Stochastic Processes in Continuous Time Joseph C. Watkins December 14, 2007 Contents 1 Basic Concepts 3 1.1 Notions of equivalence of stochastic processes . 3 1.2 Sample path properties . 4 1.3 Properties of filtrations . 6 1.4 Stopping times . 7 1.5 Examples of stopping times . 10 2 L´evyProcesses 12 3 Martingales 16 3.1 Regularity of Sample Paths . 18 3.2 Sample Path Regularity and L´evyProcesses . 23 3.3 Maximal Inequalities . 27 3.4 Localization . 28 3.5 Law of the Iterated Logarithm . 30 4 Markov Processes 33 4.1 Definitions and Transition Functions . 33 4.2 Operator Semigroups . 35 4.2.1 The Generator . 35 4.2.2 The Resolvent . 38 4.3 The Hille-Yosida Theorem . 40 4.3.1 Dissipative Operators . 40 4.3.2 Yosida Approximation . 41 4.3.3 Positive Operators and Feller Semigroups . 45 4.3.4 The Maximum Principle . 46 4.4 Strong Markov Property . 47 4.5 Connections to Martingales . 49 4.6 Jump Processes . 53 4.6.1 The Structure Theorem for Pure Jump Markov Processes . 53 4.6.2 Construction in the Case of Bounded Jump Rates . 55 4.6.3 Birth and Death Process . 57 4.6.4 Examples of Interacting Particle Systems . 59 1 CONTENTS 2 4.7 Sample Path Regularity . 61 4.7.1 Compactifying the State Space . 62 4.8 Transformations of Markov Processes . 66 4.8.1 Random Time Changes . 66 4.8.2 Killing . 67 4.8.3 Change of Measure . 69 4.9 Stationary Distributions . 71 4.10 One Dimensional Diffusions. 74 4.10.1 The Scale Function . 76 4.10.2 The Speed Measure . 77 4.10.3 The Characteristic Operator . 80 4.10.4 Boundary Behavior . 82 5 Stochastic Integrals 84 5.1 Quadratic Variation . 85 5.2 Definition of the Stochastic Integral . 90 5.3 The ItˆoFormula . 95 5.4 Stochastic Differential Equations . 98 5.5 ItˆoDiffusions . 101 1 BASIC CONCEPTS 3 1 Basic Concepts We now consider stochastic processes with index set Λ = [0, ∞). Thus, the process X : [0, ∞) × Ω → S can be considered as a random function of time via its sample paths or realizations t → Xt(ω), for each ω ∈ Ω. Here S is a metric space with metric d. 1.1 Notions of equivalence of stochastic processes m As before, for m ≥ 1, 0 ≤ t1 < ··· < tm, B ∈ B(S ), the Borel σ-algebra, call µt1,...,tm (B) = P {(Xt1 ,...,Xtm ) ∈ B} the finite dimensional distributions of X. We also have a variety of notions of equivalence for two stochastic processes, X and Y . Definition 1.1. 1. Y is a version of X if X and Y have the same finite dimensional distributions. 2. Y is a modification of X if for all t ≥ 0, (Xt,Yt) is an S × S-random variable and P {Yt = Xt} = 1. 3. Y is indistinguishable from X if P {ω; Yt(ω) 6= Xt(ω) for some t ≥ 0} = 0. Recall that the Daniell-Kolmogorov extension theorem guarantees that the finite dimensional distribu- tions uniquely determine a probability measure on S[0,∞). Exercise 1.2. If A ∈ σ{Xt; t ≥ 0}, then there exists a countable set C ⊂ [0, ∞) so that A ∈ σ{Xt; t ∈ C}. R T Thus, many sets of interest, e.g., {x; sup0≤s≤T f(xs) > 0} or {x : 0 f(xs) ds < ∞} are not measurable subsets of S[0,∞). Thus, we will need to find methods to find versions of a stochastic process so that these sets are measurable. Exercise 1.3. If X and Y are indistinguishable, then X is a modification of Y . If X is a modification of Y , then X and Y have the same finite dimensional distributions. 1 BASIC CONCEPTS 4 1.2 Sample path properties Definition 1.4. We call X 1. measurable if X is B[0, ∞) × F-measurable, 2. (almost surely) continuous, (left continuous, right continuous) if for (almost) all ω ∈ Ω the sample path is continuous, (left continuous, right continuous). Focusing on the regularity of sample paths, we have Lemma 1.5. Let x : [0, ∞) → S and suppose xt+ = lim xs exists for all t ≥ 0, s→t+ and xt− = lim xs exists for all t > 0. s→t− Then there exists a countable set C such that for all t ∈ [0, ∞)\C, xt− = xt = xt+. Proof. Set 1 C = {t ∈ [0, ∞); max{d(x , x ), d(x , x ), d(x , x )} > } n t− t t− t+ t t+ n If Cn ∩ [0, m] is infinite, then by the Bolzano-Weierstrass theorem, it must have a limit point t ∈ [0, m]. In this case, either xt− or xt+ would fail to exist. Now, write ∞ ∞ [ [ C = (Cn ∩ [0, m]), m=1 n=1 the countable union of finite sets, and, hence, countable. Lemma 1.6. Let D be a dense subset of [0, ∞) and let x : D → S. If for each t ≥ 0, + xt = lim xs s→t+,s∈D exists, then x+ is right continuous. If for each t > 0, − xt = lim xs s→t−,s∈D exists, then x− is left continuous. Proof. Fix t0 > 0. Given > 0, there exists δ > 0 so that + d(xt0 , xs) ≤ whenever s ∈ D ∩ (t0, t0 + δ). Consequently, for all s ∈ (t0, t0 + δ) + + + d(xt , xs ) = lim d(xt , xu) ≤ . 