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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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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
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0answers
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 ...
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1answer
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
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1answer
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 ...
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2answers
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 ...
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0answers
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) =...
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0answers
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 ...
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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} ...
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0answers
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
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2answers
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 \...
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0answers
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 $h_i$...
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0answers
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
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1answer
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
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1answer
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
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1answer
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{...
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1answer
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
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0answers
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
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0answers
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 $$ \dot u = -u + f(a) $$ $$ \...
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0answers
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 ...
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0answers
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 ...
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0answers
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
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1answer
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
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0answers
65 views

stochastic differential equations via Hida distributions?

Can one prove the solvability of stochastic Ito differential equations via Hida distributions?
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0answers
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
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1answer
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 ...
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0answers
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
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1answer
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
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1answer
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
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1answer
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
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1answer
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, \...
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0answers
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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0answers
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 ...
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2answers
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
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2answers
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(...
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0answers
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
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0answers
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
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1answer
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
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0answers
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
1answer
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
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2answers
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
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0answers
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(...
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2answers
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
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0answers
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 ...
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0answers
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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0answers
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
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1answer
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
1answer
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
2answers
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
1answer
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
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0answers
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) $\mathcal F$...