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I'm reading the course notes of a probability course about martingales currently and I'm trying to solve some of the exercises, however I'm very much stuck with the following exercise: Let $\left\{ ... 2answers 43 views ### Problem about martingale convergence in$L^p$I'm trying to do the following exercise: I have a martingale$Z_n=A^{S_n}Q_A^{-n}$where$A>1$,$Q_A=\frac{1}{2}(A+A^{-1})$and$S_n=X_1+\cdots+X_n$with$X_k$r.v.'s iid such that ... 0answers 37 views ### Why is the Stopping Theorem interesting? The theorem for discrete-time martingales is as follows: Let$X=(\Omega,\mathcal{F},(\mathcal{F}_n)_n,(X_n)_n,\mathrm{P})$be a supermartingale and$\tau_1,\tau_2$two a.s. bounded stopping times on ... 0answers 34 views ### An exponential martingale Let$H_{t}$be a bounded continuous and$\textbf{F}^{B}_{t}$an adapted process.$B$Brownian motion. Show that$M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$... 1answer 68 views ### Proving it's a martingale and more conditions. Let$(X_{n})_{n>0}$be a sequence of random variables in$[0, 1]$and assuming that ($X_{0}=a) \epsilon [0, 1]$then:$Pr\left(X_{n+1}=\frac{X_{n}}{2}|\mathcal{F}_{n}\right)=1-X_{n}$and ... 1answer 84 views ### Rigorous Book on Stochastic Calculus I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ... 3answers 102 views ### What is a fair game? Suppose$X_n$is the fortune of a gambler after$n$th game. Then the game is called fair (Breiman 1968) if $$E[X_{n+1} \mid X_1, \dots, X_n] = X_n \forall n$$ My question is why a fair game is not ... 0answers 20 views ### Applications of martingale to gambling In an unfavourable or fair sequence of games (where one has to bet a minimum amount if he wants to bet) one cannot keep betting indefinitely. There must be a last bet. The intuition behind this is not ... 0answers 32 views ### Azuma inequality with probabilistic bound for the increments Let$(X_i)$be a super-martingale and suppose their differences are bounded ''with high probability'', that is $$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$ ... 1answer 94 views ### martingale and stochastic Integral Let${W_t}$be 1 dimension Brownian motion and$X_t:=\exp(t/2)\cos W_tt\in[0,T]$. Show that$X_t$is martingale. I understood$df(t,W_t)=-\exp(t/2)\sin xdW_t$, but I don't know why it become ... 1answer 23 views ### Characterization of conditional independence Definition: Let$\mathcal{G},\mathcal{K},\mathcal{H}$be$\sigma $-subalgebras of$\mathcal{F}$, where$\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$is a given probability space. We say that ... 0answers 26 views ### Finding a pre-visible process Question: Let$W_t$be a standard brownian motion under P with filtration$\mathscr F_t$. Let: $$M_t=\mathbb E[W_T^2|\mathscr F_t]$$ Show that$M$is a P martingale. This is simple enough using ... 1answer 160 views ### Lookback option with floating strike: boundary condition I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let$v(t,x,y)$denote the price at time$t$of the option under the ... 2answers 61 views ### Uniformly integrable martingale I have the following martingale.$M_n=\exp\left(aB_n-\frac{1}{2}a^2n\right)$for$n\geq0$and$a\neq0$,$B_n$is a BM. I have to show that for$a>0$,$M_n\rightarrow0$in probability. Is$M_n$... 2answers 51 views ### Bounded (from below) continuous local martingale is a supermartingale Suppose$M(t)$is a continuous local martingale. That is, there exists a sequence of stopping times$T_n$which almost surely increase to$\infty$, and such that$M(t\wedge T_n)$is a martingale for ... 0answers 24 views ### Decomposition of noisy measurements What can be a good intuition behind decomposing a sequence$\{Y_n\}$of noisy measurements (i.e. random variables) into two parts: one unpredictable and the other depending on the past. $$Y_n = ... 1answer 53 views ### The Vacisek Model and the short rate process I am trying to do some calculations related to the Vacisek model, but I think I am mixing up concepts and I'm not getting to any solution. Let me explain what the problem is. The Vacisek model ... 1answer 46 views ### Expectation of e^{-4B_\tau}, where \tau is an extended stopping time This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time:$$ \tau = \inf\{t \geq 0; B_t < t-2 \} $$This is a clear ... 2answers 124 views ### Submartingale example: proof I am trying to prove if the process M_t = e^{W_t^2-t} is a submartingale (W_t is the Wiener Process). The proof becomes a bit difficult, to the point where I am unsure how to move forward. Let ... 