic process with one-time PDF f(V ; t). We introduce N times t1 < t2 < . . . < tN , (with t1 > t0 ), and consider the PDF of U(tN ) conditioned on U(t) at the earlier times {U(tN−1 ), U(tN−2 ), . . . , U(t1 )}, which is denoted by fN−1 (VN ; tN |VN−1 , tN−1 , VN−2 , tN−2 , . . . , V1 , t1 ). The PDF of U(t) conditioned on a single past time is denoted by, for example, f1 (VN ; tN |VN−1 , tN−1 ). By definition, if U(t) is a Markov process then these conditional PDFs are equal: fN−1 (VN ; tN |VN−1 , tN−1 , VN−2 , tN−2 , . . . , V1 , t1 ) = f1 (VN ; tN |VN−1 , tN−1 ). (J.1) This means that, given U(tN−1 ) = VN−1 , knowledge of the previous values U(tN−2 ), U(tN−3 ), . . . , U(t1 ) provides no further information about the future value U(tN ). 713 Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 714 J Diffusion processes The Chapman–Kolmogorov equation For any process, from the definition of conditional PDFs, we have ∞ f1 (V3 ; t3 |V1 , t1 ) = −∞ f2 (V3 ; t3 |V2 , t2 , V1 , t1 )f1 (V2 ; t2 |V1 , t1 ) dV2 , (J.2) (see Exercise J.1). For a Markov process, Eq. (J.1) can be used to replace f2 by f1 (V3 ; t3 |V2 , t2 ) which leads to the Chapman–Kolmogorov equation ∞ f1 (V3 ; t3 |V1 , t1 ) = −∞ f1 (V3 ; t3 |V2 , t2 )f1 (V2 ; t2 |V1 , t1 ) dV2 . (J.3) Increments A useful concept is the increment in a process: the increment in a positive time interval h is defined by Δh U(t) ≡ U(t + h) − U(t). (J.4) It is important to note that h is positive and that the increment is defined forward in time. A process can be considered as a sum of its increments, e.g., U(tN ) = U(t0 ) + Δt1 −t0 U(t0 ) + Δt2 −t1 U(t1 ) + . . . + ΔtN −tN−1 U(tN−1 ). (J.5) The PDF of the increment Δh U(t), conditional on U(t) = V , is denoted by g(V̂ ; h, V , t). If h is taken to be t3 − t2 , then U(t2 ) can be re-expressed as U(t2 ) = U(t3 ) − Δh U(t2 ), (J.6) and the first conditional PDF on the right-hand side of Eq. (J.3) is f1 (V3 ; t2 + h|V3 − V̂ , t2 ) = g(V̂ ; h, V3 − V̂ , t2 ). (J.7) Thus the Chapman–Kolmogorov equation can be rewritten as ∞ f1 (V ; t2 + h|V1 , t1 ) = −∞ g(V̂ ; h, V − V̂ , t2 )f1 (V − V̂ ; t2 |V1 , t1 ) dV̂ . (J.8) Diffusion processes There are qualitatively different kinds of continuous-time Markov processes, which are distinguished from each other by the behaviors of their increments Δh U(t) in the limit as h tends to zero. One defining property of a diffusion process is that its sample paths are continuous. More precisely, for every ǫ > 0, 1 lim P {|Δh U(t)| > ǫ|U(t) = V } = 0. (J.9) h↓0 h Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 J Diffusion processes 715 If they exist, the infinitesimal parameters of a process are defined by 1 Bn (V , t) ≡ lim [Δh U(t)]n |U(t) = V , h↓0 h 1 ∞ n = lim V̂ g(V̂ ; h, V , t) dV̂ , h↓0 h −∞ (J.10) for n = 1, 2, . . .. In addition to Eq. (J.9), the defining properties of a diffusion process are that the drift coefficient, a(V , t) ≡ B1 (V , t), (J.11) b(V , t)2 ≡ B2 (V , t), (J.12) and the diffusion coefficient, exist, and that the remaining infinitesimal parameters are zero: Bn (V , t) = 0, for n ≥ 3. (J.13) A differentiable deterministic process governed by the ordinary differential equation dU(t) = a(U(t), t) (J.14) dt is a degenerate diffusion process, with drift a(V , t) and diffusion coefficient b(V , t)2 = 0. A non-degenerate diffusion process (i.e., b(V , t) > 0) is clearly nowhere differentiable, for the fact that [Δh U(t)]2 /h tends to a positive  2 limit implies that  Δh U(t)/h  tends to infinity. The Kramers–Moyal equation In the Chapman–Kolmogorov equation (Eq. (J.8)), both g and f1 on the right-hand