# Tagged Questions

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### How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
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### Showing that the first hitting time of a closed set is a stopping time.

I found this exercise online: I am stuggling with the last part of the second exercise, that is I am not able to show that $\tau = \sup_i \tau_i$. Obviously we have that $\tau \ge \sup_i \tau_i$, ...
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### Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
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I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0$ and $\... 0answers 344 views ### stopping time expectation for gambler's ruin 2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin. I let$X_n = x + \sum_{i=1}^{n}\xi_i$where$\xi_i$are i.i.d. ... 2answers 2k views ### Expected stopping time of brownian motion I've been looking at this for some time now and still have no sensible solutions, can somebody help me out please. Say I define the stopping time of a Brownian motion as followed: $$\tau(a) = \min (t ... 2answers 2k views ### Sum of two stopping times is a stopping time? Let \sigma and \tau be two stopping times in \mathscr{F}_t and let this filtration satisfy all the usual conditions. Question: Is \sigma + \tau a stopping time? Attempt at a solution: I ... 1answer 61 views ### Exist \alpha < \infty, \beta > 0 such that \mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}? Let B_t be a standard one-dimensional Brownian motion. Suppose \lambda > 0 and let$$T_\lambda = \min\{t : |B_t| = \lambda\}.$$Do there exist \alpha < \infty and \beta > 0 (which may ... 2answers 32 views ### Simple question regarding stopping times. I have this exercise regarding stopping times, but I am not able to solve it. You have a probability space (\Omega, \mathcal{F},P), with a filtration \{\mathcal{F}_t\}}. You have two stopping ... 1answer 98 views ### how to prove (X_{n})_{n\in \mathbb N} and (Y_{n})_{n\in \mathbb N} are supermartingale.and (Y_{n})_{n\in \mathbb N} is convergence to -7 Let p \in [0 , \frac{1}{2}] and \eta_{i} be i.i.d random variables and P(\eta_{i}=1)=p and P(\eta_{i}=-1)=1-p and \mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n}) and X_{n}=\sum_{i=1}^{n}\... 1answer 133 views ### Find the distribution of T_a=\inf\{n\ge 0: R_{n}\gt a\} for fixed number a\gt 0 Suppose R_{n}=\sum_{i=1}^{n} X_{i} for n\ge 1 and R_{0}=0 , where X_{i}\gt 0 are independent and identically distributed. Find the probability law of the stopping time T_a=\inf\{n\ge 0: R_{n}... 2answers 268 views ### coupon collector problem for different number of copies of each coupon type I would like to pose a question on a variation on the classical coupon collector's problem: coupon type i is to be collected k_i times. What is the expected stopping time or the expected number of ... 1answer 40 views ### A martingale characterization I saw the following characterization of martingales (without proof) in some lecture notes I found on the web and I haven't been able to produce a proof it. Let X be an adapted process. If E[X_{\... 1answer 50 views ### Determine E\sum_0^\infty X_n1_{(T=n)} X_T = \sum_0^\infty X_n 1_{(T=n)} where T is a stopping time and (X_n) is a martingale. Show that if T is bounded then EX_T = EX_0: T \leq N, and then consider X_T = X_{T\wedge N} = \... 1answer 65 views ### Measurability of the zero-crossing time of Brownian motion I have the following random time \tau = \inf\{t > 0: W_t = 0\} where (W_t)_{t\geq 0} is Brownian motion with almost surely continuous paths and W_0 = 0 a.s. I need to prove that \tau is ... 1answer 56 views ### On the proof of lemma 1.2.4 of Stroock and Varadhan A question concerning stopping times In the book Multidimensional diffusion processes, of Stroock and Varadhan one reads (page 23): This is the proof of (i). Here the authors say Define f_t on (\{\tau \leq t\}, \mathcal{F}_t ... 1answer 108 views ### Markov and strong Markov properties In my study of strong Markov property of an RCLL canonical Markov process I encounter the following definition: Suppose Y_t:\omega\rightarrow \omega(t) is canonical Markov process with respect to ... 1answer 128 views ### Brownian motion proof of Dirichlet problem I am reading the proof of the Dirichlet theorem stated in the following form: Theorem: Let D be a bounded domain in \mathbb{R}^d such that every boundary point satisfies the Poincare cone ... 1answer 92 views ### Tower Property for Expectations and Stopping Times Let (\Omega,(\mathcal{F_t})_{t\geq0},P) be a filtered proability space with X\in L^1(P) and two stopping times S and T. Show that \begin{equation*} \mathbb{E}(\mathbb{E}(X|\mathcal{F}_T)|\... 1answer 242 views ### The expected time until reaching a specified set in a Markov chain I am reading an article in which they discuss a specific Markov chain in an example, and it turns out I need to sharpen up my Markov knowledge. First the setup. I have a continuous time Markov chain ... 1answer 68 views ### Determining if some random variable is a stopping time I am stuck on this issue: Let (B_t) be a Brownian motion. We know that since \{0\} is a closed set in \mathbb{R} and that (B_t) is a continuous adapted process,$$ \tau:= \inf \{ t\geq 0 : ... 1answer 272 views ### A martingale with bounded increments either converges or diverges to both infinities a.s. I am reading page 236 "Probability : theory and examples" by R. Durrett. Theorem 31. Let$X_1, X_2,\ldots$be a martingale with$|X_{n+1}-X_n|\leq M<\infty$. Let$C=\{\lim X_n \;\;\; \text{exists ...
Am i correct in my derivation? I want to calculate $\mathbb{E}B_{\tau_a}$. From the definition of the hitting time i get $B_{\tau_a}=a$, so $$\mathbb{E}B_{\tau_a}=\mathbb{E}a=a$$ I am new to the ...
Say we have a filtration $(\mathbb{F}_s)$, and a stopping time $\tau$ w.r.t. to that filtration.Let $X_t$ be a continuous stochastic process (not required to be adapted to the mentioned filtration), ...