LECTURE 7

Diffusions and stochastic differential equations. A diffusion X_{t} on R^{d} is a Markov process with continuous paths which ‘locally’ looks like Brownian motion W_{t} dilated by a factor σ(X_{t}, t) plus a ‘drift’ b(X_{t}, t). It is written formally as a stochastic differential equation

dX_{t} = σ(X_{t}, t)dW_{t} + b(X_{t}, t)dt.

The easiest case is when σ and b are just constants in which case the answer is X_{t} = σW_{t} + bt. Note that in more than one dimension, σ is a matrix and b is a vector.

You can think of the equation as some sort of limit of

X_{\frac{j+1}{2^{n}}} − X_{\frac{j}{2^{n}}} = σ(X_{\frac{j}{2^{n}}} , \frac{j}{2^{n}})(W_{\frac{j+1}{2^{n}}} − W_{\frac{j}{2^{n}}}) + b(X_{\frac{j}{2^{n}}} , \frac{j}{2^{n}})\frac{1}{2^{n}} .

Note the right hand side is, given X_{\frac{j}{2^{n}}}, just a normal mean b(X_{\frac{j}{2^{n}}} , \frac{j}{2^{n}}) and variance a(X_{\frac{j}{2^{n}}} , \frac{j}{2^{n}}) where a = σ^{T} σ.

Stochastic integrals. The stochastic differential equation doesn’t make ordinary calculus sense because W_{t} is not differentiable. What it means is

X_{t} − X_{0} = ∫_{0}^{t} σ(X_{s}, s)dW_{s} + ∫_{0}^{t} b(X_{s}, s)ds

where the second integral is just a normal integral, but the first integral is a stochastic integral defined by

∫_{0}^{t} σ(X_{s}, s)dW_{s} = lim_{n→∞} ∑_{j=0}^{⌊2^{n}t⌋−1} σ(X_{\frac{j}{2^{n}}} , \frac{j}{2^{n}})(W_{\frac{j+1}{2^{n}}} − W_{\frac{j}{2^{n}}}).

Note that the integrand is always evaluated at the left hand end point or in other words, the Brownian increments always look into the future. Therefore if X_{t} is adapted to the Brownian filtration F_{t} we have that

∫_{0}^{t} σ(X_{s}, s)dW_{s} is a martingale.

The stochastic integral can also be defined for σ depending on ω through more than just X_{s}.

∫_{0}^{t} σ(ω, s)dW_{s} = lim_{n→∞} ∑_{j=0}^{⌊2^{n}t⌋−1} σ(ω, \frac{j}{2^{n}})(W_{\frac{j+1}{2^{n}}} − W_{\frac{j}{2^{n}}}).

If you want this to be a martingale it is clear that you want σ(ω, t) to be measurable with respect to F_{t} for each t ≥ 0. This is called non-anticipating (there are other words like progressively measurable and previsible which mean more or less the same thing).

One of the reasons this is useful is that in fact every continuous martingale arises in this way.

Diffusions are continuous but not necessarily martingales because they have a drift term. So they are martingales + a process of bounded variation, sometimes refered to as semi-martingales. They are martingales exactly when the drift b = 0. On the other hand not all continuous semi-martingales are diffusions. Diffusions come from the special case where σ only depends on the history of the path through its present value. Or in other words, they are the Markov cases.

Equivalent definitions of diffusions. 1. X_{t} is a Markov process with transition densities p(s, x, t, y) satisfying, in the t, y variables (with x and s fixed) the Fokker-Planck or forward equation

\frac{∂p}{∂t} = \frac{1}{2} ∑_{i,j=1}^{d} \frac{∂^{2}(a_{ij}p)}{∂y_{i}∂y_{j}} − ∑_{i=1}^{d} \frac{∂(b_{i}p)}{∂y_{i}}

for t > s with initial condition p(s, x, t, y) = δ_{x}(y) when t = s; and in the s, x variables the Kolmogorov or backward equation

\frac{∂p}{∂s} = −\frac{1}{2} ∑_{i,j=1}^{d} a_{ij} \frac{∂^{2}p}{∂x_{i}∂x_{j}} − ∑_{i=1}^{d} b_{i} \frac{∂p}{∂x_{i}}

for 0 ≤ s < t with final condition p(s, x, t, y) = δ_{y}(x) when s = t.

