Write $N_t=M_t-M_0$ and $V_t=⟨M{⟩}_t=[N{]}_t$. The Dambis-Dubins-Schwarz theorem supplies a standard Brownian motion $W$ such thatOn the event $V_{\infty }=\infty$, continuity and monotonicity imply that the range of $t\mapsto V_t$ is all of $[0,\infty )$.

For completeness, Brownian motion hits every positive integer almost surely. The Brownian reflection principle gives, for $a>0$,Intersect over integer $a$. The times of these increasingly large values must escape every compact interval because each Brownian path is continuous there. Therefore ${\mathrm{limsup}}_{u\to \infty }{W}_u=\infty$. The unbounded clock transfers these values to arbitrarily late times for $N$, and adding the finite initial value $M_0$ preserves unboundedness. HenceThis is an instance of infinite quadratic variation forces unbounded oscillations; applying the same argument to $-W$ also gives negative oscillations.

Take $f$ in the generator domain $D(L)$. The semigroup commutation with its unbounded generator implies $P_{t}f\in D(L)$ andIndeed, applying the bounded operator $P_t$ to the difference quotient at zero gives the right derivative $P_{t}Lf$; strong continuity gives the full orbit derivative for $t>0$. Now integration by parts in the Bochner integral yieldsThe boundary term at infinity vanishes because $\Vert e^{-\lambda t}{P}_{t}f\Vert \le e^{-\lambda t}\Vert f\Vert$. The derivative integral exists as a Bochner integral, since its integrand is bounded in norm by $e^{-\lambda t}\Vert Lf\Vert$.

Multiplying this identity by $\lambda$ givesPart (b) applies to the fixed vector $Lf\in E$, soThe bounded operators $L_{\lambda }={\lambda }^2{R}_{\lambda }-\lambda I$ are the Yosida approximation of the semigroup generator. The proof only applies the unbounded operator $L$ to $f\in D(L)$; the contraction and limit arguments then act on the ordinary vector $Lf$.

The convergence is in the strong operator topology: for each fixed $f\in E$, one must prove $\Vert \lambda R_{\lambda }f-f\Vert \to 0$. First, the contraction semigroup bound givesThus $\lambda R_{\lambda }$ is a contraction. This is the useful scaling of the contraction estimate in the source's hint. Since ${\int }_0^{\infty }\lambda e^{-\lambda t}dt=1$,Fix $\epsilon >0$. By strong continuity, choose $\delta >0$ so that $\Vert P_{t}f-f\Vert <\epsilon$ for $0\le t\le \delta$. Split the integral there and use $\Vert P_{t}f-f\Vert \le 2\Vert f\Vert$ for the tail:Let $\lambda \to \infty$ and then $\epsilon \downarrow 0$. ThereforeThe distinction from operator norm convergence is essential. For example, the multiplication semigroup on ${\ell }^2(\mathbb{N})$ defined by $(P_{t}f{)}_n=e^{-nt}{f}_n$ is a strongly continuous semigroup of contractions: coordinatewise convergence and dominated convergence applied to ${\sum }_n|f_n{|}^2$ prove strong continuity. Its resolvent operator is $(R_{\lambda }f{)}_n=f_n/(\lambda +n)$, andThus the claimed convergence cannot in general be read as convergence in operator norm.

The diffusion generator is $\mathcal{L}=\mu {\partial }_x+\frac{1}{2}{\sigma }^2{\partial }_{xx}$. By the Feynman-Kac formula, the required conditional characteristic function solvesUse separation of variables in the form $F(t,x)=e^{iux}h(t)$. Since $F_x=iuF$ and $F_{xx}=-u^{2}F$, the temporal factor satisfies the ordinary differential equationThus $h(t)=\exp [(iu\mu -\frac{1}{2}{\sigma }^2{u}^2)(T-t)]$, andThis is bounded in absolute value by one, so it lies in the uniqueness class of the Feynman-Kac formula. As an independent check, integrating the stochastic differential equation gives $X_T=x+\mu (T-t)+\sigma (B_T-B_t)$. The independent Brownian increments have the normal distribution with mean zero and variance $T-t$, yielding exactly the same characteristic function. For $\sigma =0$, the conditional distribution is a point mass at $x+\mu (T-t)$, and the same boxed expression still applies.

For the stochastic differential equation $d{X}_s=b(s,X_s)ds+\Sigma (s,X_s)d{B}_s$, the diffusion generator on smooth functions isOne form of the Feynman-Kac formula is the following. Suppose the stochastic differential equation is well posed and nonexplosive, $c\ge 0$ is bounded, the terminal data $g$ and source $h$ are bounded, and $F$ is a classical solution bounded on finite time intervals ofWith the regularity needed for the Itô formula, the parabolic Feynman-Kac formula with a source statesHere ${\mathbb{E}}_{t,x}$ means expectation for the diffusion process started at $X_t=x$. Under hypotheses ensuring existence of a classical solution, this representation also identifies that solution uniquely in the stated bounded class. In the remaining parts, $c=h=0$, so $F(t,x)={\mathbb{E}}_{t,x}g(X_T)$ solves the terminal-value Kolmogorov backward equation $F_t+{\mathcal{L}}_{t}F=0$. Complex terminal data are allowed by applying the real formula to their real part and imaginary part separately.