# Tagged Questions

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### For a Brownian motion prove that (a) $N (t) -λt$ and (b) $e^{(\log(1-u) N (t) + uλt)}$, are martingales

For a Brownian motion ${z (t)}$ and for any $β ∈ R$, be $V (t) = \exp\{ βz (t) - (t β ^ 2) / 2\}, t≥0$ Show that ${V (t)}$ is a martingale with respect to a Brownian filtration. Also ${N (t)}$ be a ...
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### Is $W_{2t}-W_t$ a brownian motion?

Is $W_{2t}-W_t$ a brownian motion? $(W_t)_{t\geq 0}$ is a brownian motion, I have to show that $X_t:=W_{2t}-W_t$ is a brownian motion as well. $$W_{2t}= 1/\sqrt{2} W_t$$ (by scaling property) then ...
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### Stopping times problem: $\tau_+ = \inf \{t \ge 0 \mid W_t>0\}$

Stopping times problem, $\tau_+ = \inf \{t \ge 0 \mid W_t>0\}$ I can not prove the following : P/S: When I look at the stopping time, I feel that $\{W_0 > 0\} = \{\tau_+ = 0\}$ , is that ...
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### Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
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### Hitting time for Brownian Motion Surplus Process

I'm trying to solve this question for a continuous surplus process. The surplus process is $$U_s=U_0+s-B_s$$ where $B_t$ is a Brownian motion representing payouts, $U_0$ is starting capital, $s$ is ...
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### Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
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### Combination of Wiener Processes

If $W_s$ and $W_t$ are wiener processes, we have that the probability that $W_s$ and $W_t$ attain maximum is (I am concluding this from "running maximum", but I am not sure) ...
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### Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$dV_t = - \beta V_t dt + \sigma dW_t$$ with $V_0 = v$, where $W_t$ is a Wiener ...
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### Stopping times, open sets and Brownian Motion

Let $B_t$ be a brownian motion started at 0. I am trying to prove that $\tau$, defined as: $$\tau = \inf\{t > 0 \mbox{ }|\mbox{ } \left|B_t\right| \geq \frac{1}{1+t} \}$$ is a stopping time with ...
I am trying to prove that $\tau$, defined as: $$\tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \}$$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
Which Brownian motion property is the most important? A standard Brownian motion is a stochastic process $(W_t, t\geqslant 0)$ indexed by nonnegative real numbers t with the following properties: ...