Ito Process; Ito Formula

Advanced Stochastic Processes. LECTURE 14 Ito process. Ito formula Lecture outline • Ito process. • Ito formula. • Integration by parts. 14.1. Ito process From now on we look at IT (X) as a� process indexed by time T . So instead we use a lower case t t and consider the process It (X) = 0 XdB. In the previous lecture we computed � t t 1 It (B) = B(s)dB(s) = B 2 (t) − , 2 2 0 or � t 2 (14.1) B (t) = 2 B(s)dB(s) + t, 0 �t On the other hand, trivially 0 dB(s) = B(t). We see that when we apply the functional �t transformation g(x) = x2 to a process of the form 0 X(s)dB(s) we do not necessarily get a �t �t process�of the form 0 Y (s)dB(s), but rather 0 X(s)dB(s) plus something else. (Why t doesn’t have a XdB(t) representation?) Let us assume now that H 2 is the space of processes X such �t that E[ 0 X 2 (s)ds] < ∞ for all t ∈ R+ . Definition 14.2. An Ito process or stochastic integral is a stochastic process on (Ω, F , P) which can be represented in the form � t � t (14.3) X(t) = X(0) + U (s)ds + V (s)dB(s), 0 0 1 2 where U and V are some processes in H 2 . As a shorthand notation, we will write (14.3) as dX(t) = U (t)dt + V (t)dB(t). Thus we can rewrite (14.1) as d(B 2 (t)) = dt + 2B(t)dB(t). Note the difference from the usual differentiation: dx2 = 2xdx. The additional term dt here, arises because Brownian motion B is not differentiable and instead has quadratic variation. Notation Given an Ito process dX(t) = U (t)dt + V (t)dB(t), let us introduce the notation (dX(t))2 which stands for V 2 dt. Equivalently (dX(t))2 is (dX(t)) · (dX(t)) which is computed using the rules dt · dt = dt · dB(t) = dB(t) · dt = 0, dB(t) · dB(t) = dt. 14.2. Ito formula We now introduce the most important formula of Ito calculus: Theorem 14.4 (Ito formula). Let X(t) be an Ito process dX(t) = U (t)dt+V (t)dB(t). Suppose g(t, x) ∈ C 2 ([0, ∞) × R) is a twice continuously differentiable function (in particular all second partial derivatives are continuous functions). Then Y (t) = g(t, X(t)) is again an Ito process and dY (t) = 1 ∂ 2g ∂g ∂g (t, X(t))(dX(t))2 (t, X(t))dt + (t, X(t))dX(t) + 2 ∂t ∂x 2 ∂x Using the notational convention for dX(t) and (dX(t))2 , we can rewrite the Ito formula as � ∂g � ∂g 1 ∂2g ∂g 2 dY (t) = (t, X(t)) + (t, X(t))U (t) + (t, X(t))V (t) dt + (t, X(t))V (t)dB(t). 2 2 ∂x ∂x ∂t ∂x Thus, we see that the space of Ito processes is closed under twice­continuously differentiable transformations. Before we establish validity of this formula, let us double check it on our example. In order to re­derive (14.1) let us consider g(t, x) = 12 x2 . Then by the Ito formula we have ∂g 1 2 ∂g 1 ∂ 2g dt + dB(t) + (dB(t))2 d( B (t)) = dg(B(t)) = 2 2 ∂t ∂x 2 ∂x 1 = B(t)dB(t) + (dB(t))2 2 dt = B(t)dB(t) + , 2 which matches (14.1). Proof sketch of Theorem 14.4. We will do this for a very special case. First g(t, x) = g(x). Namely g(X(t)) depends on t only via X(t). Second, we assume that the partial derivatives ∂g ∂ 2 g as well as U and V are all bounded simple processes. ∂x ∂x2 The general case is then obtained by approximating U and V by bounded simple processes in a way similar to how we defined the Ito integral. LECTURE 14. ITO FORMULA 3 Let Πn : 0 = t0 < t1 < · · · < tn = t be a sequence of partitions such that Δ(Πn ) → 0. We use the notation ΔB(tj ) = B(tj+1 ) − B(tj ), ΔX(tj ) = X(tj+1 ) − X(tj ). Using Taylor expansion of g we obtain � g(X(tj+1 )) − g(X(tj )) g(X(t)) = g(X(0)) + j 0. Therefore the first sum converges to zero provided Δ(Πn ) → 0. To analyze the second sum, we square it and take expected value: � � ∂2g �2 (X(t ))U (t )V (t )(t − t )ΔB(t ) ] E[2 j j j j+1 j j 2 ∂x j