Questions tagged [martingales]

For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

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Why is conditional expectation of brownian motion with a negative sign?

I have been reading up on the below thread: conditional expected value of a brownian motion but I cannot understand how 𝔼[π΅π‘ βˆ’π‘ π‘‘π΅π‘‘+𝑠𝑑𝐡𝑑 | 𝐡𝑑]=𝔼[π΅π‘ βˆ’π‘ π‘‘π΅π‘‘| 𝐡𝑑]βˆ’π”Ό[𝑠𝑑𝐡𝑑 | 𝐡𝑑]. ...
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Inequality on expectation of exponential martingale

Let $X$ be a continuous local martingale with $X_0=0$. Define the exponential local martingale $$\mathcal{E}(X)=e^{X-\frac{1}{2}[X]}.$$ For any $p,q>1$, establish the identity $$\mathcal{E}(X)^p=\...
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Expected value of exponential Brownian motion

I want to prove that $$E[e^{2B_t}] = e^{2t}$$ where $B_t$ is a Brownian motion. I have been reading up on Mean of exponential Brownian motion but it does not show how the rest of the log-normal ...
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Is it true that $ \sum_{i=1}^{\infty}\left(M_{i-2} M_{i}-M_{i-1} M_{i-2}\right)<\infty, \text { a.s? } $

Prove or disprove. Suppose that $\left(M_{n}\right)_{n}$ is a martingale with $M_{n} \geqslant-10 \quad \forall n$, a.s. Is it true that $$ \sum_{i=1}^{\infty}\left(M_{i-2} M_{i}-M_{i-1} M_{i-2}\right)...
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1 answer
70 views
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Almost sure convergence of martingale increment

Prove or disprove. Suppose that $\left(M_{n}\right)_{n}$ is a martingale with $M_{n} \geqslant-10 \quad \forall n$, a.s. Is it true that $$ \sum_{i=1}^{\infty}\left(M_{i}-M_{i-1}\right)^{4}<\infty \...
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Proof of $X^{(n)}_t$ converges to 0

I recently came across this question and reckon it should be a direct application of Doob's inequality (correct me if I am wrong). But I struggle to write formal proof. For each $n \in \mathbb{N}$, ...
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Showing that a process is a supermartingale using Ito's formula

Consider a stock with price dynamics $$dS_t=S_t\sigma_tdW_t$$ where $(W_t)_{t\geq0}$ is a Brownian motion and $(\sigma_t)_{t\geq0}$ a bounded and continuous process adapted to the filtration $(\...
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1 vote
1 answer
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Uniform Integrability of a Martingale up to time T

I am trying to prove that a certain martingale $(R_t)_{t\geq 0}$ is uniformly integrable over a finite time interval $[0,T]$. Now I know that the definition of uniform integrability is that $\lim_{a\...
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Find a process $\left(A_{n}\right)_{n}$ such that $\left(\sum^{n} X_{i}\right)^{2}-A_{n}$ is a martingale.

Let $\left(X_{i}\right)_i$ be a sequence of $U[-1,1]$, i.e. $f x_{i}^{(x)}=\left\{\begin{array}{ll}1 & -1<x<1 \\ 0 & \text { otherwise. }\end{array}\right.$ Assume that $\left(X_{i}\...
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1 vote
1 answer
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Tail Probabilities of a martingale difference sequence

I'm currently facing the problem, that I can neither prove nor find a counterexample for the following statement. Let $q \in (1,2)$ and let $(D_n)_{n \in \mathbb{N}}$ be a martingale difference ...
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3 votes
1 answer
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Law of the square of a martingale divided by its bracket

Let $(M_t)_{t\geq 0}$ be a continuous martingale such that $M_0=0$ almost surely. There exists an increasing process $(\langle M\rangle_t)_{t\geq 0}$ which is called the bracket of $M$ such that $M^2-\...
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Computing $E(X_n | X_n \geq 0)$, where $X_n$ is an asymmetric lazy random walk on $\mathbb{Z}$

Suppose $X_n$ is a ''lazy'' asymmetric random walk such that, at each step, $X_{n+1} = X_n + 1$ with probability $\alpha$, $X_{n+1} = X_n - 1$ with probability $\beta$, and $X_{n+1} = X_n$ with ...
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Stopping times for martingale

