The stochastic-calculus tag has no wiki summary.

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### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,
\end{equation}
where ...

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### Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...

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### a special filtration satisfying $0$-$1$ law

Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let ...

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### Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent.
1): $X$ is $L^p$-integrable.
2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...

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### On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)

Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...

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### a question about Dambis, Dubins-Schwarz Theorem

Let $M=(M_t)_{0\le t\le 1}$ be a continous $\mathbb{F}=\{\mathcal{F}_t\}_{0\le t\le 1}$-martingale s.t. $M_0=0$. Now my question is whether there exists a Brownin motion $B$ s.t.
...

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### Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:
$$dX_t = f(X_t)\,dt + dW_t,$$
where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...

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### Discrete version of Ito's lemma

Could anyone give me some references where I could find
(a) discrete version(s) of Ito's lemma
(b) a proof how it converges to the continuous form in the limit
(c) its usage within stochastic ...

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### explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition:
$$V_t=V_0+\int_0^tH_sdX_s-K_t$$
where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...

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### a question about the modification of a supermartingale

Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process
...

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### Looking for a limit related to the series in a previous post

Can any one show that the following limit?
$$
\lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1.
$$
If one uses the ...

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### Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$,
the sequence $(M^n)_n$ converges.
Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices ...

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### How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration?
$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question here , ...

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71 views

### a question about integration by parts

Let $X$ be a cadlag martingale et $Y$ be a process of bounded variation, do we have the integration by parts formulae?
$$\int_0^1Y_tdX_t=X_1Y_1-X_0Y_0-\int_0^1X_tdY_t,~ a.s.$$
Thanks for the reply!

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### question about Doob-Meyer decomposition

Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:
...

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### Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...

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### Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...

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### Ito Integral of a SDE [on hold]

I would like to get help in solving the following It\^o stochastic equation:
$dY_t=-dW_t \, (Y_t^2+1)$
The process $W_t$ is the standard Brownian motion.
If possible I would like to get a path ...

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### Law of the $L^2$ norm of a Brownian motion and related

Let $B_t$ be a Brownian motion with variance 1. We know that $\int_0^1 B(t) \mathrm{d} t \sim \mathcal{N}(0,1/3)$. I am interested to know what we can say about the law of the two random variables
$X ...

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### log-likelihood of ito diffusion

Consider a diffusion process:
$ \text{d}X_t = f(X_t)\text{d}t + \text{d}W_t$
I've seen it given that the log-likelihood of the path is proportional to the Onsager-Machlup functional
$ \int_0^T ...

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### Stieltjes integrals of predictable processes

I am looking for a direct proof of the fact that, roughly speaking, if $S=S_0+A+M$ is an $L^2$ semimartingale, and $M$ (the martingale part) has the martingale representation property, then for any ...

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### On the solution of a stochastic partial differential equation

Consider a simple SPDE as follows:
$\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$,
$u(t,0)=u(t,1)=0$,
$u(0,x)=v(x)$,
where $V$ is a bounded, smooth ...

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### The probability of Levy process staying at a point

Assume $X_{t}$ is a 1-dimensional Levy process on a probability $(\Omega, \mathcal{F}, P)$. For a fixed point $x$ in the state space and fixed $t\neq 0$, what's the value of $ P(\omega: ...

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257 views

### Can every discrete martingale be embedded in a continuous martingale?

Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...

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### a dominated convergence theorem for martingale (II)

The question is presented in
a dominated convergence theorem for martingale
Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability space. (which means ...

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### What are the difference between modeling with stochastic differential equations (SDE) and ordinary differential equations (ODE) with a random force?

There are lots of differences between SDE and ODE. From the theoretical point of view an also from the numerical algorithms used for simulations. But I am interested in knowing if there is a point ...

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### The jump and the left martingale of semimartingale

Let $X_{t}$ be a semimartingale. Define
$\Delta X_{t} = X_{t}- X_{t-}$.
For fixed $s> 0$, $\Delta X_{s}$ and $X_{s-}$ are two random variable. Are they independent to each other? I think the ...

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### The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$.
This property is called the ...

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### The distribution of Jump gaps of Levy process

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...

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### Can this two-dimensional process self intersect?

I would like to know more about the two-dimensional processes derived from Brownian motion by the following stochastic differential equation (in the Ito sense)
$$dX_t = f(X_t) dt + ...

