# Tag Info

### Showing a basic market admits no arbitrage

The idea is actually really simple... presumably the difficulty is in cutting through the abstract formalism. We have an asset whose price is $S_0^1$ and tomorrow it can either go to $\beta S_0^1$ or ...
• 58.8k
Accepted

### Show that this stopped process converges ucp to the original process

You have $\sup_{s\in[0,t]}|M^{\tau_n}_s-M_s|=0$ on the event $\{\tau_n\ge t\}$, and so $$P\left(\sup_{s\in[0,t]}|M^{\tau_n}_s-M_s|>\epsilon\right)\le P(\tau_n<t)\to 0,\quad n\to\infty,$$ ...
• 26.1k
Accepted

### A question about Markov chain's definition

It is not necessary: if you know that $$P(Z_{n+1}=y \mid Z_0=x_0,\dots,Z_{n-1}=x_{n-1},Z_n=x)=P(Z_1=y \mid Z_0=x)$$ then you can show that both of those are equal to $P(Z_{n+1}=y\mid Z_n = x)$. To ...
• 143k
Accepted

### Demonstrate a (continuous time) chain that increases by one, and then randomly returns to the origin, is transient

Say we start at state $0$. What events need to happen for us to never return to $0$? For this particular chain, there's only one path that doesn't return. From $0$, we need to step to $1$ (prob. $1$)....
• 111

### Longest Increasing Subsequence Upper Bound

For (1), using some classical bounds of binomial coefficients, e.g., Proof for the upper bound and lower bound for binomial coefficients., we get  \binom{n}{k}/k! \leq \frac{n^k}{(\frac{k}{e})^{2k}}....
• 2,016
1 vote
Accepted

### A strange equality of distribution in Stochastic Processes

This is a consequence of a result in conditional expectation, see e.g Conditional expectation of Borel function of two independent variables for the proof: Given $X, Y$ are two independent random ...
• 1,355
1 vote
Accepted

### Fastest growing renewal process

The exponential bound is rather slack. Suppose your renewal process has interarrival times $T_1, T_2,\ldots$. Fix $c>0$ and truncate to $T_k^{(c)}:=T_k\wedge c$. This results in a renewal process ...
• 26.1k
1 vote

### Stochastic Processes with Dynamically Changing Parameters

First of all, I am going to make a slight modification to the information presented: OP talks about probability of flip and probability of landing on the right side. Here, it seems as though flip ...
• 908

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