Lecture 19: Ito applications
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چکیده
As early as 1900, Louis Bachelier had proposed using Brownian motion as a model for security prices. As insightful as it was, this model is somewhat re strictive. For example it allows stock price to be negative. A more appropriate and accepted model is to assume that stock prices follow a general Ito process Xt = X e t e t 0 + μsds + σ dB s s 0 0 , where μ, σ are adapted processes. Suppose we make several investment decisions with this stock. In particular at times t1 < t2 < · · · < tk we decide to hold θtk stocks of this security. What would be our gain/loss at some later time t = tk+1 > tk? It is simply j θ (X X ). It makes sense to assume that our trading ≤k tj tj+1 − tj strategy θ is an adapted process (including the possibility that θ is deterministic), since otherwise it means we can predict the market. Thus we can think of θ as a simple adapted process. The simplicity means that we make only k trading decisions. But there is no reason to bound the number of trading decisions a priori (unless we take trading fees into account). So we may think of θ as an arbitrary adapted process θt ∈ Ft. For technical reasons we do assume that θ ∈ L2. This is needed so that we exclude the ”doubling” strategy pathology which as know may bring a positive gain with probability one.
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