Consider the random walk that in each Δt time unit either goes up or down the amount √(Δt) with

Consider the random walk that in each Δt time unit either...

Consider the random walk that in each $\Delta t$ time unit either goes up or down the amount $\sqrt{\Delta t}$ with respective probabilities $p$ and $1-p$, where $p=\frac{1}{2}(1+\mu \sqrt{\Delta t})$. (a) Argue that as $\Delta t \rightarrow 0$ the resulting limiting process is a Brownian motion process with drift rate $\mu$. (b) Using part (a) and the results of the gambler's ruin problem (Section 4.5.1), compute the probability that a Brownian motion process with drift rate $\mu$ goes up $A$ before going down $B, A>0, B>0$

Consider the random walk that in each $\Delta t$ time unit either goes up or down the amount $\sqrt{\Delta t}$ with respective probabilities $p$ and $1-p$, where $p=\frac{1}{2}(1+\mu \sqrt{\Delta t})$.
(a) Argue that as $\Delta t \rightarrow 0$ the resulting limiting process is a Brownian motion process with drift rate $\mu$.
(b) Using part (a) and the results of the gambler's ruin problem (Section 4.5.1), compute the probability that a Brownian motion process with drift rate $\mu$ goes up $A$ before going down $B, A>0, B>0$

Key Concepts

Random Walk and Scaling Limit

The random walk is a discrete-time stochastic process where at each time increment a step is taken, which in this case is scaled by the square root of the time interval. As the time step ?t tends toward zero and the step size is appropriately scaled, the random walk converges in distribution to a continuous-time process. This procedure, often referred to as taking the scaling limit, is a critical idea in probability theory and forms the basis for deriving continuous models such as Brownian motion from discrete models.

Brownian Motion with Drift

Brownian motion with drift is a continuous-time stochastic process characterized by both random fluctuations and a deterministic linear trend. In the limiting process obtained from the random walk, the drift parameter ? represents the average directional movement per unit time, while the random fluctuations are captured by the standard Brownian motion component. This combination results in a process which is Gaussian and exhibits independent, stationary increments, with slight bias in the drift direction.

Gambler's Ruin Problem in Continuous Settings

The gambler's ruin problem, originally formulated in a discrete setting, examines the probability of reaching a certain gain or loss before being ruined. In the context of Brownian motion with drift, the problem is reformulated to compute the probability that the process reaches an upper threshold A before a lower threshold -B. This extension relies on solving boundary value problems for stochastic differential equations, effectively linking the discrete random walk results with continuous diffusion scenarios.

Transcript

00:01 Okay, so we're going to look at, in particular, example of 7 .2, where we have an apple stock, which is worth $580 at time t equal to zero.

00:22 We also have the stock value increasing by 25 cents per day, and the variating.

00:37 So the variation of the increasing trend can be described by what is called a brownian motion process.

00:45 So a random variable of this form here, which is normally distributed with mean zero.

00:55 And standard deviations, or let's say the variance is alpha times t, where alpha is just some parameter.

01:04 So in this question, we set alpha to be equal to 20.

01:09 So what we're going to do first, so this is part a, is we're going to write an expression for x of t where t is in days, and x of t describes the value of a stock after t date.

01:35 So obviously t is continuous.

01:38 So if, you know, if you're looking at the stock value after, say, 12 hours, t would be 0 .5.

01:44 So i just wanted to sort of point out that this is at a continuous time process.

01:49 As well as a continuous space process.

01:54 Okay, so x of t will be equal to the current value of the stock.

02:00 I don't want to use dollars for now.

02:02 So we already assumed that x is in dollars.

02:06 So it would be 580 plus the increase of the stock valuation, which is 0 .25 times t.

02:21 So every day the stock value increases by 25.

02:28 And the variation around that, that we're going to add it by some variation factor, you can think of it as like a signal plus noise process, which is sort of what this kind of is, plus some noise process, which is brown in motion, which is distributed normally with means zero and variance alpha t.

02:56 So in other words, standard deviation is a square root of alpha t.

03:02 So that is that.

03:04 So let me just point out that b of t is normally distributed with mean zero and standard deviation square root of 20t, which is really just two times the square root of 5t if you really want to write like that.

03:31 So obviously the t goes inside the square root.

03:35 Okay.

03:39 So that's what x of t is.

03:43 So to sort of look at some example sample functions.

03:52 Let me get to red again.

03:55 I'm not sure why it's not giving me red.

03:59 No, not third time lucky.

04:08 Okay, there we go.

04:15 So is t.

04:18 I'm actually going to not go do it like this.

04:22 I'll have to do it like this because it's pretty obvious that x of t can't be negative.

04:30 At least in the context of this question.