Stochastic Differential Equations

Author

John Robin Inston

Published

September 25, 2026

1 Preliminaries

The study of stochastic differential equations and stochastic calculus builds upon strong foundational understanding in Stochastic Processes including: Stochastic Processes; Filtrations; Martingales; Stopping Times; Lebesgue Integrals and \(L_{p}\)-Spaces and \(p\)-variation.

1.1 Lebesgue Spaces

A Lebesgue space (\(L^p\) space) is a space of space of measurable functions whose absolute value ### p-Variation

The \(p\) -variation of a real-valued stochastic process \((X_{t})\) on \([0,T]\) is a process \(\left\langle X,X \right\rangle_{t}^{(p)}\) such that for all \(t>0\) and sequence of partitions \[ \Delta^{(n)}:=\{ 0 = t_{0}^{(n)}<t_{1}^{(n)}<\dots<t_{K_{n}}^{(n)}=t \}, \] of interval \([0,t]\) such that \[ \delta^{(n)}:=\max_{1\leq i\leq K_{n}}\lvert t_{i}^{(n)}-t_{i-1}^{(n)} \rvert \to 0\quad \text{as}\quad n \to \infty, \] the following holds \[ S_{n}^\Delta = \sup_{\Delta}\sum_{i=0}^{K_{n}-1} \lvert X_{t_{i}}-X_{t_{i-1}} \rvert ^p \stackrel{\mathcal{L}_{2}~\&~\mathbb{P}}\to \left\langle X,X \right\rangle_{t}^{(p)}. \]

For \(p=1\) we call this the total variation. For \(p=2\) we call this the quadratic variation and often will drop the \(p\) notation \(\left< X,X \right>_{t}\). We primarily deal with quadratic variance when we are working in the \(\mathcal{L}_{2}\) space and is key to our construction of an integral in this space.

2 Brownian Motion

Recall that a random variable \(X\) is said to be normally distributed or Gaussian with mean \(\mu\) and variance \(\sigma^2\) if for all \(x \in \mathbb{R}\) \[ \mathbb{P}(X>x)= \frac{1}{\sqrt{ 2\pi \sigma^2 }}\int_{x}^\infty \exp\left( - \frac{{(u-\mu)^2}}{2\sigma^2} \right)~du. \]

A Brownian motion (BM) or a Wiener process with variance parameter \(\sigma^2>0\) is a stochastic process \((B_{t})_{t \geq 0}\) taking values in \(\mathbb{R}\) that satisfies: 1. \(B_{0}=0~a.s.\) (starts at zero); 2. \(\forall~0 < t_{1}<\dots<t_{n},~B_{t_{1}}-B_{0}, B_{t_{2}}-B_{t_{1}}, \dots, B_{t_{n}}-B_{t_{n-1}}\) are independent (independent increments); 3. \(\forall s < t, B_{t}-B_{s}\sim\mathcal{N}(0,\sigma^2(t-s))\) (normal stationary increments); 4. \(\omega\mapsto B_{t}(\omega)\) is a continuous function of \(t\) on \(\mathbb{R}_{+}\) almost surely (continuous paths no jumps).

If \(\sigma^2=1\) then \((B_{t})\) is called the standard BM. Furthermore, we use the notation \(B_{t}^x=B_{t}+x\) for BM starting from \(x\).

2.1 BM Properties

Here we have defined BM as a stochastic process, i.e. a family of (uncountably many) random variables \(\omega \mapsto B(t, \omega)\) defined on a single probability space. Alternatively, we could interpret BM as a random function with the sample functions defined by \(t \mapsto B(t, \omega)\), in which case it is sometimes useful to assume that the mapping \((t, \omega)\mapsto B(t, \omega)\) is measurable on the product space \([o, \infty)\times \Omega\).

BM satisfies the following properties: 1. \(B_t\) is a Gaussian Processes, i.e. for all \(0 \leq t_{1}\leq \dots \leq t_{k}\) the random variable \(Z=(B_{t_{1}}, \dots, B_{t_{k}})\in \mathbb{R}^{nk}\) has a Multivariate Normal Distribution. 2. \(B_{t}\) is both Markov and a martingale. 3. \(B_{t}\) satisfies the stronger Markov property whereby for any bounded real function \(f\) we have \(\mathbb{E}[f(B_{t})|\mathcal{F}_{s}]=\mathbb{E}[f(B_{t})|B_{s}]\).

BM is the only Lévy Process (processes with independent stationary increments) with continuous paths. Although the process has continuous paths, it is not differentiable due its roughness. We summarize this and the other key path properties of BM in the following theorem.

Brownian paths, considered to be functions of \(t\), have the following properties. Almost every sample path \(\{ B_{t}, t \in [0,T] \}\): 1. is a continuous function of \(t\); 2. is not monotone on any interval, no matter how small 3. is not differentiable at any point 4. has infinite total variation on any interval, no matter how small 5. has quadratic variation on \([0,t]\) equal \(t\), for all \(t\).

2.2 BM Transition Probabilities

To find the transition density function for \(B_{t}\) we consider Borel set \(A \in \mathcal{B}(\mathbb{R})\) and compute \[ \begin{align} \mathbb{P}_{t}^B(x,A) & = \mathbb{P}(B_{t+s}\in A|B_{s}=x) = \mathbb{P}(B_{t+s}\in A|B_{s}=x) \\ & = \mathbb{P}(B_{t+s}\in A|B_{s}=x) \\ & = \mathbb{P}(B_{t+s}-B_{s}\in A-x|B_{s}=x) \\ & = \mathbb{P}(B_{t+s}-B_{s}\in A-x), \end{align} \] where the last line holds from independent increments. Then, using the fact that \(B_{t+s}-B_{s}\sim\mathcal{N}(0, \sigma^2t)\) we have \[ \begin{align} \mathbb{P}_{t}^B(x,A) & =\int_{A-x} \frac{1}{\sqrt{ 2\pi \sigma^2t }}\exp\left( -\frac{y^2}{2\sigma^2t} \right)~dy \\ & = \int_{A} \frac{1}{\sqrt{ 2\pi \sigma^2t }}\exp\left( -\frac{(z-x)^2}{2\sigma^2t} \right) ~dz, \end{align} \] where we have used the substitution \(z=y+x\). Hence, the transition density function is \[ p_{t}^B(x,z)= \frac{1}{\sqrt{ 2\pi \sigma^2 t }}\exp\left( -\frac{(z-x)^2}{2\sigma^2t} \right). \] The transition probabilities of BM satisfy the Chapman-Kolmogorov Equations \[ p_{s+t}(x,y)= \int_{\mathbb{R}}p_{s}(x,z)p_{t}(z,y)~dz. \]