side involve the argument V − V̂ . Expanding these quantities in a Taylor series about V yields f1 (V ; t2 + h|V1 , t1 ) = f1 (V ; t2 |V1 , t1 ) ∞ ∞   (−V̂ )n ∂n  | + , t ) dV̂ . (J.15) g( V̂ ; h, V , t )f (V ; t V 1 1 2 1 2 n! ∂V n −∞ n=1 By dividing by h, taking the limit h → 0, and using Eq. (J.10), we obtain the Kramers–Moyal equation ∞  (−1)n ∂n ∂ [Bn (V , t)f1 (V ; t|V1 , t1 )]. f1 (V ; t|V1 , t1 ) = ∂t n! ∂V n n=1 (J.16) Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 716 J Diffusion processes This equation applies to processes for which the parameters Bn (V , t) exist, and for t ≥ t1 . The appropriate initial condition is f1 (V ; t1 |V1 , t1 ) = δ(V − V1 ). (J.17) The Fokker–Planck equation For a diffusion process, all of the parameters Bn are zero, except for the drift, B1 = a, and the diffusion, B2 = b2 . In this case, Eq. (J.16) reduces to the Fokker–Planck or forward Kolmogorov equation: ∂ ∂ f1 (V ; t|V1 , t1 ) = − [a(V , t)f1 (V ; t|V1 , t1 )] ∂t ∂V 1 ∂2 + [b(V , t)2 f1 (V ; t|V1 , t1 )]. (J.18) 2 ∂V 2 This equation determines the evolution of the conditional PDF. The corresponding equation for the marginal PDF f(V ; t) is obtained by multiplying by f(V1 ; t1 ) and integrating over V1 . Since, in Eq. (J.18), only f1 has any dependence on V1 , the result is simply  ∂ 1 ∂2  ∂ f(V ; t) = − [a(V , t)f(V ; t)] + b(V , t)2 f(V ; t) . 2 ∂t ∂V 2 ∂V (J.19) For the deterministic process governed by the ordinary differential equation Eq. (J.14), the diffusion coefficient is zero, and hence the last terms in Eqs. (J.18) and (J.19) vanish. The resulting equations are called the Liouville equations. The stationary distribution If the coefficients a and b are independent of time, it is possible for the diffusion process to be statistically stationary. In this case, the Fokker– Planck equation (Eq. (J.19)) reduces to 0=− which has the solution  d 1 d2  b(V )2 f(V ) , [a(V )f(V )] + 2 dV 2 dV  C f(V ) = exp b(V )2 V Vo  2a(V ′ ) ′ dV , b(V ′ )2 (J.20) (J.21) where the lower limit Vo can be chosen for convenience, and the constant C is determined by the normalization condition. (If the integral of the right-hand side of Eq. (J.21) over all V does not converge, then U(t) does not have a stationary distribution.) Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 717 J Diffusion processes 4 3 W(t) 2 1 0 –1 –2 –3 –4 –5 0 2 4 6 8 10 12 14 16 t Fig. J.1. Three sample paths of the Wiener process. The Wiener process The most fundamental diffusion process, from which all others can be derived, is the Wiener process, denoted by W (t). This is defined (for t ≥ 0) by the initial condition W (0) = 0, and by the specification of the drift and diffusion coefficients, b(V , t)2 = 1. a(V , t) = 0, (J.22) Some sample paths of W (t) are shown in Fig. J.1. As may readily be verified, the solution to the Fokker–Planck equation (Eq. (J.18)) with a = 0 and b2 = 1 from the initial condition Eq. (J.17) is   1 (V − V1 )2 1 2 f1 (V ; t|V1 , t1 ) = √ , (J.23) exp − t − t1 2π(t − t1 ) i.e., a normal distribution with mean V1 and variance t − t1 . Thus, for all h > 0, the increment Δh W (t) is normally distributed with mean zero and variance h: D Δh W (t) = N (0, h). (J.24) D (The symbol = is read ‘is equal in distribution to,’ and N (µ, σ 2 ) denotes the normal with mean µ and variance σ 2 , Eq. (3.41).) Some important properties (not all independent) of Wiener-process increments are Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 718 J Diffusion processes W (t2 ) − W (t1 ) = 0, [W (t2 ) − W (t1 )]2  = var [W (t2 ) − W (t1 )] = t2 − t1 , D W (t2 ) − W (t1 ) = N (0, t2 − t1 ), D h−1/2 Δh W (t) = N (0, 1), W (t2 ) − W (t1 ) is independent of W (t) for t ≤ t1 , [W (t3 )−W (t2 )][W (t2 )−W (t1 )] = 0 – increments in non-overlapping time intervals are independent, (vii) [W (t4 ) − W (t2 )][W (t3 ) − W (t1 )] = t3 − t2 – the covariance of increments equals the duration of the overlap of the time intervals, N 2 (viii) n=1 [W (tn ) − W (tn−1 )]  = tN − t0 , and (ix) W (t) is a Gaussian process: the joint PDF of W (t1 ), W (t2 ), . . . , W (tN ) is a joint normal. (i) (ii) (iii) (iv) (v) (vi) Several other interesting properties of the Wiener process are deduced in Exercise J.2. Stochastic differential equations Because diffusion processes are not differentiable, the standard tools of differential calculus cannot be applied. Instead of differential calculus, the appropriate method is the Ito calculus; and, instead of being described by ordinary differential equations, diffusion processes are described by stochastic differential equations. The infinitesimal increment of the process U(t) is defined by dU(t) ≡ U(t + dt) − U(t), (J.25) where dt is a positive infinitesimal time interval. For the Wiener process in particular, we have D dW (t) = W (t + dt) − W (t) = N (0, dt). (J.26) Now consider the process U(t) defined by the initial condition U(t0 ) = U0 , and by the increment dU(t) = a[U(t), t] dt + b[U(t), t] dW (t), (J.27) for given functions a(V , t) and b(V , t). It is readily verified that the process U(t) defined by this stochastic differential equation is a diffusion process; and, as implied by the notation, the drift and diffusion coefficients are a(V , t) and b(V , t)2 . A random variable is fully characterized by its PDF; and two random variables with the same PDF are statistically identical. Similarly, a diffusion process is fully characterized by its drift and diffusion coefficients; and two Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 J Diffusion processes 719 diffusion processes with the same coefficients are statistically identical. Thus the stochastic differential equation Eq. (J.27) provides a general expression for a diffusion process. The stochastic differential equation Eq. (J.27) shows that the infinitesimal increment of a diffusion process is Gaussian, i.e., dU(t) = N (a[U(t), t] dt, b[U(t), t]2 dt). (J.28) This Gaussianity is not a defining property of diffusion processes, but rather a deduction from their definition. White noise Prior to the development of the theory of stochastic differential equations, diffusion processes were commonly expressed as ordinary differential equations involving white noise. On dividing Eq. (J.27) by dt we obtain dU(t) = a[U(t), t] + b[U(t), t]Ẇ (t), dt (J.29) where the white noise Ẇ (t) is dW (t)/dt. Since neither dU/dt nor dW /dt exists, this equation cannot be interpreted in the usual way. Consequently it is preferable not to use the concept of white noise, but instead to express diffusion processes as stochastic differential equations. The evolution of moments Equations for the evolution of the unconditional moments U(t)n  can be derived from the Fokker–Planck equation (Eq. (J.19)), or from the stochastic differential equation (Eq. (J.27)). The latter approach is instructive. Taking the mean of Eq. (J.25) and substituting Eq. (J.27), we obtain U(t + dt) − U(t) = dU(t) = a[U(t), t]dt + b[U(t), t] dW (t). (J.30) Now dW (t) has zero mean, and it is independent of U(t′ ) for t′ ≤ t. Thus the last term vanishes, leading to d U(t) = a[U(t), t]. dt (J.31) Similarly, the mean of the square of U(t + dt) is U(t + dt)2  = [U(t) + dU(t)]2  = U(t)2  + 2U(t) dU(t) + dU(t)2 . (J.32) Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 720 J Diffusion processes The cross-term is 2Ua dt, while the final term is dU(t)2  = a(U[t], t)2  dt2 + b(U[t], t)2 dW (t)2  = b(U[t], t)2 dW (t)2  + o(dt) = b(U[t], t)2  dt + o(dt), (J.33) where o(h) denotes a quantity such that lim h↓0 o(h) = 0, h (J.34) (e.g., h1+ǫ = o(h), for all ǫ > 0). Thus the mean square of U(t) evolves by d U(t)2  = 2U(t)a[U(t), t] + b[U(t), t]2 . dt (J.35) Notice that, for a differentiable process dU(t)2 /dt is zero, but, for a diffusion process, it is b2 , and this leads to the final term in Eq. (J.35). The Ornstein–Uhlenbeck (OU) process The OU process is the simplest statistically stationary diffusion process. It is defined by the linear drift coefficient V , T (J.36) 2σ 2 , T (J.37) a(V , t) = − the constant diffusion coefficient b(V , t)2 = and the initial condition D U(0) = N (0, σ 2 ), (J.38) where T is a positive timescale, and σ is a constant. The corresponding stochastic differential equation is the Langevin equation  2 1/2 dt 2σ dU(t) = −U(t) + dW (t). (J.39) T T For the OU process, the Fokker–Planck equation (Eq. (J.18)) for the PDF of U(t) conditional on U(t1 ) = V1 (for t > t1 ), f1 (V ; t|V1 , t1 ), is ∂f1 σ 2 ∂ 2 f1 1 ∂ . (J.40) = (V f1 ) + ∂t T ∂V T ∂V 2 With the deterministic initial condition Eq. (J.17), the solution to this equation is the normal distribution  $ % f1 (V ; t|V1 , t1 ) = N V1 e−(t−t1 )/T , σ 2 1 − e−2(t−t1 )/T , (J.41) Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 J Diffusion processes 721 (see Exercise J.4). This solution, which fully characterizes the process, shows that the conditional mean U(t)|V1  decays from V1 to 0 on the timescale T ; while the conditional variance increases from 0 to σ 2 on the timescale 1 T . At large times, the conditional PDF tends to the stationary distribution 2 N (0, σ 2 ). An important deduction from Eq. (J.41) is that the OU process is a Gaussian process. A consequence of the Markov property is that the joint PDF of U(t) at the N + 1 times {t0 = 0, t1 , t2 , . . . , tN } can be written as the product of the marginal PDF at t0 , and the N conditional PDFs f1 (Vn ; tn | Vn−1 , tn−1 ) for n = 1, 2, . . . , N. Since each of these PDFs is normal (see Eqs. (J.38) and (J.41)), the (N + 1)-time joint PDF is joint normal, satisfying the definition of a Gaussian process. Since the mean U(t) is zero, the autocovariance (for s ≥ 0) is R(s) = U(t1 + s)U(t1 ). (J.42) This is readily evaluated from the conditional mean U(t1 + s)|V1  = V1 e−s/T by R(s) = U(t1 + s)|U(t1 ) U(t1 ) = U(t1 )2 e−s/T = σ 2 e−s/T . (J.43) For any statistically stationary process, the autocovariance R(s) and the autocorrelation function ρ(s) are even functions. Hence, for the OU process, we have R(s) = σ 2 e−|s|/T , (J.44) ρ(s) = e−|s|/T . (J.45) ∞ Notice that the integral timescale defined by 0 ρ(s) ds is T . The second-order structure function D2 (s) and the frequency spectrum E(ω) contain the same information as does the autocovariance R(s). For the OU process, the structure function is D2 (s) ≡ [U(t + s) − U(t)]2  = 2[σ 2 − R(s)] 2σ 2 = 2σ 2 (1 − e−|s|/T ) = |s| + O(s2 ), T (J.46) while the spectrum (twice the Fourier transform of R(s)) is E(ω) = (2/π)σ 2 T . 