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Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

181 questions
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Relative entropy for martingale measures

I need some help understanding a note given in a lot of papers I've read. Let $(\Omega,\mathcal{F},P)$ be a complete probability Space, $\mathbb{F} = (\mathcal{F}_t)_{t\in[0,T]}$ a given filtration with usual conditions, $S$ be a locally bounded…
Gono
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Exit and hitting times for the Bessel process $\textrm{d}X_t=\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t$

I am trying to analyse the exit time $T_1:=\inf\{t:X_t\notin[\alpha,2]\}$ and hitting time $T_2:=\inf\{t:X_t=0\}$, where $\alpha<1$ is a constant, and $X_t$ follows the Bessel process defined by the…
user107224
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How to show this is not a martingale.

Be advised that I cross-posted this question on MathOverflow. You can find it in this link: https://mathoverflow.net/questions/352152/show-that-this-process-is-not-a-martingale Assume we have the following stochastic process: $$X_t=\int_0^t…
Chaos
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Uniformly integrable local martingale

Can someone give me an example of a uniformly integrable local martingale that is not a martingale? Or are all U.I. local martingales true martingales (continuous, of course).
Protawn
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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…
verygoodbloke
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Prove that a martingale with a spatial parameter is differentiable

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $M:\Omega\times[0,\infty)\times\mathbb R^d\to\mathbb R$ such…
0xbadf00d
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Proving martingale property of $N_t = Z(M_{t\wedge s} - M_{t \wedge r})$ for martingale $M$

(Stochastic calculus and Brownian motion, LeGall, page 80). Suppose $M = (M_t)$ is a martingale. Also, let $Z$ be a bounded random variable which is $\mathcal{F}_r$ adapted. Then we like to show that for any $0 \leq r < s$, $$N_t = Z(M_{t\wedge…
Nick
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Show that $(S_t - M_t) \phi(S_t) = \Phi(S_t) - \int_{0}^{t} \phi(S_s) dM_s$

Show that $(S_t - M_t) \phi(S_t) = \Phi(S_t) - \int_{0}^{t} \phi(S_s) dM_s$ (This is from Le Gall's book, Brownian Motion, Martingales, and Stochastic Calculus.) Here, $M$ is a continuous local martingale, $S_t = \sup_{0 \leq s \leq t} M_s$,…
cgmil
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Stopped local martingale as a martingale

I am reading the Dubins-Schwarz theoem on Brownian motion and stochastic calculus. It says, given a continuous local martingale $M$ such that $\lim_{n\to\infty}[M]_t = \infty$ a.s., where $[M]_t$ denotes the quadratic variation of $M$. For each $s…
Kenneth Ng
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stochastic exponential uniformly integrable martingale

$N$ is a continuous local martingale and $T_c:=\inf\left\{t>0:\left[N\right]_t>c\right\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable martingale if and only if $$ \liminf_{c\rightarrow…
Mathfreak
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Show local martingale

I have $\exp(\lambda X_t-\frac{\lambda ^2}{2}t)$ is a local martingale, now i have to know if $X_t$ is also a local martingale. Can anybody help me how i can show this correctly?
daniäla
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The expected squared increment of a continuous local martingale

Suppose $M=\{ M_t\}_{t\geq 0}$ is a continuous local martingale, and $M_0=0$. Then I often see the following equation $$\mathbb{E}M_t^2=\mathbb{E}\sum_i(M_{t_{i+1}}^2-M_{t_i}^2)=\mathbb{E}\sum_i(M_{t_{i+1}}-M_{t_i})^2.$$ I am new to stochastic…
Yuyi Zhang
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Potential Local Martingale property derived from its quadratic variation

Suppose we have a continuous local martingale $M$ such that $\langle M \rangle_t =o(t)$ - i.e. $$\lim_{t \rightarrow 0} \frac{\langle M \rangle_t}{t} = 0$$ Does this imply that $$\lim_{t \rightarrow 0} \frac{M_t}{t} = 0 $$ as well? i.e. is $M$ also…
qp212223
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local martingale remains local martingale when lowering the localizing stopping times

Assume we have a stochastic process $(X_t)_{t \geq 0}$ which is a local martingale in respect to some filtration $F=(F_t)_{t \geq 0}$. That means by definition that there exists an almost surely increasing sequence of stopping times $(T_k)_{k \in…
Myrkuls JayKay
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Interpretation of Gisarnovs theorem

Ive found a statement of Girsarnovs theorem that looks as follows "Every $P$-semimartingale is a $Q$ semimartingale, in particular if $M$ is a local martingale then $\hat{M}_{t}=M_{t}-D_{t}^{-1}[M,D]_{t}$ is a $Q$ local martingale. Let…
Number4
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