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For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, wh …

Consider a stochastic differential equation (SDE) on some filtered probability space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$ : for all $t>0$ $$dX_t = u_tf(X_{t-})dt+ u_t g(X_{t-})dW_t + u_t\int_{ …

Let $X$ be the solution to some stochastic differential equation (unidimensional or multidimensional) : $$dX_t = b(t,X_t)\,dt + a(t,X_t)\,dW_t\quad \forall t\ge 0,$$ where $b, a$ are both Lipschitz. L …

Let $X$ be the solution to (real-valued) stochastic differential equation : $$dX_t = b(t,X_t)dt + a(t,X_t)dW_t, \quad \forall t\ge 0.$$ Let $\Delta t>0$ be given. Under suitable conditions (on $b,a, X …

Consider the stochastic differential equation $$dX_t = {\bf 1}_{\{0<X_t<1\}} a(t,X_t)dW_t, \quad \forall t\in [0,T],$$ where $a$ is continuous on $[0,T)\times [0,1]$ and is Holder continuous with resp …

Consider the SDE (stochastic differential equation) as follows: $$dX_t=X_t\big(b(X_t)dt+a(X_t)dW_t\big)$$ where $b,a:\mathbb R\to\mathbb R$ are Lipschitz and bounded and $W$ is a real-valued Brownian …

Consider two SDEs (stochastic differential equations) as follows: $$dX_t=b^-(t,X_t) \, dt+a(t,X_t) \, dW_t;\quad dY_t = b^+(t,Y_t)\,dt+a(t,Y_t)\,dW_t,$$ where $b^-,b^+,a$ are Lipschitz such that $b^-< …

Let $X$ be the solution to some stochastic differential equation $$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$ Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ denot …

Consider the delayed stochastic differential equation as below: $$dX_t^\theta=X_{(t-\theta)^+}^\theta(1-X_{(t-\theta)^+}^\theta)(dt+dW_t),\quad \forall t>0$$ $$dY_t^\theta=Y_{(t-\theta)^+}^\theta(1-Y_ …

Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate tran …

Consider the stochastic differential equation as follows: $$X_t = x + \int_0^t b(X_s)\,ds + \int_0^t a(X_{s-})\,dL_s,\quad \forall t\ge 0,$$ where $L=(L_t)_{t\ge 0}$ denotes some Lévy process. What ar …

Let $W$ be a Brownian motion and consider the SDE $$dX_t = b(t,X_t) \, dt + a(t,X_t)\,dW_t,\quad \forall t\ge 0. \tag{$\ast$} $$ Assume that $x\mapsto b(t,x), a(t,x)$ are locally Lipschitz in $x$ unif …

Let $W$ be a one-dimensional Brownian motion. Consider the stochastic differential equation (SDE) $$dX_t = C(t)(1-X_t)dW_t,\quad \forall t\ge 0,$$ where $C$ is a continuous and bounded function. Under …