# 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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119 views

### Ito lemma for manifold semimartingales

I'm looking for a generalization of the usual Ito lemma to manifolds $M$, preferably not under the assumption that $M$ is embedded in $\mathbb{R}^d$. Unfortunately any reference I've found either ...

**2**

votes

**1**answer

93 views

### Modified square root process

I am dealing with the following stochastic differential equation (SDE)
$ \begin{cases} dS_t &= \mu S_t dt + \sigma_1 S_tdW^1_t\\dG_t &= kS_t(\alpha - G_t)dt + \sigma_2\sqrt{G_tS_t}dW^2_t
\end{...

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**0**answers

65 views

### stochastic differential equations via Hida distributions?

Can one prove the solvability of stochastic Ito differential equations via Hida distributions?

**0**

votes

**1**answer

113 views

### Probability that SDE visits any point

This is a reference request question.
Statement: I am interested in an SDE of the form
\begin{equation}\fbox{1}~~~
{\rm d}X_t = f(X_t)\,{\rm d} t + g(t) \, {\rm d} B_t
\end{equation}
Where we ...

**1**

vote

**0**answers

24 views

### Generalized polynomial chaos with 2D independent uniform variables for first order equation

I'm trying to transform the first order differential equation
$\dot{x} = k x(t)$
into the corresponding set of stochastic differential equations. I have two independent uniformly distributed ...

**1**

vote

**0**answers

49 views

### Matching Numbers in Ito McKean

Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as
$e_1 = \lim_{b \...

**2**

votes

**1**answer

231 views

### Any modern/recent version of Ito & McKean?

This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their ...

**0**

votes

**0**answers

35 views

### first eigenvalue and relaxation time for ergodic diffusions

As a prototype example of ergodic diffusion, consider a real valued Ornstein Uhlenbeck process of the form
$$
d X_t = -\alpha X_t d t + \sigma d W_t, \quad X_0 = x
$$
where $\alpha, \sigma > 0$ ...

**2**

votes

**1**answer

218 views

### Weak convergence of sum of log normal random variables

Let $S_t$ be the Geometric Brownian Motion, we know that
$$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$
and the distribution of $S_t$ is known explicitly. Please see the ...

**2**

votes

**1**answer

649 views

### Variance of Multi-Dimensional Ornstein Uhlenbeck process

I am trying to compute the asymptotic variance of OU process
$$
d X_t = - H X_t dt + S dW_t
$$
where $X_t$ takes value in $R^d$. $H$ and $S$ are $d\times d$ matrices that does not have $HS = SH$ in ...

**1**

vote

**1**answer

154 views

### Solutions to linear SDE with many noise sources

It is well known how to solve the linear stochastic ODEs with one source of noise
$$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$
See, for instance, https://math.stackexchange.com/questions/1788853/...

**3**

votes

**1**answer

87 views

### inhomogeneous Ornstein-Ulhenbeck process / invariant probability measure

Let $\gamma$ be a continuous function from $\mathbb{R}^+$ to $\mathbb{R}^+$ and consider the real valued inhomogeneous Ornstein-Uhlenbeck process satisfying
$$
d X_t = -\gamma_t X_t d t + d W_t, \...

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votes

**0**answers

85 views

### Gaussian Processes and Paths

As we know, the only stationary Gaussian Markov process with continuous
autocorrelation function is the stationary OU process (Doob's theorem). To show this, one shows that the autocorrelation ...

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vote

**0**answers

30 views

### numerical scheme for SDE and empirical estimation of rate of convergence

Consider $\{X_t , t \geq 0 \}$ real valued diffusion satisfying
$$
d X_t = b(X_t) d t + \sigma (X_t) d W_t, \quad X_0 = x \in \mathbb{R}
$$
where $b, \sigma$ are well-behaved functions and $W$ is a ...

**5**

votes

**2**answers

310 views

### Reference for Feynman-Kac

I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...

**1**

vote

**1**answer

111 views

### Non-commutative Ito Formula

Does there exist a formula of the Ito lemma for matrix valued processes under matrix multiplication?
That is where
$$
\Delta X_{t+\Delta t} \neq X_{t+\Delta t} - X_t
$$
but instead
$$
\Delta X_t = ...

**1**

vote

**2**answers

104 views

### A solution to stochastic PDE $du(t)= a(t)u(t)\,dt +s(t)\,dz$

Is there a general (integral) solution to $du(t)= -a(t)u(t)\,dt +\sigma(t)\,dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s) \, ds}u(t_0)+\int_{t_0}^t \sigma(v) e^{-\int_v^t a(s) \, ds} \, dz(...