2.3 Variation of BM

If a process is differentiable on \([0,T]\) then its total variation will be finite. This is a crucial motivation for constructing stochastic intervals since we can show that BM is not differentiable by taking \(T=n\cdot\Delta t\) and computing \[ \sum_{i}\lvert B_{t_{i}}-B_{t_{i-1}} \rvert \sim n\mathbb{E}\lvert B_{\Delta t} \rvert \sim \frac{T}{\Delta t}\sqrt{ \Delta t }\to \infty\quad \text{as}\quad n \to \infty,~\Delta t \to 0. \] Hence BM has an infinite total variation. Intuitively, this means that BM is too rough for Reimann integral approximations to work (think like fractals, no matter how much we zoom in th surface doesn’t get smoother). However, BM does have a finite quadratic variation which explains the difference between the chain rule in stochastic calculus and the classical result.

For BM we have \(\left< B,B \right>_{t}=t\), that is, for partition \(\Delta^{(n)}\) of an interval \([a,b]\) such that \(\delta^{(n)}\to 0\) as \(n \to \infty\) then \(S_{n}^\Delta \stackrel{\mathcal{L}_{2}~\&~\mathbb{P}}\to b-a\).

Proof: We wish to show that \(\mathbb{E}(S_{n}^\Delta)^2\to(b-a)^2\) as \(n \to \infty\). First note that \[ \mathbb{E}S_{n}^\Delta = \mathbb{E}\left[ \sum_{i=0}^{K_{n}-1} \left\lvert B_{t_{i+1}^{(n)}}-B_{t_{i}^{(n)}} \right\rvert^2 \right]=\sum_{i=0}^{K_{n}-1}(t_{i+1}^{(n)}-t_{i}^{(n)})=b-a. \] We proceed to compute \[ \begin{align}\mathbb{E}\left[(S_{n}^{\Delta})^2\right] & =\mathbb{E}\left[ \left( \sum_{i=0}^{K_{n}-1}\lvert B_{t_{i+1}^{(n)}}-B_{t_{i}^{(n)}} \rvert ^2 \right)^2 \right] \\ & =\sum_{i=0}^{K_{n}-1}\mathbb{E}\left\lvert B_{t_{i+1}^{(n)}}-B_{t_{i}}^{(n)} \right\rvert ^4+2\sum_{i<j}\mathbb{E}\left[ \left\lvert B_{t_{i+1}^{(n)}}-B_{t_{i}}^{(n)} \right\rvert^2\cdot \left\lvert B_{t_{j+1}^{(n)}}-B_{t_{j}}^{(n)} \right\rvert \right] \\ & = \sum_{i}3\left(t_{i+1}^{(n)}-t_{i}^{(n)}\right)^2+2\sum_{i<j}\left(t_{i+1}^{(n)}-t_{i}^{(n)}\right)\cdot \left(t_{j+1}^{(n)}-t_{j}^{(n)}\right) \\ & = 2\sum_{i}\left( t_{i+1}^{(n)}-t_{i}^{(n)} \right) ^2+\left( \sum_{i}\left( t_{i+1}^{(n)}-t_{i}^{(n)} \right) \right) ^2 \\ & \leq 2\delta^{(n)}(b-a)+(b-a)^2 \\ & \to(b-a)^2\quad\text{as}\quad n \to \infty,\end{align} \] where we have used \(\mathbb{E}X^4=3[(\sigma^2_{X})^2]\) and the stationary independent increments of Brownian motion. We have showed that \[ \forall\varepsilon>0,~\exists N:\forall n\geq N,~\mathbb{E}[(S_{n}^{\Delta})^2]\leq (b-a)^2+\varepsilon. \] Additionally, we have shown that \(\mathbb{E}[(S_{n}^\Delta)^2]-(b-a)^2=\text{Var}(S_{n}^\Delta)\geq 0\) i.e. \(\mathbb{E}[(S_{n}^\Delta)^2]\geq (b-a)^2\) hence we obtain equality. Hence we have convergence in \(\mathcal{L}_{2}\) which gives convergence in probability. \(\square\)

3 Itô Integral

Consider the probability space \((\Omega, \mathcal{F}, \mathbb{P})\) and let \((\boldsymbol{B}(t))=(B_{1}(t), \dots, B_{n}(t))\) be \(n\)-dimensional (standard) Brownian Motion. Define \(\boldsymbol{\mathcal{F}}(t)=\boldsymbol{\mathcal{F}}^{(n)}(t)\) to be the filtration generated by \((B_{i}(s))_{{1\leq i \leq n}, 0 \leq s \leq t}\). Recall that we call the natural filtration of \((\boldsymbol{B}(t))\) and interpret as the history of the process \(\boldsymbol{B}(t)\) up to time \(t\).

Our aim is to construct the integral \(I(f)=\int_{S}^{T}{f_{t}}~d{B_{t}}\) for a general process \(f_{t}\) in order to define an Itô process however this is somewhat challenging since \(B_{t}\) can be shown to be nowhere differentiable and have infinite variation meaning that we cannot use a Riemann Integral or Lebesgue Integral. Instead we construct the Itô integral as the limit in mean square (or probability) of simple processes (akin to Riemann sums) where the integrand is evaluated at the left endpoint of each subinterval. This ensures the process is non-anticipating (adapted to the filtration) which preserves martingale properties.

3.1 Itô Integral Construction

We begin by defining the space of all measurable (actually progressive), adapted and \(\mathcal{L}_{2}\) processes for which we can define the Itô integral. Let \(\mathcal{V}=\mathcal{V}(S,T)\) be the class of functions \[ \mathcal{V}(S,T):=\{f(t,\omega):[0,\infty) \times \Omega \to \mathbb{R}\}, \] such that: 1. \((t, \omega)\to f(t, \omega)\) is \(\mathcal{B}\times\mathcal{F}\)-measurable, where \(\mathcal{B}\) denotes the Borel σ-Algebra on \([0,\infty)\); 2. \(f(t,\omega)\) is \(\mathcal{F}_{t}\)-adapted; 3. \(\mathbb{E}[\int_{S}^T f(t,\omega)^2dt]<\infty\).

