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Suppose $X$ is a process given by - $dX_t = db_t$ where $b_t$ is a standard Brownian motion with its filtration $(\mathcal{F}_t)$. Suppose an agent earns a payoff given by $V(x) = \mathbb{E} [\in …

I am having an SDE for which I would be in trouble if there were no strong solution. The SDE is - $ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$ where $b_1$ and $b_2$ are two indepe …

I have a conjecture that seems rather obvious but the proof seems elusive. Consider a diffusion given by, $dX_t = \mu(X_t) dt + \sigma(X_t) db_t$ where $b_t$ is a standard Brownian motion. $\mu,\ …

Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\mat …

Consider a diffusion given by, $d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$ $X_0 = x$. Suppose the functions $\mu$ and $\sigma$ are as follows - $f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \text{ …

We know that for if $X$ is a stochastic integral of the form below - $X_t = \int_0^t v(s,\omega) db(s,\omega)$. then we can use time change formula to claim that $X_t = W_{\alpha(t)}$ where $W$ is …

Suppose $B_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb P\rangle$. Consider two SDEs below. Suppose, $X_0 = Y_0 = 0$ \beg …

Suppose that $X_t$ is a strong solution to the SDE, $$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by t …