Here are some definitions. The nonnegative integer set is denoted by $\mathbb{Z}_+$. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space and $\{\mathcal{F}_{n}\}_{n\in{\mathbb{Z}_+}}$...
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Stopped version of UI martingale is UI

Suppose $(X_n)_{n\geq0}$ is a uniformly integrable martingale. Let $T$ be any stopping time. Show that the stopped process $(X_n^T)_{n\geq0}=(X_{n\wedge T})_{n\geq0}$ is uniformly integrable. My only ...
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3 votes
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Return of Brownian motion to zero

From chapter 4 of Bulinskiy & Shiryaev's Theory of Random Processes (ISBN 5-9221-0335-0): [Exercise 26] Let $W = \{W_t, t \geqslant 0 \}$ be a $m$-dimensional Brownian motion, where $m \geqslant ...
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Show that $\left(V_{n}\right)_{n \geq 1}$ converges in $L_{1}$

Let consider the Galton-Watson process with immigration which is given by the following recursion $$ Z_{n+1}=\sum_{k=1}^{Z_{n}} \xi_{k}^{(n+1)}+\eta_{n+1} $$ where $\left(\xi_{k}^{(n)}\right)_{k \geq ...
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2 votes
1 answer
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Exponential submartingale inequality

In a paper I am reading I found the following: "Applying the exponential martingale inequality we derive that $$P\Big(\omega: \sup_{0 \leq t \leq k}[M(t)-1/2 \epsilon \langle M(t) \rangle] \leq 2 ...
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  • 1,771
4 votes
2 answers
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A question about martingale and Brownian motion

It is well known that the Brownian motion $B=(B_t)_{t\ge 0}$ is a martingale with respect to its natural filtration $\mathscr{F}_t$ and the fixed probability measure $\mathbb{P}$, i.e. $$\mathbb{E}(...
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Brownian motion, submartingale

I'm trying to prove whether a given process is a submartingale. Let $W = \{W_t, t β‰₯ 0\}$ - m-dimensional Brownian motion. Prove that $( ||W_t||, F_t )_{t β‰₯ 0}$ is a submartingale (with a.s. continuous ...
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Bounded increments of a martingale converges proof.

Prove or disprove: There exists a martingale $\left(M_{n}\right)_{n}$ with $\mathbb{P}\left(M_{0}=1\right)=\mathbb{P}\left(M_{0}=-1\right)=1 / 2$ and $\left|M_{n}-M_{n-1}\right| \leq n^{-3}$ for all $...
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Proof check for Problem 1.5.12 on Brownian motion and stochastic calculus

The problem is that the continuity condition seems to be a dummy condition. Here is my proof. Let $X$ be in $\mathscr{M}_2^c$ (Continuous square-integrable martingale and $X_0=0$), and $T$ be a ...
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4 votes
2 answers
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Best strategy to reach $500 for a gambling situation in a casino

Suppose a gambler has \$100 to start with. Each time he/she has 0.4 chances of winning and 0.6 chances of losing a bet. If he/she wins he gets twice the money he put in and loses what he bet if he ...
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Martingale with bounded increments converge?

Prove or disprove the following arguments: There exists a martingale $\left(M_{n}\right)_{n}$ with bounded increments such that $\lim _{n \rightarrow \infty} M_{n}=\infty$ There exists a martingale $...
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1 answer
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Attempt to improve from $L^1$-boundedness to uniform integrability

Let $(X_n)_{n\geq0}$ be a discrete-time martingale, and let $T$ be an almost surely finite stopping time such that $$\mathbb{E}(|X_T|)<\infty,\hspace{2cm}\lim_{n\to\infty}\mathbb{E}(|X_n|1_{\{T>...
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Bound on expectation of minimum of two stopping times (proving integrability)

Let $(M_n)_{n\geq0}$ be a non-negative martingale with filtration $(\mathcal{F}_n)_{n\geq0}$. Suppose $M_0=1$ and set $$T=\min\{n\geq0:M_n=0\}.$$ Also, for $R>0$, consider the stopping time $$T_R=\...
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Attempt to show that local martingale is a true martingale

Consider the process $X_t=e^{\frac{1}{2}t}\cos(B_t)$, where $B$ is a Brownian motion in $\mathbb{R}$. Using Ito's formula (unless I'm mistaken) implies that $$dX_t=-e^{\frac{1}{2}t}\sin(B_t)dB_t,$$ ...
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2 votes
1 answer
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Future of martingale after stopping time

Let $(M_n)_{n\geq0}$ be a non-negative martingale with filtration $(\mathcal{F}_n)_{n\geq0}$. Set $$T=\min\{n\geq0:M_n=0\}.$$ Show that $M_n=0$ for all $n\geq T$ almost surely. As $(M_n)_{n\geq0}$ is ...
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1 vote
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How to actually apply martingales when conditioning on a random variable (not filtration)?

