# Questions tagged [stochastic-differential-equations]

Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.

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### Divergence form degenerate pde and Feynman Kac

Consider
$$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$
and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...

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58 views

### How to calculate the probability of 2 events happening in time series under only cdf information?

In time domain $0\rightarrow T$, there are two independent events $A$ and $B$.
$B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...

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86 views

### Reference from the article “Random Ordinary Differential Equations”, by J.L. Strand

In the article Random Ordinary Differential Equations, Journal of differential equations 7, 538-553 (1970), by J.L. Strand, reference number 6 refers to his PhD thesis: Stochastic Ordinary ...

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84 views

### When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$
$$\mathcal{F}^{\...

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### Can there be a explicit expression of g as defined in the link

This is related to the paper in the link :https://arxiv.org/pdf/1610.08468.pdf titled Algebraic normalisation of regularity structures. In the method of re- normalization the functional $g$ shown in ...

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30 views

### Characterization of Time-homogeneous flows for conditional expectation

Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...

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27 views

### Show that the transition semigroup of the strong solution to a Langevin-type SDE is immediately differentiable

Let
$\varrho\in C^1(\mathbb R)$ with $\varrho>0$
$\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$
$\mu$ denote the measure with density $\varrho$ with respect to $\lambda$
$b:=2^{-...

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37 views

### Stability of the Langevin semigroup under $C_c^\infty(\mathbb R)$

Let
$h\in C^2(\mathbb R)$
$(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$ be a continuous process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$X^x_t=x-\frac12\int_0^th'(X^x_s)...

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### Application of Gram-Charlier expansion for Swaption pricing with drift extension

I am doing a project for university and I'm stuck at some point.
The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension.
I already found out how ...

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36 views

### Reference: Stochastic Optimal Control with cost functional

There are a variety of control problems for controlled diffusions $X_t^u$, with the terminal cost given by
$$
J(u)\triangleq \mathbb{E}\left[g(X_T,u)+\int_0^t h(X_t,u_t)ds\right],
$$
function $g$ and ...

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91 views

### Additive stochastic heat equation and Markov property

For the distribution-valued field called the GFF (Gaussian free field) over a domain $\Omega\subset \mathbb{R}^{2}$:
$$ h_{\Omega}=\sum a_{k}f_{k,\Omega}(z),$$
where $a_{k}\sim N(0,1)$ and $f_{k},\...

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54 views

### invariant measure theory for spdes with distributional solutions (Hilbert versus Polish)

SPDEs such as the stochastic heat equation for $d\geq 2$ with space-time white noise and the stochastic quantization equation have distributional solutions and we still try to make sense of their ...

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72 views

### Vanishing viscosity method and random forcing

The vanishing viscosity method consists in viewing problem:
$$(A) \hspace{1cm}
u_t+g(u)_x = 0,\\[2ex]
$$
as the limit of the problem:
$$(B) \hspace{1cm}
u_t+g(u)_x+\nu \varDelta u = 0,\\[2ex]...

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60 views

### Ornstein-Uhlenbeck type process with thresholding

(Edited) I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding:
$$
dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0,
$$
where $...

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21 views

### Pathwise closeness of solutions of SDE's

Given two SDE's, if the diffusion coefficients are pathwise uniformly close, can we say the same about the solutions to corresponding SDE's?
More precisely, consider
$$
dX^1_t = b(X^1_t) dt + \sigma^...

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25 views

### Conditions for a affine term structure

I have an stochastic process $r_t$ (short rate model) with dynamics
$dr(t)=a(t,r(t)) \, dt + b(t,r(t)) \, dW(t)$,
where $W$ is a standard brownian motion.
I want to show:
If there is a function $F(...

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57 views

### Solution of stochastic ODE stationary?

Consider the following ODE:
$$\frac{ d \gamma(x,t;\tau)}{d \tau} = R(\gamma(x,t;\tau)) ; \qquad \gamma(x,t,t)=x. $$
$R$ is smooth enough, bounded away from zero and a stationary process. Is there a ...

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133 views

### stochastic recurrence relation “convergence”

Consider a recurrence relation $x_n = f(x_{n-1})$. Suppose $f(x_n)<0$ for some "large enough" $n$. Now consider a "stochastic variant" $x_n = f(x_{n-1})+y_n$ where $y_n$ is a sequence of random ...

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52 views

### Parametric distribution where the parameter follows a diffusion process

I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$:
$$\mu(\theta)\...

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57 views

### If $(Φ^x)_{x∈ℝ}$ is a family of real-valued stochastic processes and $B$ is a Brownian motion, then $\int_0^tΦ^x_s\:dB_s=(\int_0^t\Phi_s\:dB_s)(x)$

Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$
$B$ be a (standard, real-valued) $\mathcal F$...

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89 views

### Embedding a martingale by SDE

Let me reformulate my question. Let $(X_0,X_T)$ be a martingale on $\mathbb R$, then it is known that one has a SDE:
$$Z_t=Z_0+\int_0^t\sigma(s,Z_s)dB_s, \mbox{ for all } t\in [0,T]~~~~~~~~~~~~~~(\...

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81 views

### hitting time of an Ornstein-Ulhenbeck

If we consider a nice Ornstein Uhlenbeck process
$d x (t) = - \gamma x(t) dt + \sigma d w (t)$
with $x(0) = x_0 \in (-L,L)$.
Here $\gamma, \sigma$ are positive constant and $w(t)$ is a Wiener process....

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75 views

### Weak solutions of linear parabolic PDEs and corresponding SDEs

It is well known that for an Stochastic differential equation (on the real line) of the form:
$dX_t = \mu(X_t)dt + \sigma(X_t)dW$
where $W$ is the standard Wiener process, the transition probability ...

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73 views

### Law of motion when initial condition is perturbed

We know how to find the law of motion (Ito process) of the value function:
$$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$
such that
$$dX_t=\mu(t,X_t)...