# Winning money from random walks?

Mathematics Asked by Maximilian Janisch on December 15, 2020

The same question was posted on MathOverflow.

# Informal problem description

Assume that we have a stock whose price behaves exactly like a Wiener process. (There are multiple reasons why this is not the case in real life, but bear with me). I invest $$1$$ into the stock. Then I wait until the stock rises to $$1.1$$. If the stock price hits $$0$$ before it hits $$1.1$$, I go bankrupt and lose my investment.

Is this a viable strategy? And if yes, why is it possible to make money off of a random walk?

# Formal Problem setup

Assume we are given a standard Wiener process $$(W_t)_{tin[0,infty[}$$, i.e. a family of real random variables on a probability space $$(Omega,mathcal A, mathsf P)$$ (event space, $$sigma$$-algebra and the probability measure, respectively) such that for every $$tgeq0$$, we have

• $$W_0=0$$,
• for every $$t>0$$ and $$sgeq0$$, we have that $$W_{t+s}-W_t$$ is independent of every $$W_u$$ for $$ule t$$,
• for every $$tgeq0$$ and $$sgeq0$$, the difference $$W_{t+s}-W_t$$ is normally distributed with expected value $$0$$ and variance $$s$$,
• for every $$omegainOmega$$, the function $$begin{split}W(omega): [0,infty[&tomathbb R,\ t&mapsto W_t(omega)end{split}$$ is continuous.

# Question

My strategy, I will call it $$S$$, is to wait until I make $$0.1$$ in profit, i.e. until the Wiener process hits $$0.1$$. If the process hits $$-1$$ before hitting $$0.1$$, I loose all my money.

So we have $$S =0.1$$ if there exists a $$tgeq0$$ such that $$W_t=0.1$$ and $$W_s>-1$$ for all $$sle t$$ and we have $$S=-1$$ otherwise.

What is the expected value of $$S$$? It think it is $$0$$. However, I don’t know how to prove this.

# Some attempts

It is well-known$${}^1$$ that the running maximum $$M_toverset{text{Def.}}=max_{0le sle t} W_s$$ has the cumulative distribution function $$mathsf P(M_tle m)=begin{cases}operatorname{erf}left(frac m{sqrt{2t}}right), &text{if }m geq0\0, &text{if }mle0end{cases}.$$

So for a fixed $$t_0geq0$$, we know that $$mathsf P(W_s>-1text{ for all }sle t_0) = mathsf P(M_{t_0}<1)=operatorname{erf}left(frac{1}{sqrt{2t}}right).$$

I don’t see how to compute $$mathsf E(S)$$ from that, though.

# Another idea

Maybe one can try to solve a discretized problem as was done in this great answer and then use Donsker’s Theorem to conclude?

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