**0**

votes

**1**answer

181 views

### 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 ...

**2**

votes

**1**answer

208 views

### 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 ...

**2**

votes

**0**answers

65 views

### 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 ...

**5**

votes

**0**answers

384 views

### 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 ...

**1**

vote

**0**answers

308 views

### 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 ...

**1**

vote

**1**answer

397 views

### 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 ...

**0**

votes

**1**answer

368 views

### 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 ...

**1**

vote

**0**answers

112 views

### 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}
...

**1**

vote

**0**answers

88 views

### 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 ...

**1**

vote

**0**answers

208 views

### 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 ...

**0**

votes

**2**answers

1k views

### 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 ...

**2**

votes

**3**answers

355 views

### 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 ...

**1**

vote

**1**answer

502 views

### 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 ...

**2**

votes

**1**answer

145 views

### 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 ...

**1**

vote

**1**answer

275 views

### 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)| ...

**4**

votes

**2**answers

477 views

### 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 ...

**2**

votes

**1**answer

193 views

### 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 ...

**0**

votes

**1**answer

102 views

### 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 ...

**0**

votes

**1**answer

265 views

### 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 ...

**1**

vote

**0**answers

53 views

### 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 ...

**2**

votes

**0**answers

147 views

### 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 ...

**1**

vote

**1**answer

127 views

### 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 ...

**3**

votes

**1**answer

153 views

### 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 ...

**4**

votes

**1**answer

286 views

### 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:
...

**0**

votes

**2**answers

179 views

### 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 ...

**0**

votes

**1**answer

79 views

### 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

**2**

votes

**0**answers

160 views

### Computing a density function for the integral of a stochastic process, given its transition function

$P$ is a one-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | ...

**4**

votes

**0**answers

187 views

### Integrating a Bessel Bridge

Preliminaries
An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. ...

**13**

votes

**1**answer

833 views

### Bochner integral of stochastic process = path by path Lebesgue integral?

After some helpful comments, I realized that I had to repost this question in a more systematic way.
On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...

**2**

votes

**1**answer

160 views

### contraction property for conditioned SDEs

Consider a strictly convex potential $U: \mathbb{R}^d \to \mathbb{R}$ and the Langevin diffusion $$dX = -\nabla U(X) dt + dW \qquad (*)$$ where $W$ is a standard Brownian motion. If $(X_t)_{t \geq 0}$ ...

**1**

vote

**1**answer

330 views

### Solving an Ornstein-Uhlenbeck-like SDE $y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t]$

I have asked a similar question involving some finance background some time ago here math.stackexchange, however no really good answer came up. I was able to find a solution at least for a special ...

**0**

votes

**2**answers

357 views

### Representation theorem for continuous uniformly integrable martingales

For some time $u$ and positive continuous process $a_t$ adapted to $\mathcal{F}_t$ I have a (continuous-time) martingale defined as:
$$M_t(u) = \mathbb{E}[a_u | \mathcal{F}_t]$$
for $t\leq u$. I ...

**1**

vote

**1**answer

226 views

### SDE-removal of the diffusion coefficients

from math.stackexchange
I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have
\begin{align}
dX_t=b(X_t)dt+\sigma dW_t,
\end{align}
...

**1**

vote

**0**answers

73 views

### Potentials of class D

A potential $\pi_t$ is a positive supermartingale with the condition that $\mathbb{E}[\pi_t]\rightarrow 0$ as $t \rightarrow 0$. What are the necessary/sufficient conditions for a potential to be of ...

**1**

vote

**2**answers

179 views

### market completion in stochastic volatility model

Hi all,
Consider a stochastic volatility model. As there are two sources of risk and one asset only, this is an imcomplete market. One can complete the market by considering a derivative V1 used to ...

**1**

vote

**0**answers

106 views

### stochastic volatility valuation equation

I'm trying to derive the valuation equation under a general stochastic volatility model. What one can read in the litterature is the following reasonning:
One consider a replicating self-financing ...

**2**

votes

**1**answer

336 views

### A wrong proof of Squared Bessel process

The squared Bessel process with $\delta$-dimension for $\delta>0$,
denoted by $BESQ^\delta(y)$, is given by
$$d Y_t = \delta t + 2 \sqrt{Y_t} d B_t, \ Y_0 = y\ge 0$$
where $B_t$ is BM under ...

**2**

votes

**0**answers

205 views

### Is this process strictly positive?

Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider 1-D stochastic differential equation
$$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$
We assume $\sigma(0) = 0$, and ...

**2**

votes

**0**answers

97 views

### Does this series stopping times marching forward?

Let $W_t$ is standard Brownian motion under probability measure $P$.
Consider stochastic differential equation
$$ dY_t = dt + Y_t dW_t, \ Y_0 = 0.$$
Note that, the above SDE has a strong non-negative ...

**0**

votes

**1**answer

260 views

### Stochastic processes with random matrices

I am currently working on complex networks. I consider a matrix $\cal N$ with random entries $\delta_{ik}$. These entries are varying randomly in time and so I have a sequence of random matrices that ...

**1**

vote

**1**answer

159 views

### References/Papers on analytic solutions to SDEs

Does anyone know of any good references/research papers on finding analytic solutions to stochastic differential equations and/or finding approximating solutions to such a system?
I am particularly ...

**11**

votes

**0**answers

694 views

### surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...

**2**

votes

**1**answer

110 views

### change the sign of volatility

Assume the time inhomogeneous SDE
$dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$
has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE
...

**4**

votes

**1**answer

247 views

### 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} ...

**0**

votes

**0**answers

170 views

### Expected value of a logarithm of a Levy process

I have a strictly positive Levy process $(L_t)$ with no Brownian part, drift $\gamma$ and jump measure $\nu$. Is it possible to calculate the expected value of the logarithm of this process, so ...

**3**

votes

**1**answer

639 views

### karhunen-Loeve expansion of Poisson process

Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda ...

**3**

votes

**0**answers

179 views

### Time reversibility of Stratonovich Diffusion: Reference Request

Please consider the Stratonovich stochastic differential equation (SDE)
$$
dX = b(X)\circ dB
$$
where $B$ is standard Brownian motion and $X(0)=X_0$. This corresponds to the Ito (SDE)
$$
dX = ...

**3**

votes

**0**answers

178 views

### Joint distribution of Ito integral and its quadratic varation

Any idea on solving the joint distribution of
$X_T=\int_0^T \alpha_t dZ_t$ and $Y_T=\int_0^T \alpha_t^2 dt$ ? Here $X_T$ is an Ito integral and $Z_t$ is a standard Brownian process. When $\alpha_t$ ...

**0**

votes

**1**answer

354 views

### Calculate $\mathbb{E}[\int_o^T N_{t-}dS_t]$ - what went wrong?

First note, I had asked a similar question here, but the thread seems to have died, so I'll revive it here with more details. As a simplification of my real problem, I want to compute
...

**3**

votes

**0**answers

150 views

### stochastic control / geometric mean

Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...