0 u→s+,u∈D 0 and x+ is right continuous. The left continuity is proved similarly. 1 BASIC CONCEPTS 5 − + Exercise 1.7. If both limits exist in the previous lemma for all t, then xt = xt− for all t > 0. Definition 1.8. Let CS[0, ∞) denote the space of continuous S-valued functions on [0, ∞). Let DS[0, ∞) denote the space of right continuous S-valued functions having left limits on [0, ∞). Even though we will not need to use the metric and topological issue associated with the spaces CS[0, ∞) and DS[0, ∞), we proved a brief overview of the issues. If we endow the space CS[0, ∞) with the supremum metric ρ∞(x, y) = sup0≤s≤∞ d(xs, ys), then the metric will not in general be separable. In analogy with the use of seminorms in a Frechet to define a metric, we set, for each t > 0, ρt(x, y) = sup d(xmax{s,t}, ymax{s,t}). s Then, ρT satisfies the symmetric and triangle inequality properties of a metric. However, if x and y agree on [0,T ], but xs 6= ys for some s > T , then x 6= y but ρt(x, y) = 0. However, if ρt(x, y) = 0 for all t, ¯ ¯ then x = y. Consider a bounded metric d(x0, y0) = max{d(x0, y0), 1} and setρ ¯t(x, y) = sup0≤t≤t d(xs, ys), then we can create a metric on CS[0, ∞) which is separable by giving increasingly small importance to large values of t. For example, we can use Z ∞ −t ρ¯(x, y) = e ρ¯t(x, y) dt. 0 Then, (CS[0, ∞), ρ¯) is separable whenever (S, d) is separable and complete whenever (S, d) is complete. For the space DS[0, ∞), then unless the jumps match up exactly then the distance from x to y might be large. To match up the jumps, we introduce a continuous strictly increasing function γ : [0, ∞) → [, ∞) that is one to one and onto and define γ ρt (x, y) = sup d(xmax{s,t}, ymax{γ(s),t}). s and Z ∞ 0 −t γ ρ(x, y) = inf{max{ess supt| log γt|, e ρt (x, y) dt}}. γ 0 As before, (DS[0, ∞), ρ¯) is separable whenever (S, d) is separable and complete whenever (S, d) is complete. Exercise 1.9. CR[0, ∞) with the ρ∞ metric is not separable. Hint: Find an uncountable collection of real-valued continuous functions so that the distance between each of them is 1. With a stochastic process X with sample paths in DS[0, ∞), we have the following moment condition that guarantee that X has a CS[0, ∞) version. Proposition 1.10. If (S, d) be a separable metric space and set d1(x, y) = min{d(x, y), 1}. Let X be a process with sample paths in DS[0, ∞). Suppose for each T > 0, there exist α > 1, β > 0, and C > 0 such that β α E[d1(Xt,Xs) ] ≤ C(t − s) 0 ≤ s ≤ t ≤ T. (1.1) Then almost all sample paths of X belong to CS[0, ∞). 1 BASIC CONCEPTS 6 Proof. Let T be a positive integer. Claim. 2N T X β X β d1(Xt,Xt−) ≤ lim inf d1(Xk2−N ,X(k−1)2−N ) . N→∞ 0<t≤T k=1 First note that by Lemma 1.5, the left side is the sum of a countable number of terms. In addition, note that for each n ≥ 1, {t ∈ [0,T ]; d1(Xt,Xt−) > 1/n} is a finite set. Thus, for N sufficiently large, these points are isolated by the 2−N partition of [0,T ]. Thus, in the limit, these jumps are captured. Consequently, 2N T X β X β d1(Xt,Xt−) I{d (X ,X )>1/n} ≤ lim inf d1(Xk2−N ,X(k−1)2−N ) . (1.2) 1 t t− N→∞ 0<t≤T k=1 Note that the left side of expression (1.2) increases as n increases. Thus, let n → ∞ to establish the claim. By Fatou’s lemma, and the moment inequality (1.1), 2nT X β X β E[ d1(Xt,Xt−) ] ≤ lim inf E[d1(Xk2−n ,X(k−1)2−n ) ] n→∞ 0<t≤T k=1 2nT X ≤ lim inf C2−nα n→∞ k=1 = lim inf CT 2n(1−α) = 0.