15 votes
Accepted

Reference complementing Hairer's "Introduction to Stochastic PDEs"

I like Hairer's notes, but of course these are lecture notes. And some introductory books can be easier to read since there are more explanations. There are basically three approaches to analysing ...
Cahn's user avatar
  • 4,641
7 votes

Why is the gaussian free field a distribution but Brownian motion is a function?

I would argue that the GFF is not so much a generalization of Brownian motion, but rather that the one-dimensional GFF just happens to be Brownian motion. To explore this further, let $\varphi = \{\...
Jason's user avatar
  • 15.4k
4 votes
Accepted

Hölder continuity definition through distributions.

Your steps are the right ones, in this answer I'll fill in the details. Firstly, let's check that $g$ defines an $\alpha$-Holder function. Let $\rho$ be a smooth probability density with support in $B(...
Rhys Steele's user avatar
  • 19.7k
4 votes

Scaling of space-time white noise

As I had the same question as you, and also to answer Conrado Costa's question, I tried to do said covariance calculation and came up with the following heuristic argument: Space-time white noise is ...
Andre's user avatar
  • 1,540
4 votes

Why is the gaussian free field a distribution but Brownian motion is a function?

A Gaussian free field, in any dimension, is a Gaussian field with covariance given by the Green's function, $G(x,y)$ satisfying $\Delta G(x,\cdot)=\delta(\cdot-x)$. In dimension $1$, $G(x,\cdot)$ is a ...
Kostya_I's user avatar
  • 1,235
4 votes
Accepted

Characterization of $C^{k,\alpha}$ (functions with Hölder continuous derivatives) through Taylor estimates

We have $$ |f(x+h)-P_{x}(h)|\leq c|h|^{k+\alpha}% $$ for all $x\in\mathbb{R}$ and all $h$ with $|h|\leq1$. Write $$ P_{x}(h)=\sum_{n=0}^{k}\frac{1}{n!}a_{n}(x)h^{n}. $$ Taking $h=0$ you get $f(x)=a_{0}...
Gio67's user avatar
  • 20.9k
4 votes
Accepted

Derivative of a Stochastic Integral with respect to Limit & with respect to Integrator

For question 1, you are correct that $F$ is not differentiable with respect to $t$. It's not exactly clear what it would mean for $F$ to be differentiable with respect to $W_t$ because in general we ...
user6247850's user avatar
  • 13.5k
3 votes
Accepted

Volatility in an at-the-money call option

This is a very astute observation and good question which reveals a seemingly paradoxical (and commonly known) aspect of the Black-Scholes model. In the Black-Scholes model, the underlying asset ...
RRL's user avatar
  • 90.8k
3 votes
Accepted

Student T distribution as a solution of a differential equation

The pdf of the t-distribution is a solution to the following differential equation: $$ { {\begin{array}{l}\left(\nu +x^{2}\right)f'(x)+(\nu +1)xf(x)=0,\\ \\ \text{with }f(1)={\frac {\nu ^{\nu /2}(\nu +...
alexjo's user avatar
  • 15k
3 votes
Accepted

Infinitesimal generator of the Brownian motion on a circle

Starting from $$dY_1=-\frac12 Y_1 dt-Y_2 dB_t, \\ dY_2=-\frac12 Y_2 dt+Y_1 dB_t, $$ and using the general formula for the generator of a diffusion process $dX_t=f(X_t)dt+g(X_t)dB_t$, which reads $$\...
S.Surace's user avatar
  • 1,790
3 votes
Accepted

Martingale processes

Ito's formula is a wonderful tool, but it is overkill for this question. If $\alpha<0$, then $Y_t$ is undefined with positive probability for each $t>0$. If $\alpha>0$, then $Y_t>0$ a.s. ...
Yuval Peres's user avatar
2 votes
Accepted

Correlation of two processes that solve a system of SDE's

The quantity $\rho(t, S_t, Y_t)$ is the instantaneous, or local, correlation, which can be defined by $$d\langle S, Y\rangle_t/dt,$$ or $$corr(S_{t+\Delta}|{\mathscr{F}_t}, Y_{t+\Delta}|{\mathscr{F}_t}...
Gordon's user avatar
  • 4,461
2 votes
Accepted

Can a function $\varphi:\mathbb{R}^d\to \mathbb{R}$ be evaluated at a point in $\mathbb{R}$?

If you read correctly, it's not $(\varphi^n_x)\varphi^n_x(y)$. It's $(\Pi_xf(x))(\varphi^n_x)$ multiplied by $\varphi^n_x(y)$
Tryss's user avatar
  • 14.3k
2 votes

Stochastic differential equations

For each fixed $t \in [0,T]$, the first equality implies that $P(\{ \omega : X_t(\omega)=\hat{X}_t(\omega) \})=1$. (This is just saying that a nonnegative r.v. $X$ has mean zero only if it is $0$ a.s.,...
Ian's user avatar
  • 102k
2 votes

Solve $0=f'(x)-cf(x)$

we have $$\int \frac{f'(x)}{f(x)}dx$$ the soltion is $$\ln(|f(x)|)+C$$ if $f(x)\ne 0$ for all real $x$
Dr. Sonnhard Graubner's user avatar
2 votes
Accepted

Checking a proof using Markov's Inequality

As mentioned in the comments, this is obviously false as written, but is true if we restrict the supremum to be over $r\ge0$. As the comments also noted, Fernique's theorem is the way to go. Fernique'...
Jason's user avatar
  • 15.4k
2 votes

Deterministic Integral of a Predictable Process is Predictable

I think you only need the Borel measurability of $f$ for the predictability of $Y_t=f(t,X_t)$. $Y$ can be decomposed into $$ [0,\infty)\times\Omega\overset{\phi_X}{\rightarrow}[0,\infty)\times\mathbb ...
AddSup's user avatar
  • 774
2 votes
Accepted

Compute $E^{\mathbb{Q}} \left[ \exp(S_T) | \mathcal{F}_t \right] $ with exponential Vasicek Model (1978)

First, express $S_T$ in terms of $S_t$, which can be obtained from your solution in (1) \begin{eqnarray} S_T &=& e^{\alpha T} \left( S_0 + \sigma \int^t_0 e^{-\alpha s} dB_s + \sigma \int^...
Quanto's user avatar
  • 98k
2 votes

Change of variables in stochastic PDE

Proving these kind of phase shifts can indeed be difficult as the Ito calculus does not provide a nice chain rule as we have in deterministic calculus. There is however a trick to solve this problem. ...
C. Hamster's user avatar
2 votes
Accepted

The Cox-Ingersoll-Ross Model (1985)

The ODE approach is fine, but involved. There is a more convenient way to derive the results, especially so for $E_s[r_t^2]$. The trick is to work with $x_t=e^{\beta t}r_t$ and its corresponding sde ...
Quanto's user avatar
  • 98k
2 votes

The Riccatti equation for The Cox-Ingerson-Ross Model.

SOLVING YOUR ODE : $$B'(s) + \beta B(s) + \frac{1}{2} \sigma^2 B(s)^2 =1$$ Of course this is a Riccati ODE. But this is also a separable ODE. $$\frac{B'}{1-\beta B - \frac{1}{2} \sigma^2 B^...
JJacquelin's user avatar
  • 66.3k
2 votes
Accepted

Separability of Reproducing Kernel Hilbert Space of a positive definite covariance function.

From the reproducing property of the RKHS it follows that the canonical feature map $\Phi:T \to H(C):t \mapsto C(\cdot ,t)$ is continuous, since if $t_n \to t$ in $T$, then $$ ||C(\cdot,t_n)-C(\...
Andrei Kh's user avatar
  • 2,569
2 votes

A stochastic cannibalistic snail problem

This is purely a numerical answer. I used Matlab to solve the deterministic ODE and also to simulate realisations of the stochastic ODE. It seems like nothing unexpected happens, and on average the ...
David's user avatar
  • 2,464
2 votes

Space-time white noise grows at infinity?

One needs a suitable way to quantify the growth of a temperate Schwartz distribution $T$. We know the definition of being in $\mathscr{S}'$ somehow (in a rather mysterious way) says $T$ grows at most ...
Abdelmalek Abdesselam's user avatar
2 votes
Accepted

Solve the SDE $d X_{t}=X_{t} B_{t} d t-X_{t} d B_{t}$

This is similar to a geometric Brownian motion. Try the substitution $Y_t=\ln(X_t)$ (as long as $X_t>0$). Then $$ dY_t=\frac1{X_t}(X_tB_tdt−X_tdB_t)-\frac1{2X_t^2}X_t^2dt=(B_t-\tfrac12)\,dt-dB_t. $$...
Lutz Lehmann's user avatar
2 votes
Accepted

Differential of a a function of a semimartingale

You don't need to prove this from Ito's lemma because $d(A_t+B_t)=dA_t+dB_t$ holds for any two stochastic processes $A$ and $B$ by definition. If you still want to use Ito you should apply it to the ...
Kurt G.'s user avatar
  • 14.3k
2 votes
Accepted

Infinite convolution of a smooth compactly supported function converges uniformly

You have already done most of the work, well done! I think it is possible to show that $A_n$ is bounded. Let us simplify the recurrence relation. We define $B_n := \max_{k=0,\ldots,n} A_k$. Then one ...
supinf's user avatar
  • 13.4k

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