1’. For any smooth function f with compact support,

f(X_{t}) − f(X_{0}) − ∫_{0}^{t} \{\frac{1}{2} ∑_{i,j=1}^{d} a_{ij}(X_{s}, s)\frac{∂^{2}f}{∂x_{i}∂x_{j}} (X_{s}) − ∑_{i=1}^{d} b_{i}(X_{s}, s)\frac{∂f}{∂x_{i}} (X_{s})\}ds

is a martingale.

In either case one says X_{t} has generator

L = \frac{1}{2} ∑_{i,j=}^{d} a_{ij} \frac{∂^{2}}{∂x_{i}∂x_{j}} + ∑_{i=1}^{d} b_{i} \frac{∂}{∂x_{i}} .

2. For any λ ∈ R^{d},

exp \{λ · (X_{t} − X_{0} − ∫_{0}^{t} b(X_{s}, s)ds) − \frac{1}{2} ∫_{0}^{t} λ^{T} · a(X_{s}, s)λds\}

is a martingale. We call a the (infinitesimal) variance and b the (infinitesimal) drift. Explanations. 1. First lets compute from the Fokker-Planck equation lim_{h→0} \frac{1}{h} ∫ f(y−

x)p(t, x, t+h, y)dy. You get ∫ f \frac{∂p}{∂t}dy. By the Fokker-Planck equation it is ∫ f[\frac{1}{2} ∑_{i,j=1}^{d} \frac{∂^{2}(a_{ij}p)}{∂y_{i}∂y_{j}}− ∑_{i=1}^{d} \frac{∂(b_{i}p)}{∂y_{i}}]dy. Integrating by parts gives ∫ [\frac{1}{2} ∑_{i,j=1}^{d} a_{ij} \frac{∂^{2}f}{∂y_{i}∂y_{j}} +∑_{i=1}^{d} b_{i} \frac{∂f}{∂y_{i}}]pdy. Choosing f(x) = x and x^{2} we get

\begin{matrix}lim_{h→0} \frac{1}{h} ∫ (y_{i} − x_{i})p(t, x, t + h, y)dy = b_{i}(x, t) \\ lim_{h→0} \frac{1}{h} ∫ (y_{i} − x_{i})(y_{j} − x_{j})p(t, x, t + h, y)dy = a_{ij}(x, t) \end{matrix}

or, equivalently,

\begin{matrix}lim_{h→0} \frac{1}{h}E[X_{t+h}^{i} − X_{t}^{i} | X_{t}] = b_{i}(x, t) \\ lim_{h→0} \frac{1}{h}E[(X_{t+h}^{i} − X_{t}^{i})(X_{t+h}^{j} − X_{t}^{j}) | X_{t}] = a_{ij}(x, t). \end{matrix}

So it just says that X_{t} has infinitesimal mean b and variance a.

2. Pick a small h. By the martingale property

E[ exp \{λ · (X_{t+h} − X_{t}) − ∫_{t}^{t+h} λ · b(X_{s}, s)ds − \frac{1}{2} ∫_{t}^{t+h} λ^{T} · a(X_{s}, s)λds\} | F_{t}] = 1

which says that for small h, X_{t+h} − X_{t} is more or less a multivariate Gaussian with mean ∫_{t}^{t+h} b(X_{s}, s)ds and covariance ∫_{t}^{t+h} a(X_{s}, s)ds.

Computation of a stochastic integral. For practice let’s compute the stochastic integral

A_{t} = ∫_{0}^{t} W_{s}dW_{s}.

From the definition we know it should be the limit

A_{t} = lim_{n→∞} ∑_{j=0}^{⌊2^{n}t⌋−1} W_{\frac{j}{2^{n}}} (W_{\frac{j+1}{2^{n}}} − W_{\frac{j}{2^{n}}}).

It’s not exactly clear how to do it, but there is a trick. Let’s let

B_{t} = lim_{n→∞} ∑_{j=0}^{⌊2^{n}t⌋−1} W_{\frac{j+1}{2^{n}}} (W_{\frac{j+1}{2^{n}}} − W_{\frac{j}{2^{n}}}).

