I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \frac{1}{2} \Delta u$ with initial data $u(0,x)=f(x)$, considered on the whole space. Here the solution is given by $u(t,x)=\mathbb{E}(f(x+W_t))$ where $W_t$ is a […]

Let$^1$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\ge0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$ $H:=\mathbb R^d$ for some $d\in\mathbb N$ and $\xi:\Omega\times[0,\infty)\times\mathbb R^d\to\operatorname{HS}(U_0,H)$, where $U_0:=Q^{1/2}U$. Suppose we’re concerned with an SPDE $${\rm d}u_t\left(\Phi_t\left(x\right)\right)=f_t\left(\Phi_t\left(x\right)\right){\rm d}t+\nabla […]

$$ \newcommand{\mcl}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\avg}[1]{\langle#1 \rangle} \newcommand{\pth}[1]{\left( #1 \right)} \newcommand{\bck}[1]{\left\{ #1 \right\}} \newcommand{\sbck}[1]{\left[ #1 \right]} \newcommand{\bsbck}[1]{\Big[ #1 \Big]} \newcommand{\deriv}[1]{\frac{\mrm{d} #1}{\mrm{d}t}} $$ I recently learned that stochastic differential equations (SDEs) had a whole lot of theory behind them and I am genuinely surprised at the complexity that a “simple” noise term seems to introduce in the numerical […]

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are Borel-measurable functions. Furthermore suppose that we have a strong solution $X$ to the SDE with initial condition $\xi$ if $X_0 […]

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 filtration of $\mathcal A$ $B$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $b,\sigma:I\times\mathbb R\to\mathbb R$ be Borel measurable Consider the Itō equation $${\rm d}X_t=\underbrace{b(t,X_t)}_{=:\:\varphi_t}{\rm d}t+\underbrace{\sigma(t,X_t)}_{=:\:\Phi_t}{\rm d}B_t\;\;\;\text{for all }t\in I\tag1$$ and the Stratonovich equation $${\rm d}X_t=b(t,X_t){\rm […]

I want to solve the stochastic differential equation $$\mathrm dX(t)=B(t)X(t)\mathrm dt+B(t)X(t)\mathrm dB(t)$$ with condition $X(0)=1$.

slightly related:Is there a specific term for this SDE? Using $f(x_t, t ) = x_te^{at}$, then $df(x_t,t) = ax_te^{at} + e^{at}dx_t$ We can plug in the original question, $dX_t = aX_tdt + bdW_t$ which results in $df(x_t,t) = be^{-at}dW_t$ then I get, $x_te^{-at} = \int_0^t{x_te^{-rt}dW_t}$ From here, I am unsure of how to get to […]

Stock price has a classic model based on GBM: $$dS = \mu S dt + \sigma S dW$$ based on this call options values could be solve — Black-Scholes formula. But, what is the solution for the Stock price itself? is it $$S(t) = S(0) e^{\mu t + \sigma W}$$ ?

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ are given deterministic Borel functions on $\mathbb{R}_+$ that are bounded on each compact time interval. To find […]

For SDE’s of the general form $$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t \tag{1}$$ @saz taught me that there is a formula to transform it into a linear SDE, quoting from René L. Schilling/Lothar Partzsch: Brownian motion – An Introduction to Stochastic Processes, p.278. However I don’t have the book. So I’m wondering, […]

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