Lecture 8: Stochastic Differential Equations Readings Recommended: • Pavliotis (2014) 3.2-3.5 • Oksendal (2005) Ch. 5 Optional: • Gardiner (2009) 4.3-4.5 • Oksendal (2005) 7.1,7.2 (on Markov property) • Koralov and Sinai (2010) 21.4 (on Markov property) We’d like to understand solutions to the following type of equation, called a Stochastic Differential Equation (SDE): dXt = b(Xt ;t)dt + s(Xt ;t)dWt : (1) Recall that (1) is short-hand for an integral equation Z t Xt = b(Xs;s)ds + s(Xs;s)dWs: (2) 0 In the physics literature, you will often see (1) written as dx = b(x;t) + s(x;t)h(t); dt where h(t) is a white noise: a Gaussian process with mean 0 and covariance function Eh(s)h(t) = d(t − s). Each term in (1) has a different interpretation. • The term b(Xt ;t)dt is called the drift term. It describes the deterministic part of the equation. When this is the only term, we obtain a canonical ODE. • The term s(Xt ;t)dWt is called the diffusion term. It describes random motion proportional to a Brow- nian motion. Over small times, this term causes the probability to spread out diffusively with a diffu- sivity locally proportional to s 2. If the diffusion term is constant, i.e. s(x;t) ≡ s 2 R, then the noise is said to be additive. If the diffusion ¶ term depends on x, i.e. ¶x s(x;t) 6= 0 in (1), the noise is said to be multiplicative. We will see that equations with multiplicative noise have to be treated more carefully then equations with additive noise. We learned how to define the integrals in the expressions above last class. In this one we’ll look at properties of the solutions themselves. We will ask: when do solutions exist? Are they unique? And how can we actually solve them, and extract useful information? 1 Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2019 8.1 Existence and uniqueness Definition. A stochastic process X = (Xt )t≥0 is a strong solution to the SDE (1) for 0 ≤ t ≤ T if X is 1 1 2 continuous with probability 1, X is adapted (to Wt ), b(Xt ;t) 2 L (0;T), s(Xt ;t) 2 L (0;T), and Equation (2) holds with probability 1 for all 0 ≤ t ≤ T. Definition. A strong solution X to an SDE of the form (1) is called a diffusion process. Remark. To be a diffusion process, it is important that the coefficients of (1) depend only on (Xt ;t) – they can’t be general adapted functions f (w;t). n n×m Theorem. Given equation (1), suppose b 2 R , s 2 R satisfy global Lipschitz and linear growth condi- tions: jb(x;t) − b(y;t)j + js(x;t) − s(y;t)j ≤ Kjx − yj jb(x;t)j + js(x;t)j ≤ K(1 + jxj) n for all x;y 2 R , t 2 [0;T], where K > 0 is a constant. Assume the initial value X0 = x is a random variable 2 with Ex < ¥ and which is independent of (Wt )t≥0. Then (1) has a unique strong solution X. Remark. “Unique” means that if X1;X2 are two strong solutions, then P(X1(t;w) = X2(t;w) for all t) = 1. That is, the two solutions are equal everywhere with probability 1. This is different from the statement that X1, X2 are versions of each other – you should think about how. This theorem bears a lot in common with similar theorems regarding the existence and uniqueness to the solution to an ODE. Counterexamples that show the necessity of each of the conditions of the theorem that apply to ODEs, can also be used for SDEs. Example. To construct an equation whose solution is not unique, we drop the condition of Lipschitz con- 2=3 3 tinuity. Consider the ODE dXt = Xt dt, which has solutions Xt = 0 for t ≤ a, Xt = (t − a) for t > a, for any a > 0. However, b(x) = 3x2=3 is not Lipschitz continuous as 0. For an example involving a Brownian motion, consider 1=3 2=3 dXt = 3Xt dt + 3Xt dWt ; X0 = 0: 3 This has (at least) two solutions: Xt = 0, Xt = Wt . But, again, the coefficients of the SDE are not Lipschitz continuous. Example. To construct an equation which has no global solution, we drop the linear growth conditions. Consider 2 dXt = Xt dt; X0 = x0: 1 1 The solution is Xt = , which blows up at t = . 