期货与股票ppt(1)

the derivative of Brownian motion, B’(t) does not exist in the ordinary sense, they are nowhere differentiable.
• Stochastic differential
equations
Despite the fact, the infinitesimal increment of Brownian motion, the limit of B(t+h) = B(t) as h approaches to an infinitesimal of time (dt) has earned the notation dB(t) and it has become a fundamental building block for constructing other continuous time process. It is called white noise. For P(t) define earlier we have dP(t) = dt + dB(t). This is called stochastic differential equation. The natural transformation dP(t)/dt = + dB(t)/dt doesn’t male sense because dB(t)/dt is a not well defined
3. The sample paths of P(t) are continuous. P(t) is called arithmetic Brownian motion or Winner process.
If we set =0, =1, we obtain standard Brownian Motion which is denoted as B(t). Accordingly, P(t) = t + B(t)
Consider the following moments:
E[P(t) | P(t0)] = P(t0) +(t-t0) Var[P(t) | P(t0)] = 2(t-t0) Cov(P(t1),P(t2) = 2 min(t1,t2) Since Var[ (B(t+h)-B(t))/h ] = 2/h, therefore,
• Ito’s Lemma
Although the first complete mathematical theory of Brownian motion is due to Wiener(1923), it is the seminal contribution of Ito (1951) that is largely responsible for the enormous number of applications of Brownian motion to problems in mathematics, statistics, physics, chemistry, biology, engineering, and of course, financial economics. In particular, Ito constructs a broad class of continuous time stochastic process based on Brownian motion – now known as Ito process or Ito stochastic differential equations – which is closed under general non-linear transformation.
n(2-1) T 4n(-1) 2 T
This can be accomplished by setting
= ½*(1+h /), =h
• The continuous time limit
It cab be shown that the process P(t) has the following three properties:
| dB dt dB | dt 0 dt | 0 0
Using th above rule we can calculate (dP)2 = 2dt. It is not a random variable!
• Geometric Brownian motion
If the arithmetic Brownian motion P(t) is taken to be the price of some asset, the price may be negative. The price process p(t)= exp(P(t)), where P(t) is the arithmetic Brownian motion, is called geometric Brownian motion or lognormal diffusion.
期货与股票ppt(1)
We need to adjust , such that Pn(t) will converge when n goes to infinity. Consider
the mean and variance of Pn(T):n(T)) = 4n(-1) 2
1. For any t1 and t2 such that 0 t1 < t2 T: P(t1)-P(t2) ((t2-t1), 2(t2-t1))
2. For any t1, t 2 , t3, and t4 such that 0 t1 < t2 t1 < t2 t3 < t4 T, the increment P(t2)- P(t1) is statistically independent of the increment P(t4)- P(t3).
We wish to obtain a continuous time version of the random walk, we should expect the mean and variance of the limiting process P(T) to be linear in T. Therefore, we must have
(althrough dB(t) is).
The moments of dB(t): E[dB(t)] =0 Var[dB(t)] = dt E[ dB dB ] = dt Var[dB dB ] = o(dt) E[dB dt] = 0 Var[dB dt ] = o(dt)
If we treat terms of order of o(dt) as essentially zero, the (dB)2 and dBdt are both non-stochastic variables.