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### Prove $\tau=\inf\{t\in[0,T]:M_t=0\}\wedge T$ a stopping time for a continuous martingale $(M_t)_{t \geq 0}$

I have a question about a positive continuous martingale. Let $(M_t)_{t\in[0,T]}$ be a continuous martingale such that $P(M_t>0)=1$ for all $t\in[0,T]$. Set $\tau=\inf\{t\in[0,T]:M_t=0\}\wedge T$. ...
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Consider a discrete random walk taking values +1 or -1 with probabilities p and q, respectively. Let $S_n = \sum_{k=1}^{n}X_k$. Let $[-A,B]$ be an interval, $A,B \geq 1$. Now define $$\tau =\min(n:n ... 0answers 14 views ### Requesting references on the number of times a random process hit a target. Let X_t be a continuous process, A be some event, and N(x) be the number of times X_t(x) enter A in a fixed time window t \in [a,b]. Are there results concerning N(x)? Like, what its ... 0answers 19 views ### How to find intersection of moving circle and line? Say I have a point, with position (x1,y1) at time t=0, with velocity dx1 and dy1 in the x and y directions respectively, which may or may not collide with a circular entity with radius r, centered at ... 0answers 20 views ### If (F_t)_t is a filtration, T is a stopping time and Y is F_T-measurable, then 1_{\left\{T=s\right\}}Y is F_s-measurable Let (\Omega,\mathcal A) be a measurable space I\subseteq[0,\infty) \mathbb F=(\mathcal F_t)_{t\in I} be a filtration on (\Omega,\mathcal A) \tau be a \mathbb F-stopping time \mathcal ... 1answer 23 views ### If X is an \mathcal F_t-adapted process with countable time domain and \tau is a stopping time, then X_\tau is \mathcal F_\tau-measurable Let (\Omega,\mathcal A) be a measurable space I be an at most countable set \mathbb F=(\mathcal F_t)_{t\in I} be a filtration in (\Omega,\mathcal A) X=(X_t)_{t\in I} be an \mathbb ... 1answer 14 views ### Is there any intuition behind the statement E[X_\tau \mid \mathcal{F}_\sigma]=X_\sigma Is there any intuition behind the statement E[X_\tau \mid \mathcal{F}_\sigma]=X_\sigma a.s. I mean I know that the interpretation of the conditional expectation and how to visualize it somehow but I ... 1answer 26 views ### Why do we need optional stopping theorem? For martingale,optional stopping theorem states: Let (M_n)_{n\in \mathbb{N}} be adapted with M_n\in L^1 for all n and if (M_n)_{n\in \mathbb{N}} is a martingale, then E[M_T]=E[M_0], for all ... 1answer 35 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 ... 1answer 22 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 ... 1answer 15 views ### Stopping time proof with discrete martingale My professor gave me a very unclear proof of this theorem. It was so messy and unclear, I was unable to write down all the details of the proof. Theorem: Suppose \tau \in T, where T is the set ... 0answers 15 views ### Time integral of Brownian motion's running maximum Let \mu \geq 0 and consider B_{\mu}(t) := B(t) + \mu t a one-dimensional BM with drift \mu, and let M_t := \max_{0 \leq s \leq t} B_{\mu}(t) be its running maximum. My question involves two ... 1answer 36 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 71 views ### Does this game make you arbitrarily rich with probability one? We toss a coin. If it's heads we win \ 1, otherwise we lose  \ 1. Fix some large sum. Will we be winning this amount with probability one at some point? We assume that we have infinitely many ... 2answers 101 views ### Stochastic variables independent given Tau 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), ... 1answer 30 views ### Independence of a hitting time and the underlying stochastic process While I was playing around with the Girsanov's Theorem I stumbled upon the following absurdity and I couldn't resolve it with the current knowledge of stochastic analysis that I have. B being ... 