1 −t x0 x0 Proof (Uniqueness, 1d). (from Evans (2013), section 5.B.3) Let Xt ;Xˆt 2 V ([0;T]) be two strong solutions to (1). Then Z t Z t Xt − Xˆt = b(Xs;s) − b(Xˆs) ds + s(Xs;s) − s(Xˆs;s) dWs : 0 0 1Actualy we ask for something slightly stronger, namely that X be progressively measurable with respect to F , the filtration generated by (Wt )t≥0 2 Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2019 Square each side, use (a + b)2 ≤ 2a2 + 2b2, and take expectations to get Z t 2 Z t 2 2 EjXt − Xˆt j ≤ 2E b(Xs;s) − b(Xˆs;s) ds + 2E s(Xs;s) − s(Xˆs;s) dWs : 0 0 We estimate the first term on the right-hand side using the the Cauchy-Schwarz inequality, which implies R t 2 R t 2 that 0 f ds ≤ t 0 j f j ds. We then use the Lipschitz continuity of b. The result is Z t 2 Z t Z t 2 2 2 E b(Xs;s) − b(Xˆs;s) ds ≤ TE b(Xs;s) − b(Xˆs;s) ds ≤ K T EjXs − Xˆsj ds : 0 0 0 Now we estimate the second term using the Itoˆ isometry and the Lipschitz continuity of s: Z t 2 Z t Z t 2 2 2 E s(Xs;s) − s(Xˆs;s) dWs = Ejs(Xs;x) − s(Xˆs;s)j ds ≤ K EjXs − Xˆsj ds : 0 0 0 Putting these estimates together shows that Z t 2 2 EjXt − Xˆt j ≤ C EjXs − Xˆsj ds 0 2 for some constant C, for 0 ≤ t ≤ T. If we let f(t) ≡ EjXt − Xˆt j , then the inequality is Z t f(t) ≤ C f(s)ds for all 0 ≤ t ≤ T : 0 Now we can use Gronwall’s Inequality, which says that if we are given a function f and nonnegative numbers a;b ≥ 0, then Z t f (t) ≤ a + b f (s)ds =) f (t) ≤ aebt : 0 2 The proof is given in the appendix. Applying Gronwall’s Inequality with f (t) = f(t) = EjXt − Xˆt j and a = 0, b = C shows that 2 EjXt − Xˆt j = 0 for all 0 ≤ t ≤ T : Therefore for each fixed t 2 [0;T] we have that Xt = Xˆt a.s. We have to show this holds for all t simultane- ously, i.e. the whole trajectory is equal, except for w in a set of measure 0. We can argue that Xr = Xˆr a.s. for all rational 0 ≤ r ≤ T, i.e. P(Xt = Xˆt 8t 2 Q \ [0;T]) = 1. This is because can extend the equality to a ˆ countable set of t-values, say ft1;t2;:::g, because for each ti, the “bad” w-values fw : Xti (w) 6= Xti (w)g form a measure-zero set, and a countable union of measure-zero sets is measure zero. By assumption X, Xˆ have continuous sample paths almost surely, so we can extend the equality to all values of t using the fact that the rationals form a dense set in R, so P(Xt = Xˆt 8t 2 [0;T]) = 1: Proof (Existence, for a simpler equation). (Based on Evans (2013), section 5.B.1) We will show existence for the simpler equation dXt = b(Xt )dt + dWt ; X0 = x 2 R ; (3) where b 2 C1 with jb0j ≤ K for some constant K. The proof in the more general case uses similar ideas, see e.g. Evans (2013) section 5.B.3 or Oksendal (2005) section 5.2. 3 Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2019 0 The proof is based on Picard iteration, as for the typical ODE existence proof. Let Xt = x, and define Z t n+1 n Xt = X0 + b(Xs ;s)ds + dWt ; 0 n = 0;1;:::. Define n n+1 n D (t) ≡ max jXs − Xs j : 0≤s≤t Notice that for a given, continuous sample path of Brownian motion we have Z s 0 D (t) = max b(x)dr +Ws ≤ C 0≤s≤t 0 for all 0 ≤ t ≤ T, where the constant C depends on the sample path via w. We claim that Kn Dn(t) ≤ C tn : n! We show this by induction. The base case n = 0 is true. Assume it holds for n − 1 and calculate Z s n n n−1 D (t) = max b(Xr ) − b(Xr ) dr 0≤s≤t 0 Z t ≤ K Dn−1(s)ds 0 Z t Kn−1sn−1 ≤ K C ds by the induction assumption 0 (n − 1)! Kn = C tn : n! Therefore for m ≥ n we have ¥ k k m n K T max jXt − Xt j ≤ C ! 0 as n ! ¥ : ≤t≤T ∑ 0 k=n k! Therefore with probability 1, Xn converges uniformly for t 2 [0;T] to a limit process X. One can check that X is continuous, adapted and solves (3).