1 + T 2ω2 (J.47) The lack of differentiability of U(t) manifests itself in the discontinuous Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 722 J Diffusion processes slope of ρ(s) at the origin, and in the variations D2 (s) ∼ s (for small s) and E(ω) ∼ ω −2 (for large ω). In summary; the Ornstein–Uhlenbeck process (which is generated by the Langevin equation, Eq. (J.39)) is a statistically stationary Gaussian process. As such, it is fully characterized by its mean (which is zero), the variance σ 2 , and the autocorrelation function ρ(s) = e−|s|/T . The Ito transformation Consider the process q(t) defined by q(t) = Q[U(t)], (J.48) where U(t) is a diffusion process (with drift a(V , t) and diffusion b(V , t)2 ), and Q(V ) is a differentiable function, with derivatives Q′ (V ), Q′′ (V ), etc. The infinitesimal increment in q is dq(t) = Q[U(t) + dU(t)] − Q[U(t)]. (J.49) By expanding Q(U + dU) in a Taylor series about U(t), and substituting Eq. (J.27) for dU, we obtain dq(t) = Q′ [U(t)] dU + 12 Q′′ [U(t)] dU 2 + o(dt) = (Q′ a + 21 Q′′ b2 ) dt + Q′ b dW + o(dt). (J.50) Thus, q(t) is itself a diffusion process, with stochastic differential equation dq(t) = aq [q(t), t] dt + bq [q(t), t] dW (t), (J.51) where the coefficients are aq = Q′ a + 21 Q′′ b2 , (J.52) bq = Q′ b. (J.53) These relations form the Ito transformation. The essential difference – compared with the transformation rule for ordinary differential equations – is the additional drift 21 Q′′ b2 . Vector-valued diffusion processes The development presented above for the scalar-valued diffusion process extends straightforwardly to the vector-valued process U (t) = {U1 (t), U2 (t), . . . , UD (t)}. Only the principal results are given here. The drift coefficient is the vector 1 a(V , t) = lim [Δh U (t)]|U (t) = V , (J.54) h↓0 h Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 J Diffusion processes 723 while the diffusion coefficient is the D × D matrix B, with elements 1 Bij (V , t) = lim Δh Ui (t) Δh Uj |U (t) = V . h↓0 h (J.55) It follows from its definition that B is symmetric positive semi-definite. The Fokker–Planck equation for the PDF of U (t), f(V ; t), and also for the conditional PDF, is ∂f 1 ∂2 ∂ (ai f) + (Bij f), =− ∂t ∂Vi 2 ∂Vi ∂Vj (J.56) cf. Eq. (J.19). The vector-valued Wiener process W (t) = {W1 (t), W2 (t), . . . , WD (t)} is simply composed of the independent scalar processes Wi (t). The increment dW (t) is a joint normal, with zero mean, and covariance dWi dWj  = dt δij . (J.57) dUi (t) = ai [U (t), t] dt + bij [U (t), t] dWj (t), (J.58) The process is statistically isotropic: if A is a unitary matrix, then Ŵ (t) ≡ AW (t) is also a vector-valued Wiener process. The stochastic differential equation for U (t) is written where (for consistency with Eq. (J.55)) the coefficients bij satisfy bik bjk = Bij . (J.59) Notice that the non-symmetric matrix b is not uniquely determined by the symmetric matrix B. Two possible choices of b are the symmetric square root of B and the lower triangular matrix given by the Cholesky decomposition of B. All choices of b (consistent with Eq. (J.59)) result in statistically identical diffusion processes. EXERCISES J.1 Consider three random variables U1 , U2 , and U3 . Conditional PDFs are defined from the joint PDFs by, for example, f3|1 (V3 |V1 ) = f13 (V1 , V3 )/f1 (V1 ), (J.60) f3|12 (V3 |V1 , V2 ) = f123 (V1 , V2 , V3 )/f12 (V1 , V2 ). (J.61) Obtain the result ∞ f123 (V1 , V2 , V3 ) f12 (V1 , V2 ) dV2 = f3|1 (V3 |V1 ), f12 (V1 , V2 ) f1 (V1 ) −∞ (J.62) and hence verify Eq. (J.2). Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 724 J.2 J Diffusion processes Let the time interval (0, T ) be divided into M equal sub-intervals of duration h = T /M. For the discrete times nh (n = 0, 1, . . . , M), the process W̃ is defined by W̃ (0) = 0 and W̃ (nh) = n  h1/2 ξi , i=1 for n ≥ 1, (J.63) where ξ1 , ξ2 , . . . , ξM are independent standardized normal random variables. (a) Show that W̃ is statistically identical to a Wiener process sampled at the same times. (b) Let SM denote the sum of the squares of the increments of the Wiener process SM ≡ M−1  [Δh W (nh)]2 . (J.64) n=0 Obtain the results SM  = T and var(SM ) = 2hT = 2T 2 /M. Hence argue that the random variable SM has the non-random limit S∞ = T . (c) Consider a plot of W̃ against t, in which successive values are connected by straight line segments. Show  that the expected length of each line segment exceeds 2h/π, and  that the expectation of the sum of the lengths exceeds 2MT /π. Hence argue that, in every positive interval, the sample path of a Wiener process has infinite arc length. (d) What is the joint PDF fn (V , V̂ ; h) of W̃ (nh) and Δh W̃ (nh)? For n ≥ 1, consider the event Cn defined by W̃ (nh) and W̃ [(n+1)h] having opposite signs. What region of the V − V̂ sample space corresponds to the event Cn ? Show that the probability of Cn is   1 1 1 −1 √ (J.65) P (Cn ) = tan ≥ √ . π n 4 n (Hint: consider the simple transformation that transforms fn to a standardized joint normal.) The expected number of M−1 times that W̃ changes sign is NM = n=1 P (Cn ). How does NM behave as M increases? Hence argue that, in any positive interval (t1 , t2 ), the Wiener process takes every value between W (t1 ) and W (t2 ) an infinite number of times. Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 J Diffusion processes 725 (e) For any positive constant c, show that, in the transformed time t̂ = ct, the process 1 Ŵ (t̂) ≡ √ W (ct) c (J.66) is a Wiener process. J.3 J.4 For the diffusion process defined by Eq. (J.27), which of the following quantities are correlated: W (t), dW (t), U(t), dU(t), and U(t + dt)? Let Ψ(s, t) (for t ≥ t1 ) be the characteristic function of the OU process from the deterministic initial condition U(t1 ) = V1 ; so that its Fourier transform is the conditional PDF f1 (V ; t|V1 , t1 ). From the Fokker–Planck equation for f1 (Eq. (J.40)), show that Ψ(s, t) evolves by   ∂ σ 2 s2 s ∂ Ψ(s, t) = − + Ψ(s, t). (J.67) ∂t T ∂s T Use the method of characteristics to obtain the solution Ψ(s, t) = Ψ(se−(t−t1 )/T , t1 ) exp(− 21 s2 Σ(t)2 ), (J.68) Σ(t)2 ≡ σ 2 (1 − e−2(t−t1 )/T ). (J.69) where Show that, with the initial condition Ψ(s, t1 ) = e−isV1 , this solution corresponds to the normal distribution with mean µ(t) = V1 e−(t−t1 )/T J.5 (J.70) and variance Σ(t)2 . Let each component of the velocity U (t) = {U1 (t), U2 (t), U3 (t)} evolve by an independent Langevin equation, i.e.,  2 1/2 dt 2σ dU = −U dW . (J.71) + T T What is the joint PDF of U ? The speed q(t) is defined by q(t) = [Ui (t)Ui (t)]1/2 . (J.72) Show that q(t) evolves by the stochastic differential equation  2   2 1/2 2σ dt 2σ dq = −q + dW . (J.73) q T T Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025 726 J.6 J Diffusion processes Show that the stationary distribution of q is ( 2 v 2 −v2 /(2σ2 ) f(v) = e . (J.74) π σ3 Let Ψ(s, t) be the characteristic function of the general vector-valued diffusion process given by Eq. (J.58). By expanding exp(is.[U (t) + dU (t)]) (see Eq. (I.23)), obtain the result Ψ(s, t + dt) = Ψ(s, t) + isj aj eis·U (t)  dt − 21 sj sk bjℓ bkℓ eis·U (t)  dt + o(dt), (J.75) where the coefficients are, for example, aj = aj [U (t), t]. Show that the Fourier transform of this equation (when it is divided by dt) is the Fokker–Planck equation, Eq. (J.56). Downloaded from https://www.cambridge.org/core. National University of Singapore (NUS), on 08 Oct 2019 at 08:41:51, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511840531.025