# 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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### 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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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{... 0answers 65 views ### stochastic differential equations via Hida distributions? Can one prove the solvability of stochastic Ito differential equations via Hida distributions? 1answer 113 views ### Probability that SDE visits any point This is a reference request question. Statement: I am interested in an SDE of the form $$\fbox{1}~~~ {\rm d}X_t = f(X_t)\,{\rm d} t + g(t) \, {\rm d} B_t$$ Where we ... 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 ... 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 \...
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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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### 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 $\alpha, \sigma > 0$ ...
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### 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 ...
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### 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 ...
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### 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/...
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### 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 ...
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### 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: dS(t) = \mu(t)S(t)dt + \sigma_{S}(t)S(t)dW^{\mathbb{P}}_{S}(t) \...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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]... 1answer 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 $$I:=\int_0^... 0answers 61 views ### Stochastic Inverse Let X_t be a semi-martingale and H_t be a predictable process and g be a measurable bijective function with measurable inverse. Does there exist a function f(h,x) satisfying$$ \int_0^Tf(H_t,... 0answers 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 ... 1answer 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 ... 0answers 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 ... 1answer 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$[...
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{...