1answer 33 views ### Derivative of a parameteric martingale — is it a martingale? Let X(t) be a martingale (in continuous time), and for each real u, let Y_u(t)=g(u,X(t)) for some infinitely-differentiable g:\mathbb{R}^2\to\mathbb{R}, and assume that Y_u(t) is a ... 1answer 27 views ### Sufficient condition for time-changed quadratic covariation to vanish in probability Let (M_t^n)_{t \geq 0} be a sequence of continuous martingales of the form M^n_t = \int_0^t X^n_s \, dB_s where B_s is a Brownian motion. Let \tau^n(t) be the time change associated to M_t^n ... 1answer 31 views ### Is the following Markov Chain a martingale? Say I have a finite, ergodic Markov chain with states {0,1,2,3} and with the following transition matrix:$$\begin{bmatrix} \frac{7}{10} & \frac{3}{10} & 0 &0\\ \frac{1}{10} & ... 2answers 25 views ### Question on Doob's martingale convergence theorem Let$(\Omega,\mathcal F,\mathbb P)$be a probability space and$(\mathcal F_k)_{k\in\mathbb N}$a filtration of$\mathcal F$such that$\mathcal F=\sigma(\mathcal F_k\mid k\in\mathbb N).$Let ... 0answers 55 views ### Is the martingale propertey preserved by taking weak$^*$-limits? Let$(\Omega,\mathcal F)$be a measurable space,$X:\Omega\rightarrow\mathbb R^d$a$\mathcal F$-measurable map. Let$(\mathcal F_k)_{k\in\mathbb N}$be a filtration of$\mathcal F$such that ... 1answer 43 views ### arbitrage free price in martingale measures Consider a one-period market with$S^1_t,\cdots,S^n_t$, with$t=0,1$the price process of$n$assets, where$S_1$is a risk-free asset:$S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ... 1answer 49 views ### Question on Martingales and Brownian Motion I've run into an issue and I am unsure of how to proceed. In the text I'm working through, the following is left "as an exercise to the reader." Normally proofs listed as such tend to be fairly simple ... 1answer 75 views ### Levy's Martingale Using Radon Nikodym Let P and Q be two probability measures on the same space$(\Omega,\mathcal{F},\mathcal{P})$and let$\mathcal{F_n}$be filtration. Assume that$Q \ll P$. Let$X_n$denote the Radon-Nikodym derivative ... 1answer 40 views ### question about martingale In my lecture notes,I found the following problem: Let$X$an$F_{t}$adapted continuous process and$G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ... 1answer 34 views ### Expectations of martingales Consider a martingale (M_n)_{n \geq 0} adapted to a filtration (\mathcal F)_{n \geq 0} on a probability space (\Omega, \mathcal F, P). Prove that, for each k \leq n;$$E(M_n M_k) = E(M_k^2)$$... 0answers 52 views ### Is Zn a Martingale with mean 1? Consider a sequence of independent tosses of a coin, and let P_h be the probability of a head on any toss. Let A be the hypothesis that P_h = a, and let B be the hypothesis that P_h = b. Let ... 1answer 43 views ### Show that E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] for a martingale with Z_0=0 I was just wondering, if we let (Z_n)_{n\geq 0}be a martingale with Z_0=0, is it true then$$ E[Z_n^2]= \sum_{i=1}^n E[(Z_i-Z_{i-1})^2] $$Please let me know and if it is true, can someone show ... 1answer 28 views ### A bound on the maximum of a submartingale I'm trying to prove the following: If \{(S_i, \mathcal{F}_i) \mid i \in [N]\} is a nonnegative submartingale with \mathbb{E}S_i^p < \infty, let M = \max_i S_i. Then \|M\|_p \leq q ... 0answers 44 views ### Martingale and Stochastic equation Using the Ito formula, I can show that the martingale$$ Z_{t}=\frac{1}{\sqrt{1-t}}\exp -\frac{B_t^2}{2(1-t)}\qquad 0\leq t<1 $$admits the following differential$$ dZ_t=-\frac{B_t}{1-t}Z_tdB_t. ... 2answers 58 views ###$4^{Brownian(t)}$martingale proof Let$B(t)$a Brownian motion. I like to prove that$4^{B(t)}$= martingale I rewrote the expression into an exponential form (like$\exp(\ln(4) B)$), but then I don't know how to proceed. 1answer 64 views ### Transforming semimartingale to local martingale by change of measure Consider a continuous$\mathbb{P}$- semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to$\mathbb{Q}$s.t. ... 2answers 88 views ### Normal variables, uniform integrability and limit of a martingale I'm stuck in the following problem! Would be great if you can help. Suppose you have$(Y_n)_{n\in \mathbb N}$, a sequence of independent and identically distributed random variables,$Y_1 \sim ...
I know that $M_t=N_t-\lambda t$ is a martingale for $N_t$ a rate $\lambda$ poisson process and that for a brownian motion, $B_t^2-t$ is a martingale. I'm wondering, is there something similar for ...
Consider the probability space $([0,1), \mathcal{B}, \lambda)$ where $\mathcal{B}$ is the Borel $\sigma$-algebra and $\lambda$ is the uniform measure. Let $A_{i,n} = [(i-1)2^{-n}, i2^{-n})$ for \$i \in ...