For a Galton-Watson process, I've shown that $\frac{Z_n}{\mu^n}$ is a martingale i.e. $E[\frac{Z_{n+1}}{\mu^{n+1}}|\mathcal{F}_n]=\frac{Z_n}{\mu^n}$. However, I want to show that, for $n>m$, $$E[Z_{...
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Necessary and sufficient condition for random sum of independent RVs to be a martingale

Let $M$ be a Poisson random measure on $(0,\infty)$ with intensity $\lambda dt$, where $\lambda\in(0,\infty)$. Let $(Y_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables, independent ...
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2 votes
1 answer
38 views

Deterministic function that gives Snell envelope

I am studying for an exam and would love some hint on this review problem. Suppose the discrete-time process $(S_t)_{0 \leq t \leq T}$ have $i.i.d.$ increments. Fix a measurable function $f:\mathbb{R}...
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2 answers
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Expectation of product of brownian motion and stochastic integral

Let $f:[0,\infty)\to\mathbb{R}$ be a deterministic continuous function and $B$ a Brownian motion with $B_0=0$. I need to prove that $$\mathbb{E}\left(B_t\int_0^tf(s)dB_s\right)=\int_0^tf(s)ds.$$ I ...
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Showing that probability of BM being in part of a boundary is harmonic

Let $D$ be a domain in $\mathbb{R}^d$ and let $A$ be a measurable subset of its boundary $\partial D$. For $x\in D$, define $$\phi(x)=\mathbb{P}(X_T\in A)$$ where $(X_t)_{t\geq0}$ is a Brownian motion ...
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1 answer
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Inequality on martingale using submartingale

Let $S_n=X_1+\dots+X_n$, where $X_1,X_2,\dots$ are independent and $\mathbb{E}(X_m)=0$, $\mathbb{E}(X_m^2)=\sigma_m^2\in(0,\infty)$ for all $m\geq1$. Let $\mathcal{F}_n=\sigma(X_1,\dots,X_n)$. It is ...
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1 vote
1 answer
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For stochastic integral $I$ of simple process $X$, $0 \le s < t< \infty$, show $E[I_t(X) | \mathscr{F}_s] = I_s(X)$ a.s.

For stochastic integral $I$ of simple process $X$, $0 \le s < t< \infty$, show $E[I_t(X) | \mathscr{F}_s] = I_s(X)$ a.s. This is Karatzas + Shreve 2nd Edition, Chapter 3.2.B, equation (2.13) ...
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Proving an upper bound to the running maximum of a martingale with mean zero and finite second moment [duplicate]

The conditions are as given in the title. I want to show, for a discrete martingale $M$ with $E[M_0]=0$ and $E[M^2]<\infty$, $P\left(\max_{0\leq s\leq t} M_s > x\right) \leq \frac{E (M_t^2)}{E (...
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2 votes
1 answer
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Does this setup imply that $E[(M_t-M_s)^4 \mid \mathcal{F}_s]$ is bounded?

Suppose $(M_t)_{t \geq0}$ is a martingale w.r.t. a filtration $(\mathcal{F}_t)_{t \geq0}$. Suppose that $$ E[(M_t-M_s)^2 \mid \mathcal{F}_s] $$ is uniformly bounded by some constant. I want to prove ...
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Understanding proof of Azuma's inequality

I am trying to understand the proof of Azuma's inequality, though one step isn't quite clear to me: To give some context: $V_1,V_2,\dots$ is a martingale difference sequence with respect to the random ...
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2 votes
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Proving $E\bigg[\bigg(\int_a^b X_s dW_s \bigg)^2 \ \bigg\vert \ \mathcal{F}_a \bigg] =\int_a^b (X_s)^2 ds $

