ExampleAvariable iscurrently 40Itfollows aMarkovprocess Process is stationary (i.e. the parameters ofthe process do not change as we movethrough time) At the end of 1 year the variable will have anormal probability distribution with mean 40and standard deviation 10
Example A variable is currently 40 It follows a Markov process Process is stationary (i.e. the parameters of the process do not change as we move through time) At the end of 1 year the variable will have a normal probability distribution with mean 40 and standard deviation 10
Questions What is the probability distribution ofthe stock price at the end of 2 years?V2 years?V4 years?At years?Taking limits we have defined acontinuousstochastic process
Questions What is the probability distribution of the stock price at the end of 2 years? ½ years? ¼ years? Dt years? Taking limits we have defined a continuous stochastic process
Variances & Standard DeviationsIn Markov processes changes insuccessive periods of time are independentThis means that variances are additiveStandard deviations are not additive
Variances & Standard Deviations In Markov processes changes in successive periods of time are independent This means that variances are additive Standard deviations are not additive
Variances & Standard Deviations(continued)In our example it is correct to say that thevariance is 100 per year.It is strictly speaking not correct to say thatthe standard deviation is 10 per year
Variances & Standard Deviations (continued) In our example it is correct to say that the variance is 100 per year. It is strictly speaking not correct to say that the standard deviation is 10 per year
A Wiener Process (See pages 282-84)Define (u,v) as a normal distribution withmean μ and variance yA variable z follows a Wiener process if豆The change in z in a small interval of time △tis △zNz = V△t where is Φ(0,1)豆The values of △z for any 2 different (non-空overlapping) periods of time are independent
A Wiener Process (See pages 282-84) Define f(m,v) as a normal distribution with mean m and variance v A variable z follows a Wiener process if The change in z in a small interval of time Dt is Dz The values of Dz for any 2 different (nonoverlapping) periods of time are independent Dz = Dt where is f(0,1)