Geometric Brownian Motion (Stochastic Population Growth)

Transcription

1 2011 Page 1 Analytical Solution of Stochastic Differential Equations Thursday, April 14, :58 PM References: Shreve Sec. 4.4 Homework 3 due Monday, April 25. Distinguished mathematical sciences lectures today and Friday, 4 PM in Amos Eaton 214 (neuroscience, wave turbulence) Geometric Brownian Motion (Stochastic Population Growth) More elaborate version: stochastic volatility models where itself obeys a stochastic differential equation. See: Fouque, Papanicoloau, and Sircar, Derivatives in Financial Markets with Stochastic Volatility (2000) Fouque, Papanicoloau, Sircar, and Solna, Multiscale Stochastic Volatility for Equity, Interest-Rate and Credit Derivatives (2011) Since the coefficient of the noise depends on the state, we must declare an interpretation, and we'll start with the Ito interpretation and then revisit how the answer would change if instead we had taken a Stratonovich interpretation. With the Ito intepretation, how can we solve the SDE?

2 2011 Page 2 By contrast, the solution to the Stratonovich stochastic differential equation: Repeat the same steps but recall that the chain rule for Stratonovich calculus does not have the second order correction involving the quadratic variation, so one just gets the naïve solution in analogy with the solution of ordinary linear differential equations: So we see that if we would have taken the Stratonovich interpretation of the SDE, then the solution would have an effective growth rate:

3 2011 Page 3 would have an effective growth rate: We'll do all our analysis for the Ito solution and then comment on comparison with Stratonovich solution remembering that they differ only in this transformation of the growth rate in the model. First, we consider moments of the solution: assuming independent. This expectation can be computed using characteristic or moment generating functions: If Z is a Gaussian random variable, then: when Perform the integral by completing the square in the exponent, then one gets another Gaussian integral that can be explicitly computed, leaving the answer we have stated. Apply this formula with

4 2011 Page 4 Let's compare the means: In this example, the mean of the solution to the Ito SDE is the same as the solution to the noiseless SDE. This is not true in general. It works in this case only because the SDE is linear. For nonlinear Ito SDEs, the mean of the solution will in general be affected by the noise. What is to the point and what is general is that the noise term in an Ito SDE is a martingale (meaning it does not have any upward or downward bias regardless of the state value.) In particular, for the Ito SDE, we can write:

5 2011 Page 5 The discounted asset price is a martingale. Let's look at the long time properties of the solution of the geometric Brownian motion SDE. To this end, let us suppose: And then we see that the long time behavior of the moments of the solution behave like: For positive mean growth rates, all moments diverge at long time, as expected. For negative mean growth rates, one can have low order moments go to zero and high order moments go to infinity at long time. One can make this strange dichotomy of behavior sharper by looking at pathwise properties. Here we use the property that:

6 2011 Page 6 For similar reasons, under the assumptions we made above: So the long-time pathwise properties of the geometric Brownian motion are: We have therefore a bizarre regime: Under the Ito SDE: The reason why the trajectories are crashing into zero eventually is that an unlucky fluctuation will eventually push the solution close to zero, and because of the behavior of the coefficients near state value 0, the zero state behaves like a stochastic trap or asymptotically attracting state of the system. (See Feller classification of boundary conditions; see for example Gardiner Ch. 5.) But so long as trajectories aren't near a crash, they are growing exponentially. So the mean is growing exponentially because an ever smaller set of trajectories are still growing exponentially and the exponential growth is more than compensating for the fraction of trajectories that are still alive. Why does this model have such bizarre behavior? The effective growth rate is fluctuating in a very violent way: