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Questions tagged [stochastic-pde]

Stochastic partial differential equations are partial differential equations with a random driving force. Please do not use this tag just because there are stochastic processes and differential equations in your question. Consider if [tag: SDE] is a better choice. This tag is only to be used for PDEs driven by noise.

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Computing $dY^{-1}(t)$ using the SDE $dY(t)=\mu Y(t)dt+\sigma Y(t)dW(t)$

Let $\mu$ and $\sigma$ be constants and consider the SDE $dY(t)=\mu Y(t)dt+\sigma Y(t)dW(t)$ with $W(t)$ Brownian motion and $Y(0)=y_{0}$. Using the solution to the SDE, $Y(t)=y_{0}\exp[(\mu-\frac{\...
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39 views

Stochastic Integration $\int_0^T \exp[W(t)-t/2]\,\mathrm d W(t)$ [on hold]

Using Ito formula with time dependence integrate or any way you want $$\int_0^T \exp[W(t)-t/2]\,\mathrm d W(t)$$
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18 views

Black-Scholes formula with expression for strike price

I have the ordinary Black-Scholes formula that reads: $C = S(0) \Phi (d_1) - K e^{-r(T-t_0)}\Phi(d_2)$ Given the following strike price: $K=e^{r(T-t_0)}S(0)$ I have reduced the above formula as ...
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21 views

Transforming a stochastic Fokker Planck equation in a deterministic one

I have the following stochastic Fokker Planck equation, coming from a double well potential $d p = \left[ (x^3 - a x) \frac{\partial p}{\partial x} + (3 x^2 -a ) p + \frac{1}{2} \sigma^2 \frac{\...
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44 views

Change of variables in stochastic PDE

I have the following stochastic partial differential equation (SPDE): $d v = -\mu \frac{\partial v}{\partial x} dt + \frac{1}{2} \frac{\partial^2 v}{\partial x^2} dt - \sqrt{\rho} \frac{\partial v}{\...
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24 views

Finding relationship between solutions of two different first order linear PDEs (using Feynman-Kac)

The first partial differential equation, using the shorthand $\mathcal{L}u(t,x) = \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t)$, is: ...
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25 views

What is the Feynman Kac Formula for the Merton model

I know that for the diffusion process $$ X_t = \mu(t, X_t) dt + \sigma(t, X_t) dB_t, $$ the function $$ u(t, x) = \mathbb{E}_{x, t}[e^{\int_t^T r(s, X_s) ds} g(X_T)] $$ with the boundary condition $u(...
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25 views

Existence of the solution of the 3D Micropolar equations [closed]

Please how to show the local existence for the solution of the 3D micropolar equations in a Besov space setting ? $\left\{ \begin{array}{l} \partial_tu-(\nu+k)\Delta u-2k\nabla\times w+u\nabla u+\...
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31 views

Kolmogorov backward equation intuition

The Kolmogorov backward equation equation states that the probability density of a random variable $x$ which follows $dx= \mu dt + \sigma dw$ is given by $-p_t = \mu p_x + 0.5\sigma^2 p_{xx} $ ...
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42 views

Volatility in an at-the-money call option

I understand that the vega of the Black-Scholes equation is a positive function, which means the value of the option is an INCREASING function of the volatility, since vega is the derivative of the ...
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16 views

Existence of a unique solution of $dX_t=X_tB_tdB_t+X_tB_tdt, X_0=1$

Does this SDE have a unique solution: $$dX_t=X_tB_tdB_t+X_tB_tdt, \quad X_0=1?$$ I have to check if the Lipschitz condition holds and if the growth condition holds. I presume not, as the Brownian ...
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20 views

What is $E[X_1]$ where $dX_t=(1-X_t^2)^{-1}dB_t,$ $X_0=1$?

I have a stochastic equation $$dX_t=(1-X_t^2)^{-1}dB_t,$$ $X_0=1$ and $B$ is a Brownian motion. I know there exists a strong solution $X$. How can I compute $$E[X_1]?$$ I thought about trying showing ...
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34 views

Fokker-Planck equation for a Markov semigroup with densities

Let $(E,\mathcal E)$ be a measurable space $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$ and $$\kappa_tf:=\int\kappa_t(\;\cdot\;,{\rm d}y)f(y)$$ for $$f\in F_0:=\left\{f:E\to\mathbb ...
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32 views

Finding a strong solution to $X_t := (A_tX_t+a_t)dt+(S_tX_t+\sigma_t))dB_t$

I have an SDE $$X_t := (A_tX_t+a_t)dt+(S_tX_t+\sigma_t))dB_t,$$ $X_0=x_0$ and $A,a,\sigma,S$ are continuous stochastic processes, $B$ is a BM. Now if I define: $$Y_t:=e^{(\int_0^tA_sds+\int_0^...
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34 views

Standard Brownian motion almost surely not $0$. [duplicate]

Consider a continuous standard brownian motion $(B_t)_{t\ge 0}$. I want to show, that $$\mathcal{L}(\{t\ge 0: B_t=0\})=0$$ I also have a hint that I need to show that $B:\mathbb{R}_+\times\Omega\...
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1answer
41 views

Deterministic Integral of a Predictable Process is Predictable

I was reviewing a proof of existence of solutions to stochastic evolution equations which takes the form of a fixed point argument on the space of predictable processes such that $$ \sup_{t\leq T}\...
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52 views

What is a stationary solution to a SPDE?

I'm reading Hairer's notes on SPDEs: http://www.hairer.org/notes/SPDEs.pdf He says on page 6 that "the stationary solution to the stochastic heat equation is Gaussian free field". He never defines ...
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31 views

White noise is not a signed measure for fixed $\omega$

I'm going over some materials on stochastic analysis, and stuck with a problem on Gaussian white noise: Let $(\mathbb{R}^d,\mathcal{B},m)$ be the Borel measurable space on $\mathbb{R}^d$. A white ...
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31 views

When is it necessary to solve Kolmogorov forward equations (KFE) for a Markov Chain?

Say I have a continuous time markov chain, time homogeneous $X$ with a few states (say, 2). I want to know the distribution of where $X$ is at time $t$, call it $\mu_t$, which will be a vector of 2 ...
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21 views

Stochastic Fubini (Da Prato & Zabcyzk)

I've been studying Da Prato & Zabcyzk's text on SPDE, and there's a part of their proof of the stochastic Fubini theorem that I don't quite get. The setting is that $\Phi: (\Omega_T\times E, \...
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51 views

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

As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even ...
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86 views

Applying Itô's formula to logged Ornstein-Uhlenbeck process

I have the following O-U process: $$d \log z_t =-\nu \log z_t d_t + \bar\sigma dW_t \tag{1}$$ and want to apply Itô's formula as: $$dy=df(x)=\bigg(\mu(x)f'(x)+\frac{1}{2}\sigma^2(x)f''(x) \bigg)dt + ...
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1answer
73 views

Spectral density of stochastic partial differential equations

I am trying to understand some examples of computing the spectral density of SPDEs. The root of my confusion is that some terms appear to be missing from the calculation of the squared transform, and ...
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1answer
59 views

Solution of the non-linear Heat Equation

How to find $v$ such that $$u(x,t)=t^{-\alpha}v(xt^{-\beta})$$ is the solution of the non-linear Heat equation : $$u_t-\Delta(u^{\gamma})=0$$where $\frac{n-2}{n}<\gamma<1$ , $x$ $\in R^n$ ...
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49 views

In what sense is the product Gaussian measure on $\Bbb R^{\Bbb R}$ a Gaussian measure?

I'm reading Hairer's notes on SPDEs here. On page 17 he has remark 3.33 where he considers $\Bbb R^{\Bbb R}$ with the product $\sigma$-algebra and product Gaussian measure. He claims the Cameron ...
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29 views

Does the sde explode?

We consider the sde $\,\, d X_{t} = d B_{t} - \nabla{U(X_{t})}dt\,\, $ in $\mathbb{R}^{d}$. Is the translation invariance of the drift $\nabla{U}$ enough to ensure that the sde doesn't explode ...
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49 views

Deriving forward Fokker-Planck equation.

I am struggling understanding the derivation of the forward Fokker-Planck equation from the Chapman-Kolmogorov equation. My book is very messy and I have wasted a lot of time with it. Could you please ...
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160 views

Geometric Brownian Motion With Mean Reversion

Is it possible to generate this mean-reverting Ornstein-Uhlenbeck stochastic differential equation $$ dx_t = \theta(\mu - x_t)\text{d}t + \sigma \text{d}W_t $$ in Excel? Are there any mean-reverting ...
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May I ask why the following operator between Banach spaces is continuous please?

I am reading the paper "On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces" (Here is the link for electronic version). But when reading it, I meet a problem and would ...
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26 views

Calculate the dynamics of ZCB given forward-rate dynamics

Suppose the forward rate is given by: $$df(t,u)= \frac{\partial}{\partial u} \bigg( \frac{\sigma(t,u)^2}{2} \bigg)dt - \frac{\partial}{\partial u} \big( \sigma(t,u) \big)dW_t$$ where $W_t$ is a ...
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1answer
170 views

Reference complementing Hairer's “Introduction to Stochastic PDEs”

I want to study stochastic partial differential equations, and the standard reference seems to be Hairer's notes - after all, he got the first Fields Medal for work on this topic. However, I also ...
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109 views

How to solve Fokker-Planck PDE for Brownian particle in square potential driven by periodic time-dependent force

Problem statement: We have a Brownian particle in harmonic potential with additional time-dependent force. Langevin equation(mass taken to be unit): $\ddot{x} + \gamma \dot{x} + \omega_0^2 x = \...
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96 views

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\...
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14 views

Uniform Convergence in Expectation of Diffusion

Suppose that $X_t$ is a diffusion process on $\mathbb{R}$ and define the functionals $F_t$ from $L^2_{\mathbb{P}}(\mathfrak{F}_t)$ to $\mathbb{R}$ by $$ F_t: Y\mapsto \mathbb{E}\left[ (X_{t} - Y)^2 \...
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1answer
34 views

Is the negation regarding a measure zero set necessarily some positive measure set?

Suppose $z(t,k,\omega)$ is a sequence of index $k\in\mathbb N$ of stochastic process continuous in $t\in[0,T]$ for every $\omega\in\Omega$ where $\Omega$ is the sample space. Is following the correct ...
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66 views

Perturbation of a stochastic differential equation

Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively \begin{align} dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\ dx &= -(k_0(t)+\epsilon ...
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2answers
232 views

Prove stochastic exponential to be a martingale

I'm new in stochastic integral, and recently I met a problem as following: Let $B$ be a Brownian motion, $\mu_t,\sigma_t$ be uniformly bounded progressive processes and $\sigma_t>\epsilon>0$ ...
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23 views

Is there a good textbook on Itō integrals depending on a spatial parameter?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...
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205 views

Seriously struggling with Ito's lemma, and understanding what this paper does!!!

So, this is a theorem that I need to apply on a simpler system, and understanding it thus far has been the bane of my existence. I was wondering how exactly both Ito's formula and the Burkholder-...
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24 views

Why can we assume that the range of the generator of a $C_0$ semigroup is already the full space?

I am reading this script: http://www.hairer.org/notes/SPDEs.pdf about stochastic partial differential equations. I have a problem with the footnote on page 46. $L$ is the generator of some $C_0$ ...
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114 views

Please ignore the scary length of the proof! I only query two lines! (Brownian Motion Notation/ Doob's Theorem)

Please ignore the scary length of the proof! I only query two lines! So, I have two questions about the following part of the theorem. It is a paper looking at Stochastic Partial Differential ...
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75 views

Integral of second moment

I am trying to solve the function $$\frac{\partial \left \langle f^{2} \right \rangle}{\partial t} = \frac{\left \langle f \right \rangle - \left \langle f^{2} \right \rangle} {N}$$ where $\left \...
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320 views

Arithmetic Brownian Motion

An arithmetic brownian motion $X$ which follows $dX = \mu dt + \sigma dZt$ , where $μ$ and $\sigma$ are constants with an asset price $ S = X^2$. Use Ito's Lemma, find the DE satisfied by the process ...
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69 views

Why does the Malliavin derivative of a Markovian semigroup being strong Feller imply the semigroup strong Feller?

I am reading "Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction" by Arnaud Debussche In it, he claims that for $\varphi \in B_b(H)$, a bounded measurable function on a ...
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81 views

Martingale problems and SPDEs

It is a classical result that if $X$ is a process with values in $\mathbb{R}^d$ and \begin{align*} M_t^i = X_t^i - \int_0^t b_i(X_s) ds \\ M_t^i M_t^j - \int_0^t a_{ij} (X_s) ds \end{align*} are both (...
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49 views

If $M,N$ are martingales on a Hilbert space, then the Radon-Nikodým derivatives $\frac{{\rm d}[⟨M,e_m⟩,⟨N,e_n⟩]}{{\rm d}([M]+[N])}$ are summable

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a filtration on $(\Omega,\mathcal A)$ $U$ and $\tilde U$ be infinite-...
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1answer
65 views

Fourier transform of KPZ equation

What is the Fourier transform of $|\partial f(x,t)/\partial x|^{2}$? To put the question in some context, I was trying to write the KPZ equation in Fourier space. The KPZ equation is given by $\...
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64 views

Connection SDE and PDE problem with time dependent probability measure

I am just working my way through a talk from an Italian researcher and I don´t get one of his points. Let´s look at these two slides. I don´t understand why he says on slide 2 that $u_t$ is a ...
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143 views

How does the Kunita-Watanabe identity generalize to stochastic integration on Hilbert spaces?

Let $U,H$ be a separable $\mathbb R$-Hilbert spaces, $M$ be a $U$-valued square-integrable martingale on a filitered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$...
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1answer
96 views

Understanding the “distribution-valued” solution of a SPDE

Consider the Additive Stochastic Heat Equation (ASHE) given by $$ \begin{cases} \partial_t u = \Delta u + \mathcal{\dot{W}}\\ u(x,0) = \xi \end{cases} $$ where $\mathcal{\dot{W}}$ is a white noise and ...