For (3) say \(f \in \mathcal{L}_{2}([S,T]\times \Omega)\) since \(\left< f_{t},g_{t} \right> = \mathbb{E}\left[ \int_{S}^T f_{t}g_{t} ~dt \right]\) is the inner product on such a space. Our goal is to define the stochastic integral \(I(f)=\int_{S}^T f_{t}dB_{t}\) for a general process \(f\) in such a Hilbert space.

Our construction process is as follows, first consider defining such an integral for an elementary process and then extend it to general processes in \(\mathcal{V}\). The main thought is to find a dense subset of the Hilbert space and define the stochastic integral on it to prove that it is actually an isometry. We are then able to extend it as the isometry to the whole Hilbert space.

A process \((X_{t})\) is called an elementary adapted process if it can be written in the form \[ \varphi_{t}(\omega)=\sum_{j \geq 0} e_{j}\mathbb{1}_{\left[\frac{j}{2^n}, \frac{{j+1}}{2^n}\right)}(t), \] where \(n\in \mathbb{N}\) and each \(e_{j}\in \mathcal{F}_{t}\) and \(e_{j}\in\mathcal{L}_{2}\).

This is a slightly changed step function where \(e_j\) are now a collection of \(\mathcal{F}\) measurable random variables with finite over the dyadic number partition. For elementary functions we can define the stochastic integral \[ \int_{S}^T \varphi_{t}~dB_{t}=\sum_{j \geq 0}e_{j}[B_{t_{j+1}}-B_{t_{j}}], \] where we specify \[ t_{k}=t_{k}^{(n)}=\begin{cases} \frac{k}{2^n} & \text{if }S\leq \frac{k}{2^n}\leq T \\ S & \text{if } \frac{k}{2^n} < S \\ T & \text{if } \frac{k}{2^n}>T. \end{cases} \] At this point it is natural to approximate a given function \(f(t,\omega)\) by \[ \sum_{j}f_{t_{j}^*}(\omega)\cdot \mathbb{1}_{[t_{j},t_{j+1})}(t), \] where the points \(t_{j}^*\in[t_{j}, t_{j+1}]\) allowing us to define the integral \(\int_{S}^T f_{t}(\omega)dB_{t}(\omega)\) as the limit (in \(\mathcal{L}_{2}\)) of \(\sum_{j}f_{t_{j}^*}(\omega)[B_{t_{j+1}}-B_{t_{j}}](\omega)\) as \(n \to \infty\). However, unlike with the Riemann-Stieltjes Integral our choice of \(t_{j}^*\) impacts our final result, as shown with the following example.

Example (Choice of \(t^*\) on interval) We define \[ \varphi^{(1)}_{t}(\omega)=\sum_{j \geq 0} B_{\frac{j}{2^n}}(\omega)\mathbb{1}_{[\frac{j}{2^n}, \frac{{j+1}}{2^n})}(t)\quad \& \quad \varphi_{t}^{(2)}(\omega) = \sum_{j \geq 0} B_{\frac{j+1}{2^n}}(\omega)\mathbb{1}_{[\frac{j}{2^n}, \frac{{j+1}}{2^n})}(t). \] Then from independent increments we have \[ \mathbb{E}\left[ \int_{0}^T \varphi _{t}^{(1)}(\omega)~dB_{t}(\omega)\right]=\sum_{j \geq 0 }\mathbb{E}[B_{t_{j}}(B_{t_{j+1}}-B_{t_{j}})]=0. \] However \[ \mathbb{E}\left[ \int_{0}^T \varphi^{(2)}_{t}(\omega)~dB_{t}(\omega) \right]=\sum_{j \geq 0 }\mathbb{E}[B_{t_{j+1}}\cdot(B_{t_{j+1}}-B_{t_{j}})]=T. \] The two most useful choices have been found to be the left end point \(t_{j}^* = t_{j}\) leading to the Itô integral (the primary consideration in this note and throughout this topic) and the mid point \(t_{j}^*=(t_{j}+t_{j+1})/2\) leading to the Stratonovich integral.

To begin our extension from the elementary functions we make the following important observation.

If \(\varphi_{t}(\omega)\) is bounded and elementary then we have \[ \mathbb{E}\left[ \left( \int_{S}^T \varphi_{t}(\omega )dB_{t} \right) ^2 \right] =\mathbb{E}\left[ \int_{S}^T \varphi_{t}^2(\omega )dt \right]. \]

Proof: Define \(\Delta B_{j}=B_{t_{j+1}}-B_{t_{j}}\) and then note that \[ \mathbb{E}[e_{i}e_{j}\Delta B_{i}\Delta B_{j}]=\begin{cases}0 & \text{if }i \neq j \\ \mathbb{E}[e_{j}^2]\cdot(t_{j+1}-t_{j}) & \text{if }i = j, \end{cases} \] from independent increments. Thus we have \[ \begin{align}\mathbb{E}\left[ \int_{S}^T \varphi_{t}~dB_{t} \right]=\sum_{i,j}\mathbb{E}[e_{i}e_{j}\Delta B_{i}\Delta B_{j}]=\sum_{j}\mathbb{E}[e_{j}^2]\cdot(t_{j+1}-t_{j})=\mathbb{E}\left[ \int_{S}^T \varphi_{t}^2~dt \right],\end{align} \] completing the proof. \(\square\)

3.1.1 Step 1: Bounded Continuous Processes

Letting \(g \in \mathcal{V}\) be bounded and \(g_{\cdot}(\omega)\) continuous for each \(\omega\). Then there exist elementary functions \(\varphi^{(n)}\in \mathcal{V}\) such that \[ \mathbb{E}\left[ \int_{S}^T (g_{t} - \varphi_{t})^2dt \right]\to 0\quad \text{as}\quad n \to \infty. \] To see this define \(\varphi_{t}^{(n)}(\omega)=\sum_{j}g_{t_{j}}(\omega)\cdot \mathbb{1}_{[t_{j},t_{j+1})}(t)\) which is elementary since \(g \in\mathcal{V}\) and \[ \int_{S}^T(g-\varphi_{n})^2dt\to 0\quad \text{as}\quad n \to \infty, \] for each \(\omega\) (pointwise) since \(g_{\cdot}(\omega)\) is pointwise continuous. Hence the result follows by bounded convergence.