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### Solving a SDE with quadratic drift

I am wondering whether the following SDE can be solved explicitly?
$$
d X_t = X_t^2 d t + X_t d B_t
$$
where $B_t$ is a standard Brownian motion. If not, can we say some thing about the moments of ...

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257 views

### Martingale part of the discontinuous put payoff

I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$
$d[(S_t -K)^+ ]$ ??
I guess I need to use local times but how?

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### Ito formula for max(X,0) where X is a semimartingale

Has anyone ever applied the Ito formula on $|X^+|^2$ for $X^+ = \max(X,0)$ with
$X(t) = X(0) + M(t) + V(t)$, where $M(t)$ is a local martingale and $V(t)$ is bounded variation process. I found it in ...

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### Domino Shuffling and Warren's process

In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the ...

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### Can $<.>$ of a martingale determine it only?

Let $\Omega$ be the space of continuous functions defined on $[0,1]$. Define the canonical process $B$ by
$$B_t(\omega)=\omega_t,~ \forall\omega\in\Omega$$
Let us equip $\Omega$ with the usual ...

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### On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients

Consider the SPDE $$\frac{\partial}{\partial t}u_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u_t(x) + u_t(x)(K-u_t(x)) + \sigma u_t(x) \xi(t,x),$$
where $(t,x)\in {\mathbb R}_+\times ...

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### a $L^1$ convergence for backward martingale

I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ...

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### What is the optimal growth of the constant in BDG?

Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} ...

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### On the expected value of a random integral:

Is it possible to find the expected value of $u(t)$ in terms of the following information:
$$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$
where:
$X_s$ is a wide sense stationary process with known ...

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### Any suggestions on a rigorous stochastic differential equations book?

I have been looking through some books and they are not very rigorous. Any suggestions would be great.

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### On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form

Denote $E = C([0, 1])$. I am consider a 1-dimentional stochastic heat equation on $h$:
$\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x)$, for all $(t, x) \in (0, ...

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### How to deal with the vector norm item as a denominator in this expectation?

Hello, everyone.
I want to calculate the expectation shown in the following formula, where $X$ follows a standard $d$-dimensional multi-variable normal distribution as ...

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### Stochastic calculus for several inputs

In "On the Gap Between Deterministic and Stochastic Ordinary Differential Equations," The Annals of Probability, Vol. 6, No. 1 (Feb., 1978), pp. 19-41, Hector J. Sussmann showed that a stochastic ...

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### Multiplicative version of Novikov inequality for Ito integral

It is clear that Ito isometry
$E(∫^t_0fdW)^2=E(∫^t_0f^2dt)$
can be written in the multiplicative form as
$E(∫^t_0fdW\cdot∫^t_0gdW)=E(∫^t_0f⋅gdt).$
Is it possible to obtain the multiplicative version ...

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### When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?

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### Time change for non-homogeneous Markov processes

Background: Let $C$ be the space of continuous function on $[0,T]$, $f, \sigma \in C$ bounded with $\sigma^2 \geq \varepsilon > 0$ and let $X=(X_t)_{t\in [0,T]}$ be a diffusion process of ...

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### construction of a approximate martingale

everyone.
Given a probabilistic space $(\Omega, \mathcal{F}_t, \mathbb{P})$ and a martingale $(M_t)_{t\leq 1}$ on it. Suppose
$$M_1\stackrel{\mathbb{P}}{\sim}\mu$$
where $\mu$ is a probability ...

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### stochastic calculus

I am trying to understand Ito's lemma for Poisson type processes and here is my question:
Assume I have a jump process given by the stochastic equation:
$dp(t,T_1)= r_1p(t,T_1)1_{\tau<=t}dt + ...

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133 views

### Stochastic integral with respect to discontinuous martingale

in my research, I need to deal with a stochastic integral with respect to a compensated poisson process, namely,
$ \int_0^t f(X_t) dM_t,$
where $M(t) = N(t) - \int_0^t \lambda(s)ds$.
The integrand ...

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### Example of Girsanov change of density with finite relative entropy, but with infinite integral over squared changed drift

Let $(\Omega, (\mathcal F_t), \mathbb P)$ denote the usual Wiener space where $\Omega = C[0,\infty)$, etc., and where $(W_t)_{t \geq 0}$ denotes the Wiener process.
Let $Z \in L^1(\mathbb P)$ with $Z ...