Stochastic Processes and Financial Mathematics
(part two)

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Chapter 14 Stochastic processes in continuous time

We’ve already made several comparisons between ordinary differential equations and SDEs. We make a further connection in this section: we show how SDEs can be used to represent solutions to a particular family of partial differential equations. We’ll also develop a little bit of general theory concerning the Markov property.

14.1 The Feynman-Kac formula

Consider $F(t,x)$ where $t\in [0,T]$ and $x\in \R $. We will begin by looking at the partial differential equation

\begin{align} \frac {\p F}{\p t}(t,x)+\alpha (t,x)\frac {\p F}{\p x}(t,x)+\frac {1}{2}\beta (t,x)^2\frac {\p ^2F}{\p x^2}(t,x)&=0\label {eq:fk1a}\\ F(T,x)&=\Phi (x)\label {eq:fk1b} \end{align} Here, $\alpha (t,x)$ and $\beta (t,x)$ are (deterministic) continuous functions, and $\Phi (x)$ is a function known as the boundary condition.

Remark 14.1.1 $\offsyl $ Those familiar with partial differential equations may want to hear the ‘proper’ terminology: this is a second order parabolic PDE with a terminal boundary condition.

We will now show that the solutions of this PDE can be written in terms the solutions to a SDE. In particular, let $X$ be a stochastic process that satisfies

\begin{equation} \label {eq:fk_base} dX_u=\alpha (u,X_u)\,du+\beta (u,X_u)\,dB_u. \end{equation}

We will need to consider solutions to this SDE where we vary the initial value of $x$, and also the time at which the ‘initial’ value occurs. For given $x\in \R $ and $t\in [0,T]$ we write the subscripts $\P _{t,x}$ (and $\E _{t,x}$) to specify that $X$ represents the solution of (14.3) with the initial condition that $X_t=x$ (and we are then interested in $X_u$ during time $u\in [t,T]$). We will continue to use $\P $ and $\E $ to denote starting at time $0$ with unspecified initial value $X_0$. We assume also that $\beta (t,X_t)\frac {\p F}{\p x}(u,X_u)$ is in $\mc {H}^2$, to ensure that the Ito integrals in the proof really exist.

The connection is as follows.

Lemma 14.1.2 Suppose that $F$ is a solution of (14.1) and (14.2). Then,

\[F(t,x)=\E _{t,x}\l [\Phi (X_T)\r ]\]

for all $x\in \R $ and $t\in [0,T]$.

Proof: We apply Ito’s formula to $Z_t=F(t,X_t)$, giving

\[dZ_t=\l (\frac {\p F}{\p t}+\alpha (t,X_t)\frac {\p F}{\p x}+\frac 12\beta (t,X_t)^2\frac {\p ^2 F}{\p x^2}\r )\,dt + \beta (t,X_t) \frac {\p F}{\p x}\,dB_t\]

where, as usual, we have suppressed the $(t,X_t)$ arguments of the partial derivatives of $F$. We know that $F$ satisfies the PDE (14.1), which means the first term on the right hand side of the above vanishes. Writing the result out with integrals, and taking the time limits to be $[t,T]$, then gives

\[F(T,X_T)=F(t,X_t)+\int _t^T\beta (u,X_u) \frac {\p F}{\p x}(u,X_u)\,dB_u.\]

We now take expectations $\E _{t,x}$, and recall from Theorem 12.2.1 that the expectation of integrals with respect to $dB_t$ is zero. Note that here, to apply Theorem 12.2.1, we use that $\beta (t,X_t)\frac {\p F}{\p x}(u,X_u)$ is in $\mc {H}^2$. This leaves us with

\[\E _{t,x}[F(T,X_T)]=\E _{t,x}[F(t,X_t)]\]

Under $\E _{t,x}$ we have $X_t=x$, so $F(t,X_t)=F(t,x)$, which is deterministic. From (14.2) we have $F(T,X_T)=\Phi (T)$, which is also deterministic. Hence we have

\[\E _{t,x}[\Phi (X_T)]=F(t,x)\]

as required. ∎

Lemma 14.1.2, (14.1) is very useful, from a theoretical point of view, but it is not quite what we need for later. The PDE that will turn out to be important for option pricing is

\begin{align} \frac {\p F}{\p t}(t,x)+\alpha (t,x)\frac {\p F}{\p x}(t,x)+\frac {1}{2}\beta (t,x)^2\frac {\p ^2F}{\p x^2}(t,x)-rF(t,x)&=0\label {eq:fk2a}\\ F(T,x)&=\Phi (x)\label {eq:fk2b} \end{align} where $\alpha ,\beta , \Phi $ are as before and $r$ is a deterministic constant. We can treat this PDE in a similar style, even using the same SDE for $X$; we just need an extra term in the calculations.

Lemma 14.1.3 Suppose that $F$ is a solution of (14.4) and (14.5). Then,
$\seteqnumber{0}{14.}{5}$
\begin{equation} \label {eq:fk_2_eq} F(t,x)=e^{-r(T-t)}\E _{t,x}\l [\Phi (X_T)\r ] \end{equation}

for all $x\in \R $ and $t\in [0,T]$.