1answer 23 views ### Show that \mathbb{E}\left[c_{\tau\wedge n}X_{\tau\wedge n}-\sum_{i=1}^{\tau\wedge n}c_i\mathbb{E}(X_i-X_{i-1}\mid\mathcal{F}_{i-1})\right]\le 0 I am trying to go through a past exam paper but I don't know how to deal with stopping times since we only did 2 exercises in class... I got stuck, so I would really appreciate if someone could help ... 1answer 11 views ### Expectations of stopping times in general I have a very basic question: So for a stopping time \tau with E(\tau)<\infty we have E(\tau)=\sum_{n=0}^\infty P(\tau>n), right? Why is that? Thanks! 1answer 68 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 45 views ### Find \mathbb{E}_{X_0 = x} X_\tau for an Ornstein-Uhlenbeck process (X_t)_{t \geq 0} where \tau = \inf\{t>0 \mid X_t \notin [a,b]\} Let X_t satisfy the following SDE: dX_t = X_t dt + \sigma dB_t, \sigma is a constant and B_t is Brownian Motion. Find \mathbb{E}_{X_0 = x} X_\tau where \tau = \inf\{t>0 \mid X_t \notin ... 0answers 64 views ### A Markov Chain probability, conditioned on a random time. My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set A (if it matters you can set A=\lbrace ... 1answer 24 views ### Rewriting probabilities as expectation Consider the stopping time \tau_a:=\lbrace{t>0| W_t >a\rbrace}, where W_t is a Brownian Motion. Define: X_t:=W_{\tau_a+t}-W_{\tau_a}. We have that X_t is a Brownian Motion independent ... 1answer 29 views ### Simple Probability Inequality with Stopping Times Suppose U_1,...,U_n are independent random variable with \mathbb{E}[U_i]=0. Define Z_k:=\sum_{i=1}^k U_i. Set T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace. Clearly \lbrace T=k ... 1answer 34 views ### An Algorithmic approach to the secretary problem with unknown n I've been reading about the secretary problem these days and I got the idea for the case when we know the number of applicants n in advance. I'd like to know what would be an algorithmic approach ... 1answer 36 views ### E[|X(t)|]\leq K\implies E[|X(\tau)|]\leq K ? Let X(t) be a stochastic process. Assume that, for every t\leq M, it holds$$E[|X(t)|]\leq K, $$for some constant K. Let now \tau\in[0,M] be random (stopping time). Is it true that also ... 1answer 39 views ### Distribution Stopping time under Brownian motions Considering W the canonical process on C([0,1],\mathbb{R}) and the row filtration generated by the coordinate process of W, I want to prove that ... 1answer 32 views ### Illegal lottery problem (Merging dependent bernoulli trials) Suppose I am in a town that playing lottery is illegal. If I buy a ticket for 1 dollar, I will win the lottery with probability p. Each time I buy a ticket, the police may catch me and confiscate ... 0answers 60 views ### Secretary Problem - Optimal algorithm for expected value of candidate. I recently encountered the secretary problem and there are essentiall two problems: Maximizing the probability of choosing the best candidate. Gnedin proved in his paper ... 0answers 33 views ### Charaterize the \mathcal{F}_\tau a sigma algebra for the stopping time \tau Consider a stochastic process X: [0, \infty) \times \Omega \to \mathbb{R}^d We define \mathcal{F}_t = \sigma(X(s), 0 \leq s \leq t) The sigma algebra generated by the sets \{\omega: X(s,\omega) ... 1answer 33 views ### Martingale: Show p\{T<+\infty \}=1. Let (X_i) i.i.d. such that p\{X_i=+1\}=p\{X_i=-1\}=\frac{1}{2} and let (S_n) the martingale define by S_0=0 and S_n= X_1+...+X_n. Moreover, let$$T=\begin{cases}\inf\{n\geq 0\mid ...
Let $\{X_n:n=0,1,\ldots\}$ be a martingale with respect to a filtration $\{\mathcal F_n\}$. Let $A,B$ be nonempty, disjoint Borel sets and define $T_0=0$, \begin{align} S_n &= \inf\{m\geqslant ...