Introduction to Ito Calculus

Advanced Stochastic Processes. LECTURE 12 Introduction to Ito calculus Lecture outline • Simple processes. Ito isometry • First 3 steps in constructing Ito integral for general processes 12.1. Ito integral for simple processes. Ito isometry � � Our immediate goal is to give meaning to expressions of the form X(t)dB(t) = X(t, ω)dB(t, ω), where X(t) is some stochastic process which � is adapted to the same filtration as B. As in the case integration, the idea is to define X(t, ω)dB(t, ω) as some kind of a limit of (random) of usual � sums j X(tj , ω)(B(tj+1 , ω) − B(tj , ω)) and show that the limit exists. First we address the problem of X(t) being adapted to the same filtration as B(t). We would like to consider processes which are not necessarily continuous. Hence the Borel σ­field B would not work, since it was defined on the space of continuous functions. Thus, we consider a more general setting which for now we assume exists. Setting. We have a probability space (Ω, F, P) and a filtration Ft ⊂ F, t ∈ R+ . A Brownian motion process B(t) is adapted to this filtration. In addition, we have another stochastic process X(t, ω), t ≥ 0, ω ∈ Ω which is also adapted to the same filtration. It is assumed that for every ω the function X(ω) : [0, ∞) → R is measurable with respect to Borel σ­field on [0, ∞) (but it is not necessarily continuous). Alternatively, we think of X as a mapping X : Ω × [0, ∞) → R which is F × B measurable, where B is Borel σ­field on [0, ∞). An intuitive way to think about this setting is that for each sample ω ∈ Ω we have a sample of our process X(·, ω) and a sample of a Brownian motion B(·, ω). For example we may define X(t) = B(t), t ≤ 1 and X(t) = 0, t > 1. Then for each sample ω of a Brownian motion we have a sample X(ω). Note that the path of X(ω) up to time t is completely determined by the path of B(ω) up to t. But since these samples are discontinuous, we need to ”lift” the probability space to non­continuous processes. 1 2 In the space of processes X(t) adapted to Ft consider a subspace H 2 of square integrable processes. That is H 2 is the space of processes which satisfy the following property: for every T >0 � T E[ X 2 (t, ω)dt] < ∞, 0 where integration is in the Riemann sense. In particular, we assume that the integral exists. Our goal is to define the Ito integral � T IT (X(ω)) = X(t, ω)dB(t, ω). 0 Specifically, IT (X(ω)) is a random variable which is defined on the same probability space (Ω, F, P). Later we will see that when we vary T we obtain that IT (X) is a stochastic pro­ cess which for each T is measurable w.r.t. FT , i.e. it is adopted to the filtration Ft . Definition 12.1. A process X ∈ H 2 is called simple if there exists a partition Πn : 0 = t0 < · · · < tn = T such that X(t, ω) = X(tj , ω) for all t ∈ [tj , tj+1 ), 0 ≤ j ≤ n for all ω. The partition Πn does not depend on ω. The subspace of simple processes is denoted by S 2 ⊂ H 2 . ˆ ω) As an example, fix any partition Πn and a process X(t) and consider the process X(t, ˆ ω) = X(tj , ω), where tj is defined by t ∈ [tj , tj+1 ). In the definition it is defined by X(t, ˆ ˆ important that X(t) = X(tj ) and not X(tj+1 ). The reason is that X(t) = X(tj ) ∈ Ftj ⊂ Ft . Thus the simple process is adapted to the filtration Ft . But X(tj+1 ), while it belongs to Ftj+1 , does not necessarily belong to Ft , so it is not necessarily adapted. Given a simple process X ∈ S 2 define its integral by � X(tj , ω)(B(tj+1 , ω) − B(tj , ω)) IT (X(ω)) = 0≤j≤n−1 Theorem 12.2 (Ito isometry). For every simple process X ∈ S 2 , E[IT (X(ω))] = 0. and E[IT2 (X(ω))] T � = E[ X 2 (t, ω)dt]. 