Proof: This time we apply Ito’s formula to the process $Z_t=e^{-rt}F(t,X_t)$. We obtain

\begin{align*} dZ_t &=\l (-re^{-rt}F+e^{-rt}\frac {\p F}{\p t}+\alpha (t,X_t)e^{-rt}\frac {\p F}{\p x}+\frac 12\beta (t,X_t)^2e^{-rt}\frac {\p ^2 F}{\p x^2}\r )\,dt+\beta (t,X_t)e^{-rt}\frac {\p F}{\p x}\,dB_t\\ &=e^{-rt}\l (-rF+\frac {\p F}{\p t}+\alpha (t,X_t)\frac {\p F}{\p x}+\frac 12\beta (t,X_t)^2\frac {\p ^2 F}{\p x^2}\r )\,dt+e^{-rt}\beta (t,X_t)\frac {\p F}{\p x}\,dB_t. \end{align*} We know that $F$ satisfies (14.4), so the first term on the right hand side vanishes. Writing the result as integrals with time interval $[t,T]$ we obtain

\[e^{-rT}F(T,X_T)=e^{-rt}F(t,X_t)+\int _t^T e^{-ru}\beta (u,X_u)\frac {\p F}{\p x}(u,X_u)\,dB_u.\]

The second term on the right is an Ito integral, and hence has mean zero. Taking expectations $\E _{t,x}$ leaves us with

\[\E _{t,x}\l [e^{-rT}F(T,X_T)\r ]=\E _{t,x}\l [e^{-rt}F(t,X_t)\r ].\]

Under $\E _{t,x}$ we have $X_t=x$, so $F(t,X_t)=F(t,x)$ which is deterministic. We obtain that

\[e^{-r(T-t)}\E _{t,x}\l [F(T,X_T)\r ]=F(t,x)\]

and using (14.5) then gives us

\[e^{-r(T-t)}\E _{t,x}\l [\Phi (X_T)\r ]=F(t,x)\]

as required. ∎

Remark 14.1.4 Setting $r=0$ in Lemma 14.1.3 gets us back to the statement of Lemma 14.1.2.

We can start to see the connection to finance emerging in (14.6). If we set $t=0$, we obtain

\[F(0,x)=e^{-rT}\E _{0,x}[\Phi (X_T)]\]

which bears a resemblance to the risk-neutral valuation formula we found in Chapter 5. The connection will be explored when we come to apply Lemma 14.1.3, in Section 15.3.

Both Lemma 14.1.2 and 14.1.3 assert that, for some particular PDE, if a solution exists then it has a particular form. They assert uniqueness of solutions, without proving that a solution exists. In fact, in both cases, a (unique) solution does exist; this can be proved either using theory from the PDE world, or by using delicate real analysis to show explicitly that $\E _{t,x}[\Phi (X_T)]$ is differentiable. We don’t include this step in our course.

Lemmas 14.1.2, 14.1.3 and variations on the same theme are collectively known as ‘the’ Feynman-Kac formula. They are named after the Richard Feynman (a theoretical physicist, famous for his work in quantum mechanics) and Mark Kac (pronounced “Kats”, a mathematician famous for his contributions to probability theory).

Remark 14.1.5 $\offsyl $ There are many other families of PDEs that have relationships to other families of stochastic processes. These connections are exploited by researchers to transfer information and results between the ‘dual’ worlds of stochastic processes and PDEs.

Example 14.1.6 In cases where we can solve (14.3), we can use the Feynman-Kac formula to find explicit solutions to PDEs. For example, consider
$\seteqnumber{0}{14.}{6}$
\begin{align*} \frac {\p F}{\p t}(t,x)+\frac 12\sigma ^2\frac {\p ^2 F}{\p x^2}(t,x)&=0\\ f(T,x)&=x^2. \end{align*} Here, $\sigma \geq 0$ is a deterministic constant and $t,x\in \R $.

We have $\alpha (t,x)=0$, $\beta (t,x)=\sigma $ (both constants), and $\Phi (x)=x^2$. From Lemma 14.1.2 the solution is given by

\[F(t,x)=\E _{t,x}[X_T^2]\]

where $dX_u=0\,du+\sigma \,dB_u=\sigma \,dB_u$. This means that
$\seteqnumber{0}{14.}{6}$
\begin{align*} X_T&=X_t+\sigma \int _t^T\,dB_u =X_t+\sigma (B_T-B_t). \end{align*} Therefore,
$\seteqnumber{0}{14.}{6}$
\begin{align*} F(t,x)&=\E _{t,x}\l [\l (X_t+\sigma (B_T-B_t)\r )^2\r ]\\ &=\E \l [\l (x+\sigma (B_T-B_t)\r )^2\r ]\\ &=\E \l [x^2+\sigma ^2(B_T-B_t)^2+2x\sigma (B_T-B_t)\r ]\\ &=x^2+\sigma ^2(T-t). \end{align*} Here, we use that $X_t=x$ under $\E _{t,x}$ and that $B_T-B_t\sim B_{T-t}\sim N(0,T-t)$.

See exercises 14.1 and 14.2 for further examples of this method.

Stochastic Processes and Financial Mathematics (part two)

Chapter 14 Stochastic processes in continuous time

14.1 The Feynman-Kac formula

Stochastic Processes and Financial Mathematics
(part two)