**1**

vote

**0**answers

48 views

### Time discretization of the (stochastic) Navier-Stokes equation

Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be nonnempty and open
$\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_{L^2(\Lambda,\:\mathbb R^d)}$
I've found a thesis where ...

**2**

votes

**0**answers

84 views

### Markov chain approximates a fractional diffusion

Let assume that
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R}
$$
Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...

**3**

votes

**0**answers

87 views

### Monte-carlo estimation on the drift of SDE

On any probability space $(\Omega, \mathcal{F} , \mathbb{P})$ with a Brownian motion $W$, we consider the following one-dimensional SDE:
$$dX_t = F(t, X_t) \, dt + dW_t,$$
where $F(t,x) = \mathbb{...

**5**

votes

**1**answer

267 views

### Reference: Stochastic Analysis on Hilbert Manifolds

I'm looking for a reference to a book which develops an It\^{o} lemma for semi-martingales with values in infinite dimensional Hilbert-Manifolds. I expect the techniques to be the same but still I ...

**3**

votes

**2**answers

398 views

### Is the “hybrid” Black-Scholes Hull-White model arbitrage free?

Given a "hybrid" Black-Scholes Hull White (BSHW) model. That is, the stock price is modelled by a Black Scholes SDE:
\begin{equation} dS(t) = \mu(t)S(t)dt + \sigma_{S}(t)S(t)dW^{\mathbb{P}}_{S}(t)
\...

**2**

votes

**0**answers

115 views

### Feynman-Kac formula for *general* Sturm-Liouville operator

One way to state (omitting technical requirements) the Feynman-Kac formula that I am familiar with is as follows.
Let $u$ be a solution to the pde
$$u_t(x,t)=-\frac{\sigma^2(x,t)}2u_{xx}(x,t)-V(x,t)u(...

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votes

**2**answers

159 views

### Reference: Non-smooth Ito Lemma

Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth?
I've been looking but have not found much, any ...

**6**

votes

**0**answers

81 views

### Reference Request: Vector-Valued Ito Formula

I know that there exist Ito formulae to understand
$
f(X),
$
where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale.
However I'm ...

**0**

votes

**0**answers

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)\...

**1**

vote

**0**answers

163 views

### Mean and Variance of SDE

What is the mean and the variance of $y_t$, given the following SDE:
$dy_t = -x_t y_t dt + \sigma_1 dW^1_t$
$dx_t = -\sigma_2 y_t dW^2_t$
$W^1$ and $W^2$ are (possibly correlated) Wiener ...

**1**

vote

**1**answer

869 views

### Mixing the Ornstein-Uhlenbeck Process and Geometric Brownian Motion

The Ornstein-Uhlenbeck process with mean reversion level 0 is defined as follows:
$$dX_t=a X_t dt + \sigma_1 d W_{1t}. \tag{1} $$
Geometric Brownian motion is defined as follows:
$$dX_t= a X_t dt + ...

**5**

votes

**1**answer

355 views

### Definition of the nonlinear part of the drift in a (stochastic) Navier-Stokes equation

Let
$T>0$
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\overline{\mathcal V}^{\left\|\;\cdot\;\...

**0**

votes

**0**answers

56 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) $\mathcal F$...

**1**

vote

**0**answers

136 views

### Transforming reaction-diffusion equations to random walk processes

I have a two species reaction-diffusion system which is a Turing-type (activator-inhibitor) equation. I am trying to transform my reaction-diffusion system into a system of multiple walkers on a ...

**2**

votes

**1**answer

303 views

### On representing a continuous time Markov chain by a stochastic integral of a Poisson random measure

Let $Q=(q_{ij})$ be the transition rate matrix of a continuous time Markov chain $\{ X_t \}$ with countable state space $M$. Let $q_i = -q_{ii}=\sum_{j \neq i}q_{ij}$, and let $\Gamma_{ij}$ be defined ...

**0**

votes

**0**answers

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]~~~~~~~~~~~~~~(\...

**2**

votes

**0**answers

88 views

### I've found a representation of the Itō-Stratonovich correction term and don't understand the used notion of a “trace”

Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a ...

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votes

**0**answers

51 views

### Perscribed/Inverting Conditional Expectation

I'm having difficulty finding papers which deal with the following inversion problem.
Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...

