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      All Questions

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      1answer
      85 views

      Approximation of function in general measure space

      Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with $$ \int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
      1
      vote
      2answers
      82 views

      Backward Stochastic Differential Equation

      Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and $$ dX_t=f_tdt+B_tdW_t $$ where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...
      1
      vote
      1answer
      144 views

      Feynman-Kac formula for lattice heat equation with non-diagonal potential

      Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let $$u(t,x):=\mathbf E\...
      2
      votes
      0answers
      43 views

      Floquet stochastic process

      Let $X_t$ be defined by the SDE $$ dX_t = A(t, X_t)dt + dW_t $$ where $A(t, X_t)$ is linear in $X_t$ and periodic in $t$. Assume also that the process is stable. If $A(\cdot)$ didn't have $t$ ...
      4
      votes
      1answer
      178 views

      Is there a Feynman-Kac formula for vector-valued Schrödinger operators?

      Given a vector function $$f=(f_1,\ldots,f_n)\in L^2(\mathbb R,\mathbb R^n)$$ (for some $n\in\mathbb N$), let us define $$\Delta f:=(\Delta f_1,\ldots,\Delta f_n),$$ where $\Delta$ is the Laplacian ...
      2
      votes
      0answers
      118 views

      Stochastic Approximation in Reproducing Kernel Hilbert Space

      Consider an iterative algorithm with incremental updates \begin{align} x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}], \end{align} where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
      2
      votes
      0answers
      76 views

      Stochastic Approximation Algorithms Converging to Local Equilibriums

      Consider the stochastic iterative updates \begin{align} \theta_{t+1} \leftarrow \theta_t + \alpha_t \cdot \left [ h(\theta_t) + M_t \right ], \end{align} where $\theta_t \in \mathrm{R}^d$, $h \colon ...
      1
      vote
      1answer
      216 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/...
      2
      votes
      1answer
      244 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{...
      2
      votes
      0answers
      221 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
      1answer
      341 views

      General solution to system of stochastic linear differential equations

      Assume we are given the system of linear stochastic differential equations $$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
      1
      vote
      1answer
      140 views

      Finding a stochastic differential equation as limit of a discrete stochastic equation

      I'm dealing with the following problem: Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation: $Z_{k+1}-Z_k=P_k(1-2Z_k)$ where $P_k=0$ with probability $...
      3
      votes
      0answers
      238 views

      Lorenz attractor power spectrum

      If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
      2
      votes
      1answer
      571 views

      Branching Brownian Motion and the KPP equation

      I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
      3
      votes
      2answers
      2k views

      Any suggestions on a rigorous stochastic differential equations book?

      I have been looking through some books and they are not very rigorous. Any suggestions would be great.

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