3.1.2 Step 2: Bounded Processes

Let \(h \in \mathcal{V}\) be bounded. Then there exist bounded functions \(g^{(n)} \in \mathcal{V}\) such that \(g_{t}^{(n)}(\omega)\) is continuous for all \(\omega\) and \(n\), and \[ \mathbb{E}\left[ \int_{S}^T (h_{t} - g_{t}^{(n)})^2dt \right]\to 0. \] To see this suppose \(\lvert h_{t}(\omega) \rvert\leq M\) for all \((t, \omega)\). For each \(n\) let \(\phi^{(n)}\) be a non-negative, continuous function on \(\mathbb{R}\) such that: 1. \(\phi^{(n)}(x)=0\) for \(x \leq -\frac{1}{n}\) and \(x \geq 0\); and 2. \(\int_{\mathbb{R}}\phi^{(n)}(x)~dx =1.\) Define \[ g_{t}^{(n)}(\omega)=\int_{0}^t\phi^{(n)}_{t}(s-t)h_{s}(\omega)ds. \] Then \(g_{\cdot}^{(n)}(\omega)\) is continuous for each \(\omega\) (pointwise) and \(\lvert g_{t}^{(n)}(\omega) \rvert\leq M\). Since \(h \in \mathcal{V}\) we show that \(g^{(n)}_{t}(\cdot)\) is adapted (e.g. Karatzas and Shreve (1991); pg 133). Moreover \[ \int_{S}^T(g^{(n)}_{s}(\omega)-h_{s}(\omega))^2ds\to 0 \quad \text{pointwise as}\quad n\to \infty, \] since \(\{ \phi_{n} \}\) constitutes an approximate identity (e.g. Hoffman (1962); pg 22). Hence, by bounded convergence we obtain the result.

3.1.3 Step 3: General Process

Let \(f \in\mathcal{V}\). Then there exists a sequence \(\{ h^{(n)} \}\subset\mathcal{V}\) such that \(h_{n}\) is bounded for each \(n\) and \[ \mathbb{E} \left[ \int_{S}^T(f-h^{(n)})^2 dt \right]\to 0 ~~\text{as}~~ n \to \infty. \] To see this we simply put \[ h^{(n)}_{t}(\omega) = \begin{cases} -n & \text{if } f_{t}(\omega)<-n \\ f_{t}(\omega) & \text{if } -n \leq f_{t}(\omega)\leq n \\ n & \text{if } f_{t}(\omega)>n, \end{cases} \] and the conclusion follows by the Dominated Convergence Theorem.

3.2 Itô Integral & Itô’s Isometry

For \(f\in\mathcal{V}(S,T)\) the Itô integral from \(S\) to \(T\) is defined by \[ \int_{S}^Tf(t, \omega)dB_{t}(\omega) =\lim_{ n \to \infty } \int_{S}^T \phi_{n}(t, \omega)dB_{t}(\omega), \] where the limit is taken in \(L^2\) and \(\{ \phi_{n} \}\) is a sequence of elementary functions such that \[ \mathbb{E}\left[ \int_{S}^T (f(t, \omega)-\phi_{n}(t, \omega))^2dt \right] \to 0, \] as \(n \to \infty\).

3.3 Itô Isometry

For \(f \in\mathcal{V}(S,T)\) we have that \[ \mathbb{E}\left[ \left( \int_{S}^T f(t,\omega )dB_{t} \right) ^2 \right] =\mathbb{E}\left[ \int_{S}^T f^2(t,\omega )dt \right]. \]

The Itô integral has many of the same properties as the classical integrals which allow for ease of manipulation.

Let \(f,g\in\mathcal{V}(0,T)\) and let \(0\leq S<U<T\). Then we have that: 1. \(\int_{S}^Tf~dB_{t}=\int_{S}^U f~dB_{t}+\int_{U}^T f~dB_{t}\) almost surely. 2. \(\int_{S}^T(cf+g)dB_{t}=c\int_{S}^Tf~dB_{t}+\int_{S}^Tg~dB_{t}\) almost surely for constant \(c\) 3. \(\mathbb{E}\left[\int_{S}^T f~dB_{t}\right]=0\). 4. \(\int_{S}^Tf~dB_{t}\) is \(\mathcal{F}_{T}\)-measurable.

Proof: This clearly holds for elementary functions hence by taking limits we obtain this for all \(f,g\in\mathcal{V}(0,T)\). \(\square\)

Another important property of the Itô integral is that it is a martingale, that is it does not drift on average over time.

4 Itô Processes

4.1 Itô Process

Let \(B_{t}\) be a 1-dimensional Brownian motion on \((\Omega, \mathcal{F}, \mathbb{P})\). A (1-dimensional) Itô process (or stochastic integral) is a stochastic process \(X_{t}\) on \((\Omega, \mathcal{F}, \mathbb{P})\) of the form \[ X_{t}=x+\int_{0}^{t}{u(s, \omega )}~d{s} + \int_{0}^{t}{v(s,\omega)}~d{B_{s}}, \] where we say that \(u\) is the drift coefficient (controls the growth of the process) and \(v\) is the diffusion coefficient (controls the size of the noise). Sometimes this is written in the shorter differential form \[ dX_{t}=udt+vdB_{t}. \]

4.2 Itô Formula

Itô’s formula or Itô’s lemma is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule.

Let \(X_{t}\) be an Itô process given by \[ dX_{t}=udt+vdB_{t}. \] Further, let \(g(t,x)\in C^2([0,\infty)\times \mathbb{R})\) (i.e. twice differentiable on \([0,\infty)\times \mathbb{R}\)). Then \[ Y_{t}=g(t,X_{t}) \] is again an Itô process and \[ dY_{t}=g_{t}(t, X_{t})dt+g_{x}(t, X_{t})dX_{t}+ \frac{1}{2}g_{x x}(t, X_{t})d\left< X,X \right>_{t}. \]

