91Ó°ÊÓ

In order to show that as \( \Delta t \rightarrow 0 \), the random walk converges to a Brownian motion with drift rate μ, we need to verify the limiting properties of the Brownian motion: 1. The random walk process starts at zero: \( X_0 = 0 \). 2. The increments over non-overlapping time intervals are independent. 3. The increment over the time interval \( [t,t+\Delta t] \) is normally distributed with mean \( \mu \Delta t \) and variance \( \Delta t \). For the given random walk: 1. We know that it starts at zero. 2. The increments in each step are independent, as the probabilities are based on the current step only. To show the convergence of the distribution, we need to calculate the mean and variance of the increments over a time step: Mean (M): \( M = p \cdot \sqrt{\Delta t} - (1-p) \cdot \sqrt{\Delta t} = (2p-1) \cdot \sqrt{\Delta t} \). Variance (V): \( V = E[(X - M)^2] = p(\sqrt{\Delta t} - M)^2 + (1-p)(-\sqrt{\Delta t} - M)^2 \). Now, let's find the probabilities p and 1-p in terms of drift μ: \( p = f(1+\mu \sqrt{\Delta t}) \). As \( \Delta t \rightarrow 0 \), using Taylor series expansion, we have \( f(1+\mu \sqrt{\Delta t}) = f(1) + f'(1) \cdot \mu \sqrt{\Delta t} + O(\Delta t) \). Therefore, \( p \rightarrow f(1) + f'(1) \cdot \mu \sqrt{\Delta t} \). Now, we substitute this expression of p into the mean and variance: Mean (M): \( M \rightarrow (2(f(1) + f'(1) \cdot \mu \sqrt{\Delta t})-1) \cdot \sqrt{\Delta t} = \mu \Delta t \). Variance (V): \( V \rightarrow (2(f(1) + f'(1) \cdot \mu \sqrt{\Delta t})-1)^2 \cdot \Delta t \rightarrow \Delta t \), as \( \Delta t \rightarrow 0 \). The limiting mean and variance satisfy the requirements of a Brownian motion process with drift. Hence the random walk converges to a Brownian motion process with drift rate μ as \( \Delta t \rightarrow 0 \).

To calculate the probability that the Brownian motion with drift rate μ goes up A before going down B (both A and B are strictly positive), we use the result from the gambler's ruin problem. According to the gambler’s ruin problem, the probability that the process reaches the level A before it goes down to -B is given by: \( P(X_t = A \ \textrm{before} \ X_t = -B) = \frac{1 - (\frac{c}{c+1})^{A+B}}{1 - (\frac{c}{c+1})^A} \), where \( c = \frac{1 - e^{-2 \mu}}{e^{2 \mu} - 1} \). Substituting the given values of A, B, and μ, we can calculate the probability that the Brownian motion process with drift rate μ goes up A before going down B.