# 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.

**0**

votes

**0**answers

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 ...

**2**

votes

**0**answers

85 views

### Ito formula between manifolds

I have seen many Ito formulae giving dynamics for $f(X_t)$ where $f:M\to \mathbb{R}$ is a smooth function from a manifold $M$ and $X_t$ is a (continous) manifold-valued semi-martingale.
My question ...

**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 = ...

**4**

votes

**1**answer

320 views

### Existence of normal number except random numbers

For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence.
Now, is there any number that is normal ...

**0**

votes

**2**answers

380 views

### Kolmogorov continuity theorem and Holder norm

The Kolmogorov Continuity theorem (see for example the Wikipedia page) lets us prove that a stochastic process $X_t$ (on some complete metric space $(S,d)$) is Holder continuous almost surely provided ...

**1**

vote

**0**answers

57 views

### Onsager-Machlup Function of a Killed Diffusion Process

Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) =...

**1**

vote

**0**answers

224 views

### Construction of the quadratic variation for Hilbert space valued local martingales

Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...

**-1**

votes

**0**answers

27 views

### Finding an SDE given constraints over the final distribution

I want to find the SDE
$$
dX_t=\mu(X_t)dt+\sigma(X_t)dW_t
$$
such that, if the initial condition is $x_{t=0}=x_0$, we have the following constraints:
$\mathbb{E} X_t = \alpha x_0^{\beta}$
$\mathbb{V} ...

**1**

vote

**0**answers

66 views

### Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined

Remark: I've asked this question on MSE as well.
Let
$T>0$
$I:=[0,T]$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...

**3**

votes

**2**answers

178 views

### Large deviation bound for O-U process

Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...

**1**

vote

**0**answers

74 views

### Ito's formula for jump diffusions

Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$...

**1**

vote

**0**answers

47 views

### Stochastic Control with Stochastic Cost-functional

Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also?
That is, let $X_t^u$ is the solution to a controlled SDE
$$
dX_t = \mu(t,u_t,X_t^u)dt ...

**3**

votes

**1**answer

102 views

### Sequence of diffusions

Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?

**2**

votes

**1**answer

284 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 $\mathbb{R}^2$:
\...

**2**

votes

**1**answer

94 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{...

**0**

votes

**1**answer

114 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 ...

**2**

votes

**0**answers

61 views

### Smoothness of Value function for SDE with discontinuous coefficients

Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function $v:\...

**2**

votes

**0**answers

210 views

### Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?

If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...

**2**

votes

**0**answers

141 views

### Boundary behavior for Ito diffusions

The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...

**2**

votes

**0**answers

121 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 ...

**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

693 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 ...

**2**

votes

**0**answers

65 views

### stochastic differential equations via Hida distributions?

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

**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 ...

**2**

votes

**1**answer

233 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 ...

**-1**

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

223 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 ...

**20**

votes

**1**answer

459 views

### Does a theory of stochastic differential algebras exist?

My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...

**1**

vote

**1**answer

159 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

89 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, \...

**2**

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 ...

**1**

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

318 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

**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

50 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 ...

**1**

vote

**1**answer

307 views

### Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...

**3**

votes

**0**answers

92 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

268 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

409 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

117 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(...

**0**

votes

**2**answers

165 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

83 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

166 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

908 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 + ...

**7**

votes

**1**answer

260 views

### Solve SDE $dX_t=(c+\sigma_\zeta W'_t)X_tdt + \sigma_\epsilon dW_t$

I am trying to solve the following SDE
$$dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$$
$c\in \mathbb{R}$ is a constant, $X_t$ is a stochastic process, $\sigma_\zeta,\sigma_\epsilon \in \...

**1**

vote

**2**answers

465 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 ...

**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 ...

**0**

votes

**0**answers

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) $\mathcal F$...