Here \(d\left< X,X \right>_{t}=(dX_{t})\cdot(dX_{t})\) is computed according to the following: \[ dt\cdot dt=dt\cdot dB_{t}=dB_{t}\cdot dt=0,\quad dB_{t}\cdot dB_{t}=0. \] Proof: Substituting \(dX_{t}=udt+vdB_{t}\) into the Itô formula we get \[ \begin{align} g(t,X_{t}) & = g(0,X_{0}) + \int_{0}^t\left( \frac{{\partial g}}{\partial s}(s,X_{s})+u_{s}\frac{{\partial g}}{\partial x}(s,X_{s})+ \frac{1}{2} v_{s}^2 \frac{{\partial^2g}}{\partial x^2}(s,X_{s}) \right)ds \\ & + \int_{0}^t v_{s} \frac{{\partial g}}{\partial x}(s,X_{s})dB_{s}. \end{align} \] We may assume that \(g, g_{t}, g_{x}, g_{x x}\) are bounded for each \(n\) and converge uniformly on compact subsets of \([0,\infty)\times \mathbb{R}\) to \(g,g_{t},g_{x},g_{x x}\) respectively. Moreover, we may assume that \(u,v\) are elementary functions. Using Taylor’s theorem we get \[ \begin{align} g(t,X_{t}) & = g(0,X_{0}) + \sum_{j}\Delta g(t_{j}, X_{j}) \\ & = g(0,X_{0})+\sum_{j} \frac{{\partial g}}{\partial t} \Delta t_{j}+\sum_{j} \frac{{\partial g}}{\partial x}\Delta X_{j}+ \frac{1}{2}\sum_{j} \frac{{\partial^2g}}{\partial t^2}(\Delta t_{j}^2) \\ & \qquad + \sum_{j} \frac{{\partial^2g}}{\partial t\partial x}(\Delta t_{j})(\Delta X_{j}) + \frac{1}{2}\sum_{j} \frac{{\partial^2g}}{\partial x^2}(\Delta X_{j})^2 + \sum_{j}R_{j}, \end{align} \] where \(\frac{{\partial g}}{\partial t}, \frac{{\partial g}}{\partial x}\) etc. are evaluated at the points \((t_{j}, X_{t_{j}})\), \[ \begin{align} \Delta t_{j} & =t_{j+1}-t_{j} \\ \Delta X_{j} & = X_{t_{j+1}}-X_{t_{j}} \\ \Delta g(t_{j},X_{j}) & = g(t_{j+1},X_{t_{j+1}})-g(t_{j},X_{j}), \end{align} \] and \(R_{j}=o(\lvert \Delta t_{j} \rvert^2+\lvert \Delta X_{j} \rvert^2)\) for all \(j\). If \(\Delta t_{j}\to 0\) then \[ \begin{align} \sum_{j} \frac{{\partial g}}{\partial t}\Delta t_{j} & =\sum_{j} \frac{{\partial g}}{\partial t}(t_{j}, X_{j})\Delta t_{j} \to \int_{0}^t \frac{{\partial g}}{\partial t}(s,X_{s})~ds \\ \sum_{j} \frac{{\partial^2 g}}{\partial x^2} \Delta X_{j} & = \sum_{j} \frac{{\partial g}}{\partial x}(t_{j}, X_{j})\Delta X_{j} \to \int_{0}^t \frac{{\partial g}}{\partial x}(s,X_{s})dX_{s}. \end{align} \] Moreover, since \(u\) and \(v\) are elementary we get \[ \begin{align} \sum_{j} \frac{{\partial^2 g}}{\partial x^2}(\Delta X_{j})^2 & = \underbrace{\sum_{j} \frac{{\partial^2g}}{\partial x^2}u_{j}^2 (\Delta t_{j})^2}_{\to 0\text{ as }\Delta t_{j}\to 0} + \underbrace{2 \sum_{j} \frac{{\partial^2g}}{\partial x^2} u_{j}v_{j}(\Delta t_{j})(\Delta B_{j})}_{\to 0\text{ as }\Delta t_{j}\to 0} +\sum_{j} \frac{{\partial^2g}}{\partial x^2}v_{j}^2 \cdot (\Delta B_{j})^2, \end{align} \] where \(u_{j}=u(t_{j},\omega)\) and \(v_{j}=v(t_{j},\omega)\) and convergence in \(L_2\). We claim the third term tends to \[ \int_{0}^t \frac{{\partial^2g}}{\partial x^2}v^2~ds. \] To prove this put \(a(t)= \frac{{\partial^2g}}{\partial x^2}(t,X_{t})v^2(t,\omega)\), \(a_{j}=a(t_{j})\) and consider \[ \mathbb{E}\left[ \left( \sum_{j} a_{j}(\Delta B_{j})^2-\sum_{j}a_{j}\Delta t_{j} \right)^2 \right] = \sum_{i,j} \mathbb{E}[a_{i}a_{j}((\Delta B_{i})^2-\Delta t_{i})((\Delta B_{j})^2 - \Delta t_{j})]. \] If \(i<j\) then \(a_{i}a_{j}((\Delta B_{i})^2-\Delta t_{i})\) and \((\Delta B_{j})^2-\Delta t_{j}\) are independent so the terms vanish, and similarly if \(i>j\). This leaves \[ \begin{align} \sum_{j}\mathbb{E}[a_{j}^2((\Delta B_{j})^2-\Delta t_{j})^2] & = \sum_{j}\mathbb{E}[a_{j}^2] \cdot \mathbb{E}[(\Delta B_{j})^4-2(\Delta B_{j})^2\Delta t_{j} + (\Delta t_{j})^2] \\ & = \sum_{j}\mathbb{E}[a_{j}^2]\cdot (3(\Delta t_{j})^2 - 2(\Delta t_{j})^2+(\Delta t_{j})^2) \\ & =2\sum_{j}\mathbb{E}[a_{j}^2]\cdot(\Delta t_{j})^2 \\ & \to 0\quad (\Delta t_{j}\to 0). \end{align} \] Thus we have established that \[ \sum_{j}a_{j}(\Delta B_{j})^2 \to \int_{0}^t a(s)ds, \] which is often expressed as \[ (dB_{t})^2=dt. \] The argument above also proves that \(\sum R_{j}\to 0\) as \(\Delta t_{j}\to_{0}\). \(\square\)

Example (Itô Formula Application I) Consider the integral \[ Y_{t}=\int_{0}^tB_{s}dB_{s}. \] Assuming we can write \(Y_{t}=g(t, B_{t})\) from Itô’s formula we have \[ dY_{t}=\left[ g_{t}(t,B_{t})+\frac{1}{2}g_{BB}(t,B_{t}) \right] dt+g_{B}(t,B_{t})dB_{t}, \] which written in integral form gives \[ Y_{t}-Y_{0}=\int_{0}^t\left[ g_{t}(s,B_{s})+\frac{1}{2}g_{B B}(s,B_{s}) \right]ds+\int_{0}^t g_{B}(s, B_{s})dB_{s}. \] Hence we need some \(g(t,B_{t})\) such that \[ (1)~~g_{t}(t,B_{t})+\frac{1}{2}g_{BB}(t, B_{t})=0\quad \&\quad (2)~~ g_{B}(t,B_{t})=B_{t}. \] Since \(g_{B}=B_{t}\implies g_{BB}=1\) and so \(g_{t}=-\frac{1}{2}\). An obvious candidate therefore is \[ g(t,B_{t})=\frac{1}{2}B_{t}^2 - \frac{1}{2}t, \] which in fact is the result, as can be verified by plugging into Itô’s formula.