0 In particular, E[IT2 (X(ω))] 2 < ∞, since X ∈ H . For notational convenience we drop ω from now on. Proof. � E[IT (X)] = E[X(tj )B(tj+1 ) − B(tj )] 0≤j≤n−1 � = 0≤j≤n−1 = 0, E[X(tj )E[B(tj+1 ) − B(tj )|Ftj ]] LECTURE 12. INTEGRATION 3 where in the second equality we use the tower property and the fact Xtj ∈ Ftj . Now for the second moment we have � E[X(tj1 )X(tj2 )(B(tj1 +1 ) − B(tj1 ))(B(tj2 +1 ) − B(tj2 ))]. E[IT2 (X)] = 0≤j1 ,j2 ≤n−1 When j1 < j2 we have E[X(tj1 )X(tj2 )(B(tj1 +1 ) − B(tj1 ))(B(tj2 +1 ) − B(tj2 ))] = 0 which we obtain by conditioning on Ftj2 , using the tower property and observing that all of the random variables involved except for B(tj2 +1 ) are measurable with respect to Ftj2 (recall that Ftj1 ⊂ Ftj2 ). Now when j1 = j2 = j we have E[X 2 (tj )(B(tj+1 ) − B(tj ))2 ] = E[X 2 (tj )E[(B(tj+1 ) − B(tj ))2 |Ftj ]] = E[X 2 (tj )(tj+1 − tj )]. Combining, we obtain E[IT2 (X)] = � � � 2 E[X (tj )(tj+1 − tj )] = E[ X (tj )(tj+1 − tj )] = E[ j j 2 T X 2 (t)dt]. 0 � 12.2. Constructing Ito integral for general square integrable processes To continue our discussion we need some background regarding square integrable random vari­ ables. Given any probability space (Ω, F, P), define L2 (P) to be the space of random variables X : Ω → R which are square integrable: E[X 2 ] < ∞. One technical assumption required is that the probability space is complete. Namely, if A ∈ F is such that P(A) = 0, then for every B ⊂ A; B ∈ F , and, as a result, P(B) = 0. It turns out that this condition is w.l.g. Namely, for every probability space we can simply add all subsets of zero probability sets into F. It is an easy exercise to show that the resulting space is again a probability space. A sequence of random variables Xn is defined to be Cauchy if for every � > 0 there exists n0 such that for all n, m > n0 we have E[(Xn − Xm )2 ] < �. Theorem 12.3 (Completeness of L2 (P)). The space L2 (P) is complete. Namely, for every Cauchy sequence Xn there exists a unique random variable X ∈ L2 (P) such that limn→∞ E[(Xn − X)2 ] = 0. Now we use this to define IT (X). We do this in 3 steps. First we define it for the case when X ∈ H 2 is an a.s. bounded and continuous process. Then we construct it for a bounded process. Finally we approximate any process X ∈ H 2 by bounded processes and extend the definition of IT (X). Step 1. 4 Proposition 1. Suppose X ∈ H 2 is a bounded continuous process: |X| ≤ M . There exists a sequence of simple processes Xn ∈ S 2 such that � T (12.4) lim E[ (Xn (t) − X(t))2 dt] = 0. n 0 Proof. Fix a sequence of partitions Πn such that Δn = max(tj+1 − tj ) → 0 as n → ∞. Given process X, consider the modified process Xn (t) = X(tj ) for all t ∈ [tj , tj+1 ). This process is simple and is adapted to Ft . It is also bounded by M . Therefore Xn ∈ S 2 . By continuity of X, and since Δn → 0, we have for each realization ω and each t, Xn (t, ω) → X(t, ω). Lemma 12.5. The following holds almost surely with respect to ω ∈ Ω: � T lim (Xn (t) − X(t))2 dt = 0 n 0 Proof. This is really a result from real analysis, but we can give a probabilistic proof of it. Consider a uniform random variable U ∈ [0, T ]. Then for each sample ω note that � T (Xn (t) − X(t))2 dt = T