Suppose $(\mathcal{F}_t)_{t \geq0}$ is a filtration on a probability space and $W=(W_t)_{t \geq0}$ is a Brownian Motion with respect to this filtration. Let $(X_t)_{t \geq0}$ denote some ...
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Proof that existence of specific process implies no numeraire strategies

Consider a discrete-time market with $n$ assets and (possibly negative) prices $(P_t)_{t\geq0}$. A numeraire strategy is a self-financing investment strategy such that the wealth process is almost ...
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3 votes
0 answers
32 views

Supremum of Brownian motion increments

Let $W=(W(t))_{t\geq 0}$ be a Brownian motion. Consider the random variable $$Y(t):=\sup_{1\leq s\leq t}[W(s)-W(s-1)],$$ for some fixed instant $t\geq 0$. I am interested in this $Y(t).$ Could anyone ...
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  • 559
0 votes
1 answer
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Discrete time positive martingale with non-trivial limit

Let $X_t$ be a positive discrete time martingale. (I.e., $X_t>0$ and $\mathbb{E}_t X_{t+1}=X_t$ for all $t\in\mathbb{Z}$.) Then by Doob's supermartingale convergence theorem, there exists some non-...
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1 vote
1 answer
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Filtration of sum of independent sub-martingales

Let $x_1,\ldots, x_n$ be independent continuous martingales with filtrations $\mathcal{F}^1_{t},\ldots, \mathcal{F}^n_{t}$. Let $Y_i=(x_i)^2$ be the associated sub-martingale with each $x_i$ (we can ...
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1 vote
1 answer
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Durrett's Theorem 4.8.7 (Symmetric Simple Random Walk)

I am trying to understand the proof of Theorem 4.8.7 in Durrett's Probability: Theory and Examples. The theorem is about symmetric simple random walk: let $\xi_1, \xi_2, \cdots$ be $i.i.d.$, $S_n = ...
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1 vote
1 answer
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Is the difference of random walk a martingale

Suppose $X_i \sim i.i.d. N(0,1), i=1,...$, $S_n=X_1+...+X_n$ let's $ Y_{i}^{\left( v \right)} := X_i \textbf{1}_{ \left\{ \left| X_i \right| \leq v \right\} }, X_{i}^{\left( v \right)} := \left( Y_{i}...
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3 votes
1 answer
93 views

Stopping time and super-martingale

Consider a right-continuous super-martingale $(X_u,\mathcal{F}_u)_{u \in \mathbb{R}_+}.$ Let $\theta_1$ and $\theta_2$ be two stopping times such that $\theta_1 \leq \theta_2.$ Prove that $(X_{u\wedge\...
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3 votes
0 answers
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Relation between supremum of quadratic variation expectation with expectation of supremum of martingale

Let $X$ be a local martingale. Then the quadratic variation $[X]$ is such that $X^2-[X]$ is a local martingale. I am tasked with showing that $$\sup_{t\geq0}\mathbb{E}([X]_t)<\infty\iff\mathbb{E}\...
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2 votes
1 answer
40 views

Surprisingly simple expected time for the "range" of a Brownian motion to extend beyond $a$ - is there a martingale method?

I will call the range $R(t)$ of a standard Brownian motion the difference between its maximum $M(t) = \max_{0\leq s \leq t} B(t)$ and its minimum $m(t) = \min_{0\leq s \leq t} B(t)$. That is, I am ...
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3 votes
2 answers
124 views

How to find $E(\tau)$?

Question. Let $X_1,X_2,...$ be an i.i.d sequence of random variable with $P(X_i=0)=P(X_i=1)=1/2$. Let $\tau$ be the waiting time until the appearance of the six consecutive $1's$. That is $$ \tau = \...
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  • 1,481
3 votes
1 answer
53 views

Is Brownian motion a semimartingale?

I have read an article about semimartingales on Wikipedia and it says that: "A Brownian motion is a semimartingale". However, it is hard for me to find any proof of this statement, and I ...
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4 votes
1 answer
43 views

Showing that a function of two brownian motions is a martingale.

Let $B$ be a standard Brownian motion, let $f$ be a smooth function taking values in $[a,b]$ where $0<a<b<\infty$ and assume that the derivative $f^\prime$ is bounded. For $t\in[0,1]$ and $x\...
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