Example (Itô Formula Application II) Consider the Itô process \[ g(t,B_{t})=\exp(bt+\sigma B_{t}), \] where \(b,\sigma\) are constants for drift and diffusion. To compute \(dg\) we apply the Itô formula \[ \begin{align} dg(t,B_{t}) & =b\exp(bt+\sigma B_{t})dt+\sigma \exp(bt+\sigma B_{t})dB_{t}+ \frac{1}{2} \sigma^2\exp(bt+\sigma B_{t})d\left< B,B \right>_{t} \\ & = g(t,B_{t})\left\{ \left( b+ \frac{1}{2} \sigma^2 \right)dt+\sigma dB_{t} \right\}. \end{align} \] The process \(X_{t}=g(t,B_{t})\) defines geometric BM and is used to simulate the change of the stock price in the Black-Scholes-Merton Model (\(\mu\) as mean return and \(\sigma\) as volatility).

For a continuous function \(f(s,\omega)\) that is of bounded variation with respect to \(s \in [0,t]\) almost surely we have that \[ \int_{0}^t f(s)dB_{s}=f(t)B_{t}-\int_{0}^t B_{s}df_{s}. \]

4.3 Multi-dimensional Itô Formula

Considering the situation in higher dimensions we define a \(d\)-dimensional Itô process as the integral with respect to the \(m\)-dimensional BM as \[ X_{t}^i=x_{0}^i+\int_{0}^t\psi_{s}^ids+\sum_{k=1}^m \int_{0}^t \varphi_{s}^{i,k}dB_{s}^k, \] for \(k = 1, \dots, m\) and \(i = 1, \dots, d\) where we note that such an Itô process exists in \(\mathbb{R}^d\) and is constructed with respect to BM \(B_{s}=(B_{s}^1, \dots, B_{s}^m)\). To write this in a more compact form \[ X_{t}=x_{0}+\int_{0}^t \psi_{s}ds+\int_{0}^t\varphi_{s}\cdot dB_{s}, \] where \(X_{t}, x_{0},\psi_{s} \in \mathbb{R}^d\) and \(\varphi_{s}\in \mathbb{R}^{d \times m}\) and here \(\varphi_{s}^{i,k}\) stands for the process \(\varphi\) used to construct the \(i\)-th coordinate of the Itô process as a stochastic integral with respect to the \(k\)-th coordinate of the \(m\)-dimensional BM.

For some vector-valued function \(g:\mathbb{R}_{+}\times \mathbb{R}^d\to \mathbb{R}^p \in C^{1,2}\), then the multi-dimensional Itô formula holds \[ \begin{align}dg^i(t,X_{t}) & =\partial_{t}g^i(t,X_{t})dt+\nabla_{x} g^i(t,X_{t})\cdot dX_{t}+ \frac{1}{2}\mathrm{Tr}(\varphi_{t}^T H \varphi_{t})dt \\ & = \partial_{t}g^i(t,X_{t})dt+\sum_{k=1}^{d}\partial_{x_{k}}g^i(t,X_{t})dX_{t}^k+ \frac{1}{2}\sum_{j,k = 1}^{d}\partial_{x_{j}, x_{k}}g^i(t,X_{t})d\left< X^j, X^k \right> _{t},\end{align} \] where \(\nabla _x f\) denotes the gradient of \(f\) with respect to vector \(x\), \(\varphi_{t}\in \mathbb{R}^{d \times m}\) is a matrix and \(H_{d \times d}\) is the Hessian of \(g\) restricted on its action on \(x \in \mathbb{R}^d\) at \((t, X_{t})\). The bracket \(d\left< X^j, X^k \right>_{t}=\sum_{l=1}^{m}\varphi_{t}^{j,l}\varphi_{t}^{k,l}dt\).

Proof: The structure of the proof is still exactly the same as the 1-dimensional case. We need only verify that \(dB_{t}^pdB_{t}^q=0\) for \(p \neq q\) and \(d\left< X^j,X^k \right>_{t}=\sum_{l=1}^{m}\varphi_{t}^{j,l}\varphi_{t}^{k,l}dt\). \(\square\)

4.4 The Martingale Representation Theorem

The martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.

Let \(B(t)\) be \(n\)-dimensional BM. Suppose \(M_{t}\) is an \(\mathcal{F}_{t}^{(n)}\)-martingale (w.r.t.) \(\mathbb{P}\) and that \(M_{t}\in\mathcal{L}_{2}(\mathbb{P})\) for all \(t \geq 0\). Then there exists a unique stochastic process \(g(s,\omega)\) such that \(g \in \mathcal{V}^{(n)}(0,t)\) for all \(t \geq 0\) and \[ M_{t}(\omega)=\mathbb{E}[M_{0}]+\int_{0}^t g(s,\omega) dB(s), \] almost surely for all \(g \geq 0\).

The proof of this result is involved and is omitted from this note. Details can be found in our note on this theorem [[martingale-representation-theorem|here]].

5 Stochastic Differential Equations

A general stochastic differential equation in \(\mathbb{R}^n\) with initial condition has the form \[ \begin{cases} dX_{t}=b(t,X_{t})dt+\sigma(t, X_{t})dBt \\ X_{0}=x, \end{cases} \] where \(x,X_{t}\in \mathbb{R}^n\), \(b: \mathbb{R}_{+}\times \mathbb{R}^n\to \mathbb{R}^n\), \(\sigma:\mathbb{R}_{+}\times \mathbb{R}^n\to \mathbb{R}^{n \times m}\) and \(B_{t}\) is an \(m\)-dimensional BM. Here \(b\) is the drift coefficient and \(\sigma\) is the volatility coefficient. We note the difference between SDEs and a general Itô process lies in the fact that \(b,\sigma\) are functions of the unknown process \(X_{t}\).

