The stochastic-calculus tag has no wiki summary.

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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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### 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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### 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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### 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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69 views

### 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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### 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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### 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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### 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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### a generalization of Monge-Kantorovich Problem

I am thinking about the martingale version of Monge-Kantorovich Problem.
Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on ...

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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 ...

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### distribution of integral of exponential of wiener process

I am absolute newbie to stochastic calculus and have to solve a weighted hazard rates integral, where the hazard rates are stochastic, their logarithm governed by arithmetic Ornstein-Uhlenbeck (OU) ...

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### Is there a theory of SDEs whose coefficients are themselves adapted processes (i.e. “may depend on the past”)?

Is there an existence and uniqueness theorem for SDEs of the following type:
$dW_{t}=d\tilde{W}_{t}+\mu\left(\left(W_{s}\right)_{0\le s\le t},t\right)dt$,
where $\tilde{W}_{t}$ is say ...

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244 views

### Markov Chain: state reduction

Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following:
Firstly we have a Markov chain $\{Y_k\}$ with finite ...

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### Upper bound on the maxima of ratio of expectation of quantities under Gaussian measure

Let $\lambda,\eta >0$ be given, and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued function. Define
$$\Delta(u)= \frac{\int u(h) \exp(-\eta ...

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### Feynman-Kac for jump-diffusion

I'm looking for a more general Feynman-Kac formula that works in the case of jump-diffusion processes.
I know that, given a pure diffusion process like
$$dS_t=\mu_tdt+\sigma_tdW_t,$$ if $u(t,s)$ ...

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### Asymptotic behavior of solutions of stochastic differential equations

I am studying a risk model whose dynamic is specified by a first order differential equation with a compound Poisson process on the right hand side. I would like to know whether there are some papers ...

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### Supermartingale inequality on a particular event

Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a ...

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### Lyapunov function of exponential growth for existence of a solution of an SDE

Let
$$dX_t = a(X_t) dt + b(X_t) dW_t$$
be a one-dimensional stochastic differential equation, where the coefficients $a,b: \mathbb{R} \rightarrow \mathbb{R}$ satisfy for every ball $B_R$ the following ...

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### Malliavin calculus w.r.t $G$-Brownian motion

I wonder if it is possible to define a Malliavin calculus w.r.t $G$-Brownian motion defined on a Sublinear Expectation Space available on this link.
G–Brownian motion has a very rich and interesting ...

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311 views

### 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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### Existence of predictable quadratic covariation for a special pair of local martingales

In Limit theorems for stochastic processes, by Jacod and Shiryaev we have the existence of a predictable quadratic covariation process stated as the following theorem
$\mathbf{Theorem}$ To each ...

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### Colored noise in SDE

I want to numerically study the behavior of a system described by a set of differential equations in the presence of colored noise. It seems that the standard procedure is to use the Langevin ...

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### Conditional law of an Ito's stochastic integral

Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real ...

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### Tail for the integral of a diffusion process

I would like to compute the following tail,
$$
\mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right),
$$
assuming
$$
\mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x),
$$
and $X$ is a diffusion ...

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### When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form
$$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$
for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb ...

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### How is Kolmogorov forward equation derived from the theory of semigroup of operators?

In Lamperti's Stochastic Processes, given
a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$,
a ...

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### Iterated Ito Integral, Gaussian Volterra Process

Let me define
$$
J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1}
$$
where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice deterministic ...

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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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### Time integral of a diffusion

Define $\bar\sigma^2_t=\frac{1}{t}\int_0^t\sigma^2(X_s)ds$ where $\sigma(x)\geq0$ is a measurable function and $X_t$ a diffusion process defined by
\begin{equation}
...

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### Attractors and solutions to these generalized Ornstein–Uhlenbeck processes?

This is a question about generalized Ornstein–Uhlenbeck processes I asked on MSE, but I haven't received replies about their attractors and solutions yet. So I would appreciate if someone could give ...

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### What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?

There is a theorem as follows:
Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto ...

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### When are two operators simultaneously diagonalisable?

I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...

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### Ito formulae for stochastic processes with finite cubic, quartic … n-tic variation

Many stochastic processes that you encounter are kind of well-behaved, i.e. have infinite variation, yet finite quadratic variation.
My question revolves around stochastic processes that have ...

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### Good books on stochastic partial differential equations?

I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are ...

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### Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...

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### On martingale representation theorem

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\mathcal{F_{t}})_{0\le t\le T}$ a filtration generated by standard Brownian motion $W_t$.
Let $f(x)$ be $C^1$ function such that $|f'(x)| ...

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### Converse to Girsanov's theorem?

Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...

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### Limit of a Wiener integral

How to show that
$$ \lim_{\alpha \rightarrow \infty} \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.}$$
where $\left (B_s ...

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### Concerning Jump process (Lévy process)

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is
$$ \nu \left( dx\right) = A ...

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### Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral

Hi, I have the following expected value to compute
$E[ \int_{o}^{T} f(t) dt \int_{o}^{T} H(s) dW(s)]$,
where $f(t)$ and $H(s)$ are two stochastic processes adapted to the filtration generated by the ...

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### Maximal Principle for stochastic heat equation

Consider $\partial_{t}u=\partial_{xx}u$ with Neumannboundary conditon
$u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$.
Then up to time $T$, the maximal value of $u$ should be ...

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### Cameron-Martin like RKHS

Hello,
I know that $k(x,y)=min(x,y)$ is the reproducing kernel of the Cameron Martin space of all i.i.d. RVs of Brownian motion at different times, with the $cov$ inner product.
What is the RKHS ...

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### Can we express a one-dimensional raised Bessel Bridge as a function of a single Brownian Motion?

A Bessel Bridge is a Brownian Motion, conditioned such that $B(0) = B(1) = 0$ and $B([0, 1]) \ge 0$. A raised Bessel Bridge is a generalization of this: it's a Brownian Motion conditioned such that ...

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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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### Trajectorial version of Doob's $L^2$ inequality

In the paper http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf
you can find a trajectorial version of Doob's inequality. It is given by:
...

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### Properties of the Euler Discretization of a diffusion

Let $X$ be a continuous 1-d diffusion:
$$
dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x.
$$
W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties.
Let ...

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### Parameter Sensitivity of Stochastic Process

How do I compute the derivative \frac{\partial X_t}{\partial \sigma}? Where dX_t=\theta (\mu-X_t)dt+\sigma \sqrt{X_t}dZ_t