**2**

votes

**0**answers

182 views

### Definition of the Stratonovich integral in Hilbert spaces

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

**1**

vote

**2**answers

456 views

### Deriving the HJB equation for exponential utility

I would like to derive the HJB equation for the following stochastic optimal control problem:
$ \Phi(t,x)=\sup_{h} E \left[\exp \left\{\gamma \int_t^T g(X_s,h(s);\gamma)\ ds \right\} \right]$
where ...

**1**

vote

**2**answers

163 views

### Conditional Wiener measure continuous

consider a complete Riemannian manifold $M$ with heat kernel $p_M$ and let $U\subset M$ be an open set. Let $W_{x,t}^{y}$ be the Wiener measure associated to the Brownian motion starting at $x$ and ...

**2**

votes

**0**answers

89 views

### Well-posedness of a stochastic differential equation in the Stratonovich sense

Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$
$(B_t)_{t\in[0,\:T]}$ be an $(\mathcal F_t)_{...

**2**

votes

**1**answer

280 views

### Reflecting Brownian motion and its transition probability density

I have a question about reflecting Brownian motion on an unbounded domain.
Let us consider the reflecting Brownian motion $\{X_t\}_{t \ge 0}$ on the following domain $\bar{D}$ of $\mathbb{R}^2$:
\...

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vote

**0**answers

142 views

### Conditional Expectation of solution to SDE

Suppose $g \in C^{1,2,2}$ and $X_t$ and $Y_t$ solve some SDEs
$$
dY_t = a(t,X_t,Y_t)dt +b(t,X_t,Y_t)dW_t
$$
and
$$
dX_t = c(t,X_t)dt + d(t,X_t)d\tilde{W}_t
$$, then Iio's lemma implies
$$
g(t,X_t,...

**4**

votes

**1**answer

79 views

### Time Integral over the (positive) Innovations of a Stochastic Process

Consider the Itô-Process
$$X(t) = X_{0} + \int_{0}^{t}\mu(s,X(s))\,ds + \int_{0}^{t}\sigma(s,X(s))\,dW(s)$$
where you can safely assume that the drift $\mu$ and the volatility $\sigma$ satisfy the ...

**1**

vote

**0**answers

24 views

### Usually trivial Excursion-type process

How Can i construct a stochastic process $X_t$ which has the property that:
$X_t \in [0,1]$ for all $t \in [0,T]$ and
$m(\{t \in [0,T] : X_t>0 \})\leq \delta$, for some pre-chosen $\delta \in [0,T]...

**2**

votes

**1**answer

88 views

### Joint distribution of integrals of diffusion and driving noise

Consider a generic diffusion of the form
$$dX_t=f(t,X_t)dt+dB_t,$$
where $f$ is some nice function and $B_t$ is a standard Brownian motion.
The marginal distributions of the integrals
$$I:=\int_0^...

**1**

vote

**0**answers

61 views

### Stochastic Inverse

Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying
$$
\int_0^Tf(H_t,...

**3**

votes

**0**answers

89 views

### When we integrate with respect to a $Q$-Wiener process on $U$, why do we restrict integrands to be operators on $Q^{1/2}U$ (instead of $U$)?

When we integrate with respect to a $Q$-Wiener process $(W_t)_{t\ge 0}$ ($Q$ being a bounded, linear, nonnegative and self-adjoint operator on a separable $\mathbb R$-Hilbert space $U$ with finite ...

**4**

votes

**1**answer

246 views

### Variance and expectation of timed-change squared Bessel process

Let $X_t$ be a squared Bessel process satisfying the SDE:
$$
dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t
$$
and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...

**2**

votes

**0**answers

43 views

### Minimizer of a class of SDEs

Setup
Let $\mathscr{H}$ be a separable Hilbert space, $\mathcal{X}\triangleq \langle \Omega,\mathscr{F},\mathscr{F}_t,\mathbb{P}\rangle$ be a stochastic base and $X_t$ be an $H$-valued stochastic ...

**2**

votes

**1**answer

210 views

### Generalizing HJB equation for a terminal stopping time

The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation:
Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[...

**2**

votes

**1**answer

214 views

### Solving a matrix ODE

Consider the linear matrix differential equation
$\def\diag{\mathrm{diag}}$
\begin{align}
U(0) &= I\\
\frac{\mathrm{d}U}{\mathrm{d}t}(t) &= U(t) \phantom{.} Q(t) & & \quad(1)
\end{...