Questions tagged [martingales]

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

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2 votes
0 answers
34 views

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 ...
  • 643
6 votes
2 answers
336 views

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 ...
  • 495
0 votes
0 answers
48 views

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 $...
3 votes
1 answer
57 views

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>...
4 votes
0 answers
41 views

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=\...
1 vote
0 answers
32 views

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 $$dX_t=-e^{\frac{1}{2}t}\sin(B_t)dB_t,$$ ...
2 votes
1 answer
56 views

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 ...
1 vote
0 answers
38 views

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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2 votes
1 answer
113 views

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 ...
2 votes
1 answer
52 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}...
0 votes
2 answers
97 views

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 ...
0 votes
0 answers
16 views

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 ...
0 votes
1 answer
25 views

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 ...
1 vote
1 answer
42 views

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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1 vote
0 answers
19 views

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 (...
2 votes
1 answer
40 views

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 ...
  • 963
0 votes
1 answer
64 views

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 ...
  • 3
2 votes
0 answers
29 views

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 ...
  • 963
0 votes
0 answers
24 views

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 ...
4 votes
0 answers
58 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 ...
  • 569
0 votes
1 answer
42 views

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-...
  • 623
1 vote
1 answer
66 views

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 ...
1 vote
1 answer
83 views

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 = ...
  • 623
2 votes
1 answer
119 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\...
  • 564
3 votes
0 answers
62 views

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}\...
2 votes
1 answer
61 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 ...
  • 1,005
3 votes
2 answers
149 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 = \...
  • 1,577
3 votes
1 answer
127 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 ...
4 votes
1 answer
59 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\...
2 votes
1 answer
42 views

A local martingale subtract half its quadratic variation tends to negative infinity

Let $M$ be a continuous local martingale with $M_0=0$ and $[M]_\infty=\infty$ almost surely. I am required to show that $M_t-\frac{1}{2}[M]_t\to-\infty$ almost surely as $t\to\infty$. Unfortunately I ...
5 votes
1 answer
116 views

Probability of stopping time being finite.

Let $X$ be a continuous non-negative local martingale with $X_0=1$ and $X_t\to0$ almost surely as $t\to\infty$. For $a>1$, let $\tau_a=\inf\{t\geq0:X_t>a\}$. I am tasked with showing that $\...
0 votes
0 answers
34 views

If $X$ is a discrete supermartingale and $S$ is a bounded stopping time, then $X^{S}$ is also a supermartingale

In discrete time, suppose $X_{t}$ is a supermartingale. Consider the stopped process $X^{S}_{t}(\omega) = X_{S(\omega)\wedge t}(\omega)$, I want to show it is still a supermartingale. We have the ...
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0 votes
1 answer
100 views

betting in fair game over infinite horizon, is it possible to win?

If a gambler were to play in a fair game, lets say he wins/loses 1 dollar with equal probability in each step. Let $X_i$ denote the amount of money he has after $i$ steps. And he plays until he either ...
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1 vote
0 answers
59 views

Prove $X_t=\sqrt{t} \sin B_t$ is a martingale, if $B_t$ is a Brownian motion

I previously asked whether $X_t=\sin B_t$ is a martingale, where $B_t$ is a standard Brownian motion and the underlined filtration is the canonical one. As it can be seen in the answer provided, this ...
  • 698
2 votes
1 answer
112 views

Properities of sine(and cosine) of Brownian Motion

Suppose $(B_t, \mathcal{F}_t)_{t \geq 0}$ is a classical Brownian Motion (with the canonical filtration) and consider the process $X_t=\sin(B_t)$. What properties does our new process share with a ...
  • 698
0 votes
1 answer
56 views

A question from the proof of function of the Brownian motion is a martingale.

This is the theorem 7.5.8 in Durrett book. If $u(t,x)$ is the polynomial in $x$ and $t$ with $$ \frac{\partial u}{\partial t}+\frac{1}{2}\frac{\partial^2 u}{\partial x^2}=0 $$ Then $u(t,B_t)$ is a ...
  • 1,577
2 votes
0 answers
84 views

Is a Riccati BSDE explicitly solvable?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
  • 569
2 votes
1 answer
46 views

Why $\mathbb{E}[(S_{\tau}-\tau \mathbb{E}[\xi_1])^2]=\mathbb{E}[\tau] \sigma^2$?