EU [(Xn (U, ω) − X(U, ω))2 ], 0 where the expectation is only with respect to the uniform random variable U and is defined sample wise for ω. Now a.s. with respect to U we have |Xn (U, ω) − X(U, ω)| → 0. Since |X |, |Xn | ≤ M , then applying the Bounded Convergence Theorem, we obtain limn E[(Xn (U, ω) − X(U, ω))2 ] = 0. � �T Now we define Zn = 0 (Xn (t) − X(t))2 dt. We just showed Zn → 0 a.s. On the other hand Zn ≤ 4M 2 T a.s. Again invoking the Bounded Convergence Theorem, we obtain limn E[Zn ] = 0. This completes the proof of the proposition. � We can now define IT (X) for every bounded continuous process X as a limit of IT (Xn ), where Xn ∈ S 2 is a sequence of simple processes satisfying (12.4). In order to finalize this definition, we need to show that the limit of IT (Xn ) exists in some appropriate sense, and is independent from the choice of Xn (as there may be many such simple processes). Before we conduct this analysis, we extend the scope of processes X for which we define the limit. First we do this for a bounded, but not necessarily continuous process, and then for any process X ∈ H 2 . Step 2. Proposition 2. Suppose X ∈ H 2 is a bounded process: |X| ≤ M . There exists a sequence of processes Xn ∈ H 2 such that Xn is a.s. a bounded continuous process and � T (12.6) lim E[ (Xn (t) − X(t))2 dt] = 0. n 0 Proof. We use a certain ”regularization” trick to turn a bounded process into a bounded con­ tinuous approximation. Given any n > 0, consider any non­negative continuous function ψn (t) such that ψn (t) = 0 for t ∈ (−∞, − n1 ) ∪ [0, ∞) and � 0 ψn (t)dt = 1. 1 −n LECTURE 12. INTEGRATION 5 Define � t ψn (s − t)X(s, ω)ds. Gn (t, ω) = 1 t− n The value Gn (t, ω) is defined via values X(s) with s ≤ t and some other deterministic function ψn . Therefore Gn (t) ∈ Ft . We need to establish that Gn is bounded continuous and that it approximates X. Problem 1. Prove that a.s. Gn (t, ω) is a bounded continuous function. Also prove that a.s. � T lim (X(t, ω) − Gn (t, ω))2 ds = 0 n→∞ 0 This concludes the proof of the proposition. � Step 3. Proposition 3. Suppose X ∈ H 2 . There exists a sequence of processes Xn ∈ H 2 such that Xn is a.s. a bounded process and � T (12.7) lim E[ (Xn (t) − X(t))2 dt] = 0. n 0 Proof. Define Xn by Xn (t) = X(t) when −n ≤ X(t) ≤ n, Xn (t) = −n, when X(t) < −n and Xn (t) = n, when X(t) > n. Clearly Xn (t) ∈ Ft . It is also bounded. We now assert (12.7). From the fact X ∈ H 2 it follows � T � T � 2 2 ∞ > c � E[ X (t)dt] ≥ m E[ 1{m ≤ |X(t)| < m + 1}dt], 0 0 m≥0 Introducing a shorthand notation T � pm = E[ 1{m ≤ |X(t)| < m + 1}dt] 0 we have � m≥0 m2 pm ≤ c < ∞. On the other hand � T � T 2 E[ (Xn (t) − X(t)) dt] = E[ (X(t) − n)2 1{X(t) > n}dt] 0 0 � T + E[ (X(t) + n)2 1{X(t) < n}dt] 0 Note � T � � T 2 E[ (X(t) − n) 1{X(t) > n}dt] = E[ (X(t) − n)2 1{m ≤ X(t) < m + 1}dt] 0 0 m≥n ≤ � (m + 1 − n)2 pm m≥n ≤ � m≥n m2 pm . 6 � Fix � > 0. Since m≥0 m2 pm ≤ c is finite, then there exists n1 such that the partial sum � 2 m≥n m pm < �/2 for all n > n1 . Then for all n > n1 � T � E[ (X(t) − n)2 1{X(t) > n}dt] < . 2 0 Similarly, we show that for some n2 � T � E[ (X(t) + n)2 1{X(t) < −n}dt] < 2 0 for all n > n2 . Combining, � T E[ (Xn (t) − X(t))2 dt] < � 0 for all n > max(n1 , n2 ) and the proposition is proven. � 12.3. Additional reading materials • Course Packet. Chapter from Harrison’s book ”Brownian models and stochastic control”. • Øksendal [1], Chapter III.