Wiener increment

Exercise. Let Y0 be the Wiener process and Yt, Y2,..., Yr random walks with different step sizes and transition probabilities. Show that Y0 + Yx + Y2 + + Yr is a process with independent increments, see (IV.4.7), and find its transition probability. 

For the piuposes of employing option pricing models, the d3mamic behaviour of asset prices is usually described as a function of what is known as a Wiener process, which is also known as Brownian motion. The noise or volatility component is described by an adapted Brownian or Wiener process, and involves introducing a random increment to the standard random process. This is described next. 

The standard Wiener process is a close approximation of the behaviour of asset prices but does not account for some specific aspects of market behaviour. In the first instance, the prices of financial assets do not start at zero, and their price increments have positive mean. The variance of asset price moves is also not always unity. Therefore, the standard Wiener process is replaced by the generalised Wiener process, which describes a variable that may start at something other than zero, and also has incremental changes that have a mean other than zero as well as variances that are not unity. The mean and variance are still constant in a generalised process, which is the same as the standard process, and a different description must be used to describe processes that have variances that differ over time these are known as stochastic integrals (Figure 2.3). 

Consider a function/(S, f) dependent on two variables S and t, where S follows a random process and varies with t. If S, is a continuous-time process that follows a Wiener process W then it directly influences the function/() through the variable t in/(S t). Over time, we observe new information about W, as weU as the movement in S over each time increment, given by dS,. The sum of both these effects represents the stochastic differential and is given by the stochastic equivalent of the chain rule known as It s lemma. So, for example, if the price of a stock is 30 and an incremental time period later is 30 /i, the differential is Vi. 

The Wiener process represents one possible form of diffusion processes. It is actually the integral of what in practical applications is called a white noise. The Wiener process with drift will be used in our application. The initial mean value (drift) is p and standard deviations for each time increment have been previously calculated—see Table 1. For our model we apply Wiener process with drift given by stochastic differential equation. 

The mathematical model of a one dimensional diffusion is the Wiener process (W,), which satisfies the following three conditions (1) Wo = 0, (2) Wt is continuous with independent increments and (3) the trajectory of [Wt+st - Wj] can be sampled from a normal distribution with mean (/u.) of zero and variance (a ) of St (strong Markov property). 

Unfortunately, Eq. (4.1) is only exact when dt is infinitesimal, however it can be approximated by using the Ito interpretation of a Wiener process [7], such that the increment Wt — Ws is shown to have the property of a normally distributed random... 

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