My problem: Let $(\xi_n)_{n \geq 1}$ be a sequence of bounded independent and identically distributed random variables and $S_n = \sum_{k=1}^{n} \xi_k$. Let $\mathcal{F}_n = \sigma (\xi_k : k \leq n)$ ...
3 votes
2 answers
70 views

Solving a linear backward stochastic differential equation

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $\{B_t\}_{t\in[0;T]}$ be an adapted process and ...
  • 569
2 votes
1 answer
151 views

Why a martingale should have limit at infinity?

Problem: suppose $(M_t)_{t \geq 0}$ is right-continuous and a martingale with respect to a filtration $(\mathcal{F}_t)_{t \geq 0}$, i.e. for all $0 \leq s \leq t$ $\mathbb{E}[M_t | \mathcal{F}_s]=M_s$....
0 votes
0 answers
29 views

How to show that $E[|Y_n|]<\infty$ in martingale?

Let $X_1,X_2,..$ are i.i.d. random variables with $E(|X_1|)<\infty$. Define the partial sum $S_n = X_1+X_2+...+X_n$. Show that $Y_n=S_n-n \mu$ is a martingale where $E(X_1)=\mu$. I took $\mathcal{F}...
  • 1,577
3 votes
0 answers
62 views

How to prove that $p$ is an Hermite polynomial?

Problem: Let $$H_n(x):=(-1)^n e ^{\frac{x^2}{2}} \frac{d^n}{dx^n}(e ^{-\frac{x^2}{2}})$$ be the $n^{th}$-Hermite polynomial and $$G_n(t,x):=t^{\frac n 2}H_n(\frac {x}{\sqrt{t}})$$ Moreover, let $(B_t)...
0 votes
0 answers
33 views

Kolmogorov's Maximal Inequality: Supremum and Maximum

I feel kinda stupid for asking this, but anyway, I have a quick questionabout Kolmogorov's Maximal Inequality, which can be stated as follows: Let $ (X_n)_{n \in \mathcal{Z}} $ be a sequence of ...
1 vote
0 answers
137 views

Ito's Lemma - Stochastic differential equation

Question. Apply Ito's formula to calculate the stochastic differential of: $X_t = e^{at} \cos(b W_t)$ Determine for which values of $a, b \in \mathbb{R}$ the process $X_t$ is a martingale. Attempt Set ...
1 vote
0 answers
30 views

$\mathbb{E}[(\int_{0}^{t} V_s^2: d\langle M^{c}\rangle_{s})^{\frac{1}{2}}]<\infty$ implies $N_t:= \int_0^t V_s \cdot d M_{s}^{c}$ is a martingale.

Suppose that $M$ is a continuous martingale and $V_s$ is a progressive process with $\mathbb{E}\left[\left(\int_{0}^{t} Tr(V_sV_s^T d\left\langle M\right\rangle_{s})\right)^{\frac{1}{2}}\right]<\...
  • 409
3 votes
1 answer
134 views

Show that a local martingale is a true martingale if and only if it is a process of class DL

Let $M$ be a local martingale. Show that $M$ is a (true) martingale if and only if it is a process of class DL. Quick definitions: $\mathscr{S}_a$ is the class of all stopping times $T$ such that $P(...
  • 2,479
1 vote
1 answer
51 views

How to show that $\sup_n(|X_n|) < \infty$?

Suppose that $X_1,X_2,....$ is a martingale satisfying $E[X_1]=0$ and $E[X_n^2] <\infty$. Assume that $\sum_n E[(X_n-X_{n-1})^2] < \infty$. Prove that $X_n$ converges with probability $1$. I am ...
  • 1,577
1 vote
1 answer
60 views

How to show that $X_n$ fails to converge to $0$ in $\mathcal{L}_1$ as $ n \rightarrow \infty$?

Suppose $W_1,W_2,....$ are independent and identically distributed random variables such that $P(W_1=0)=P(W_1=2) =1/2$. For each $n=1,2,3...$ define the random variable $X_n = W_1.W_2...W_n$. Show ...
  • 1,577
0 votes
0 answers
35 views

proof verification: problem 1.3.24 from Shreve and Karatzas ii

Assume that $\{X_{t}, \mathcal{F_{t}}, 0 \leq t < \infty\}$ is a right-continuous submartingale and $S \leq T$ are stopping times of $\mathcal{F_{t}}$. Then $\mathbb{E}[X_{T\wedge t}|\mathcal{F_{S}}...
  • 347
0 votes
1 answer
35 views

Law of total expectation on product of RVs

I am working through a problem where I wish to claim that: $$ \mathbb E \left[\prod_{i=0}^n \frac{f(X_i)}{\mathbb E [f(X_i) \ | \ \mathcal F_i]} \right] = \mathbb E \left[\prod_{i=0}^n \frac{f(X_i)}{f(...

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