By Bain A.
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Additional resources for Stochastic Calculus
Sample text
1. A process X is a semimartingale if X is an adapted CADLAG process which has a decomposition X = X0 + M + A, where M is a local martingale, null at zero and A is a process null at zero, with paths of finite variation. Note that the decomposition is not necessarily unique as there exist martingales which have finite variation. To remove many of these difficulties we shall impose a continuity condition, since under this most of our problems will vanish. 2. A continuous semimartingale is a process (Xt )t≥0 which has a Doob-Meyer decomposition X = X0 + M + A, where X0 is F0 -measurable, M0 = A0 = 0, Mt is a continuous local martingale and At is a continuous adapted process of finite variation.
We obtain t f (Bt ) − f (B0 ) = 0 1 ∂f (Bs )dBs + ∂x 2 t 0 ∂2f (Bs )d B, B s , ∂x2 noting that B0 = 0 for a standard Brownian Motion we see that t Bt2 =2 0 whence we derive that 1 Bs dBs + 2ds, 2 t Bs dBs = 0 Bt2 t − . 2 2 For those who have read the foregoing material carefully, there are grounds to complain that there are simpler ways to establish this result, notably by consideration of the definition of the quadratic variation process. However the point of this example was to show how Itˆ o’s formula can help in the actual evaluation of stochastic integrals; not to establish a totally new result.
The variety of SDE to be considered here describes a diffusion process and has the form dXt = b(t, Xt ) + σ(t, Xt )dBt , (∗) where bi (x, t), and σij (t, x) for 1 ≤ i ≤ d and 1 ≤ j ≤ r are Borel measurable functions. In practice such SDEs generally occur written in the Statonowich form, but as we have seen the Itˆ o form has numerous calculational advantages (especially the fact that local martinagles are a closed class under the Itˆ o integral), so it is conventional to transform the SDE to the Itˆ o form before proceeding.