Fixing time \(T>0\) and assuming \(b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n\), \(\sigma:[0,T]\times \mathbb{R}^n \to \mathbb{R}^{n \times m}\) are deterministic functions with bounded time variable \(T\) which satisfy the following conditions: 1. \(\exists c>0,~\forall t \in [0,T],~\forall x \in \mathbb{R}^n,~\lvert b(t,x) \rvert + \lvert \sigma(t,x) \rvert\leq c(1+\lvert x \rvert)\) (growth condition); 2. \(\exists D>0,~\forall t \in[0,T],~\forall x,y \in \mathbb{R}^n,~\lvert b(t,x)-b(t,y) \rvert + \lvert \sigma(t,x)-\sigma(t,y) \rvert\leq D\lvert x-y \rvert\) (Lipschitz condition),

then the SDE has unique solution in \(L^2([0,T]\times \Omega)\) that has continuous sample path.

NEED TO ADD PROOF.

5.1 Strong and Weak SDE Solutions

A strong solution to a stochastic differential equation is defined as the solution \(X_{t}\in \mathcal{F}_{t}\) adapted to the filtration generated by Brownian Motion and the initial condition (if it is random) \[ \mathcal{F}_{t}=\sigma(B_{s}, 0 \leq s \leq t)\vee \sigma(X_{0}). \] The uniqueness and global existence theorem stated for Stochastic Differential Equations is just proving that for strong solutions.

5.2 SDE Example: Linear SDE

References: Linear Stochastic Differential Equations.

Let \(W_t\) be \(m\)-dimensional BM, \(A\) be an \(n \times n\) matrix and \(B\) an \(n \times m\) matrix. Then a linear stochastic differential equation \(X_{t}\) has dynamics given by \[ \begin{cases}dX_{t}=AX_{t}dt+BdW_{t},\\X_{0}=x.\end{cases} \]

To find the general solution of a linear SDE we first note that by setting the noise term \(B\) to the \(0\)-matrix we have the ODE \(dX_{t}=AX_{t}dt\) which has the solution \(X_{t}=X_{0}e^{At}\). Setting this as our

which are given explicitly by \[ X_{t}=e^{At}x+\int_{0}^{t}{e^A(t-s)B}~d{W_{s}}. \]

5.3 SDE Example: Black Scholes Model

The Black-Scholes model provides the structure on which we are able to price contingency claims (specifically options). Specifically, the Black-Scholes model is a mathematical model for the dynamics of a financial market containing derivative investment instruments such as stock shares or futures contracts assumed to have a lognormal distribution of prices following a random walk with constant drift and volatility.

The Black-Scholes model consists of two assets, a risk free asset \(B\) and a stock \(S\) with price dynamics given by \[ \begin{align}dB_{t}&=rB_{t}dt \\dX_{t}& =\mu X_{t}dt+\sigma X_{t}dB_{t}\end{align} \] where \(r,\mu,\sigma \in \mathbb{R}\) are deterministic constants.

So in our notation we have \(b(t,x)=\mu x\) and \(\sigma(t,x)=\sigma x\) which are both Lipschitz in in \(x\) and satisfy the growth condition.

To solve this system we note that if we ignore the stochastic term \(dX_{t}=\mu X_{t}dt\) is an ODE with solution \(X_{t}=X_{0}\cdot e^{\mu t}\). We therefore consider changing variables with \(Y_{t}=\log(X_{t})\) and applying Itô’s formula we obtain \[ dY_{t}=\frac{1}{X_{t}}dX_{t}-\frac{1}{2X_{t}^2}d\left< X,X \right> _{t}=\frac{1}{X_{t}}dX_{t}-\frac{1}{2}\sigma^2dt, \] where we have used that \(d\left< X,X \right>_{t}=\sigma^2X_{t}^2d\left< B,B \right>_{t}=\sigma^2X_{t}^2dt\). Substituting in our initial expression we have \[ \begin{align} dY_{t} & =\mu dt+\sigma dB_{t}-\frac{1}{2}\sigma^2dt=\left( \mu-\frac{1}{2}\sigma^2 \right)dt+\sigma dB_{t} \\ \implies Y_{t}-Y_{0} & =\int_{0}^t\left( \mu-\frac{1}{2}\sigma^2 \right)dt+\int_{0}^t\sigma dB_{t} \\ & =\left( \mu-\frac{1}{2}\sigma^2 \right)t+\sigma B_{t} \\ \implies X_{t} & =X_{0}\cdot e^{\left( \mu-\frac{1}{2}\sigma^2 \right)t+\sigma B_{t}}. \end{align} \] ### SDE Example: OU Process

An Ornstein-Uhlenbeck process is a mean reverting stochastic process that is a Gaussian process, is Markov and is temporally stationary. This process can be used to describe the fluctuation of interest rate around the mean interest rate \(M\) with the speed of the regression described by \(\theta\).

The Ornstein-Uhlenbeck process \((X_{t})\) with drift has dynamics described by the stochastic differential equation \[ dX_{t}=\theta(\mu-X_{t})dt+\sigma dB_{t} \] where \(\theta,\sigma>0\), \(\mu \in \mathbb{R}\) are all constant and \(B_{t}\) is a BM.

Consider such a process with a deterministic starting value \(X_{0}\). To find a solution to this SDE we first change the variable to set the regression level to 0 with \(Y_{t}=X_{t}-\mu\). From Itô’s formula we have \[ dY_{t}=dX_{t}=\theta(\mu-X_{t})dt+\sigma dB_{t}=-\theta Y_{t}dt+\sigma dB_{t}. \] Considering when \(\sigma=0\) we have the ODE \(dY_{t}=-\theta Y_{t}dt\) with the solution \(Y_{t}=Y_{0}\cdot e^{-\theta t}\) and so replacing this constant \(Y_{0}\) with some yet to define process \(C_{t}\) we have \(Y_{t}=C_{t}e^{-\theta t}\) and from Itô’s formula we have \[ dY_{t}=-\theta e^{-\theta t}C_{t}dt+e^{-\theta t}dC_{t}=-\theta Y_{t}dt+e^{-\theta t}dC_{t}. \] We therefore can define \(dC_{t}=\sigma e^{\theta t}dB_{t}\) and so \(C_{0}=Y_{0}=X_{0}-\mu\) and \[ \begin{align} C_{t}-C_{0} & = \int_{0}^t\sigma e^{\theta s}dB_{s} \\ \implies C_{t} & =X_{0}-\mu+\sigma \int_{0}^te^{\theta s}dB_{s} \\ \implies Y_{t} & = (X_{0}-\mu)e^{-\theta t}+\sigma e^{-\theta t}\int_{0}^te^{\theta s}dB_{s} \\ \implies X_{t} & = (X_{0}-\mu)e^{-\theta t}+\mu+\sigma e^{-\theta t}\int_{0}^te^{\theta s}dB_{s}. \end{align} \] ### SDE Example: Brownian Bridge

A Brownian bridge is a continuous-time gaussian process \((X_{t})\) whose probability distribution is the conditional probability distribution of a standard Brownian Motion \(B_t\) subject to the condition (when standardized) that \(W_{T} = 0\), so that the process is pinned to the same value at both t = 0 and t = T.

[!definition] Brownian Bridge The Brownian Bridge is an SDE \((X_{t})\) characterized by the dynamics \[\begin{cases}dX_{t}=\frac{b-X_{t}}{1-t}dt+dB_{t}\\ X_{0}=a,\end{cases}\] for \(a,b \in \mathbb{R}\) and \(0 \leq t < 1\).

This is the Brownian bridge starting from \(a\) and ending at \(b\) that has the same finite-dimensional distribution as the BM conditioning on starting from \(a\) at time \(0\) and ending at \(b\) at time \(1\). The Brownian bridge starting from \(0\) at time \(0\) and ending at \(0\) at time \(1\) has the simple form as \(B_t−tB_1\).

We can verify that the solution is given by \[ Y_{t}=a(1-t)+bt+(1-t)\int_{0}^t \frac{1}{1-s}dB_{s}. \] First note that since \[ G:=\int_{0}^t \frac{1}{1-s}sB_{s} \sim\mathcal{N}\left( 0, \frac{1}{1-t} \right), \] its characteristic function is \[ \phi(p)=\phi_{G}(p(1-t))=e^{\frac{1}{2(1-t)}p^2(1-t)^2}=e^{\frac{1}{2}p^2(1-t)}\to 1~~\text{as}~~t \to 1. \] Hence \((1-t)G\stackrel{\mathcal{L}}{\to} 0\) as \(t \to 1\) and since the limit is a constant \((1-t)G\stackrel{\mathbb{P}}{\to} 0\) as \(t \to 1\). Finally, we notice that this stochastic integral is the sum of countably many independent random variables, hence from Durrett 2.5.10 we have \((1-t)G\stackrel{\text{a.s.}}{\to} 0\) as \(t \to 1\) and \(Y_{t}\stackrel{\text{a.s.}}{\to}b\).

5.4 5.6 SDE Example: Geometric Mean-Reverting Process

The geometric mean-reverting process \(X_{t}\) is defined as the solution to the SDE \[ \begin{cases}dX_{t}=\kappa(\alpha-\log X_{t})X_{t}dt + \sigma X_{t}dB_{t}\\X_{0}=x>0.\end{cases} \]

By using the substitution \(Y_{t}=\log X_{t}\) we can verify that the solution to the SDE is \[ X_{t}=\exp \left\{ e^{-\kappa t}\log x+\left( \alpha- \frac{\sigma^2}{2\kappa} \right)(1-e^{-\kappa t})+\frac{\sigma^2}{2\kappa}(1-e^{-2\kappa t}) \right\}. \]

6 Diffusion Processes

7 Diffusions

  • Itô Diffusion
    • Generator of an Itô Diffusion
  • Dynkin Formula
  • Characteristic Operator
  • Kolmogorov’s Backward Equation
  • Feynman-Kac Formula
  • Martingale Problem
  • Random Time Change
  • Girsanov Theorem

A (time-homogeneous) Itô diffusion in \(\mathbb{R}^n\) is a stochastic process \((X_{t})\) defined on space \((\Omega, \mathcal{F},\mathbb{P})\) that satisfies the stochastic differential equation \[ \begin{cases} dX_{t}=b(X_{t})dt+\sigma(X_{t})dB_{t} \\ X_{s}=x \end{cases} \] where \(X_{t}\in \mathbb{R}^n\), \(B_{t}\in \mathbb{R}^m\), \(B:\mathbb{R}^n\to \mathbb{R}^n\), \(\sigma:\mathbb{R}^n\to \mathbb{R}^{n \times m}\) where it is assumed that \(b,\sigma\) are both Lipschitzs on \(\mathbb{R}^n\) (to ensure the existence and uniqueness of strong solutions).

The distinguishing feature of diffusions is that the drift and diffusion coefficients \(b,\sigma\) are time independent and the Lipschitz condition is here to ensure the existence and uniqueness of the strong solution. Next we detail the results outlining the key properties of Itô diffusions.

Denoting the solution to the SDE above with initial condition \(X_{s}=x\) as an Itô diffusion by \(X_{s+h}^{s,x}\) then \[ \forall s \geq 0,~x \in \mathbb{R}^n,~\{ X_{s+t}^{s,x} \}_{t \geq 0}\stackrel{\mathcal{L}}{=}\{ X_{t}^{0,x} \}_{t \geq 0}. \]

Proof:

For an Itô diffusion \(X\) we have \[ \forall t,h \geq 0,~ X_{h}^{0,X_{t}^{0,x}}\stackrel{a.s.}{X_{t+h}^{0,x}}. \]

Proof: By construction, both are strong solutions to the same SDE \[ X_{t+h}^{0,x}=X_{t}^{0,x}+\int_{t}^{t+h}{b(X_{s}^{0,x})}~d{s}+\int_{t}^{t+h}{\sigma(X_{s}^{0,x})}~d{B_{s}}. \] By the uniqueness in modification we conclude

Diffusion properties: 1. Time-homogeneity 2. Flow Property 3. Markov Property 4. Strong Markov Property

Ito Diffusion Generators Semi-Group Generators Infinitesimal Generators (special and general cases) Infinitesimal Generator Examples Dynkin Formula Dynkin Formula Examples Backward Kolmogorov Equation (BKE) Forward Kolmogorov Equation (FKE, Fokker-Planck Equation) Feynman-Kac Formula Itô Process as DIffusion Time Changing Processes Girsanov Theorem

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