Then B_{t} − A_{t} is just the quadratic variation process, which we know is t, and

B_{t} + A_{t} = lim_{n→∞} ∑_{j=0}^{⌊2^{n}t⌋−1}(W 2_{\frac{j+1}{2^{n}}} − W 2_{\frac{j}{2^{n}}}) = W_{t}^{2}.

So

A_{t} = \frac{1}{2}([B_{t} + A_{t}] − [B_{t} − A_{t}]) = \frac{1}{2}(W_{t}^{2} − t).

Stochastic calculus. Ordinary calculus is based on the chain rule. The chain rule for diffusions is Ito’s formula: If f is a smooth function and dX_{t} = σdW_{t} + bdt then (in one dimension)

df(X_{t}) = f′(X_{t})dX_{t} + \frac{σ^{2}}{2} f′′(X_{t})dt.

Of course this is just shorthand for saying that

f(X_{t})−f(X_{0}) = ∫_{0}^{t} f′(X_{s})σ(X_{s}, s)dW_{s} +∫_{0}^{t} f′(X_{s})b(X_{s}, s)ds+ \frac{1}{2} ∫_{0}^{t} σ^{2}(X_{s}, s)f′′(X_{s})ds.

Ito’s formula is easy to prove. You just write

f(X_{t}) − f(X_{0}) = lim_{n→∞} ∑_{j=0}^{⌊2^{n}t⌋−1}[f(X_{\frac{j+1}{2^{n}}}) − f(X_{\frac{j}{2^{n}}})].

Then you expand in a Taylor series

\begin{matrix}f(X_{\frac{j+1}{2^{n}}}) − f(X_{\frac{j}{2^{n}}}) = & f′(X_{\frac{j}{2^{n}}})(X_{\frac{j+1}{2^{n}}} − X_{\frac{j}{2^{n}}}) + \frac{1}{2}f′′(X_{\frac{j}{2^{n}}})(X_{\frac{j+1}{2^{n}}} − X_{\frac{j}{2^{n}}})^{2} \\ + \frac{1}{2 · 3}f′′′(X_{\frac{j}{2^{n}}})(X_{\frac{j+1}{2^{n}}} − X_{\frac{j}{2^{n}}})^{3} + ⋯ \end{matrix}

Now you use the stochastic integral equation for X_{\frac{j+1}{2^{n}}} − X_{\frac{j}{2^{n}}} and let n → ∞. The first term gives no special problem. The third and higher terms can easily be seen to go to zero. The only hard thing is the second term. But if you look, it is almost exactly the same as the computation of the quadratic variation of Brownian motion. This is the origin of the extra ‘Ito’ term in stochastic calculus. The rules are summarized as

\begin{matrix}× & dW & dW′ & dt \\ dW & dt & 0 & 0 \\ dW′ & 0 & dt & 0 \\ dt & 0 & 0 & 0 \end{matrix}

where W and W′ are independent Brownian motions.

The multidimensional version is just as easy, and the function f can depend on t:

df(X_{t}, t) = σ(X_{t}, t)∇f(X_{t}, t)dW_{t} + \{\frac{∂f}{∂t} (X_{t}, t) + Lf(X_{t}, t)\} dt.

The product formula is a particular case of the above. If dX_{1} = σ_{1}dW_{1} + b_{1}dt and dX_{2} = σ_{2}dW_{2} + b_{2}dt then

dX_{1}X_{2} = X_{1}dX_{2} + X_{2}dX_{1} + σ_{1}σ_{2}d(Cov(W_{1}, W_{2})).

The last term is a little cumbersome, and there is a nice way to write it. Any continuous submartingale X_{t} can be written as the difference of a continuous martingale and a continuous, increasing adapted process, called its compensator,

X_{t} = M_{t} − A_{t}.

This is called the Doob-Meyer decomposition. Also every square of a martingale is a submartingale by Jensen’s inequality. So for every continuous martingale M_{t} we have a compensator for its square which we call the quadratic variation process ⟨M_{t}, M_{t}⟩. For example, for Brownian motion ⟨W_{t}, W_{t}⟩ = t. This can be polarized ⟨M_{t}, N_{t}⟩ = \frac{1}{4}[⟨M_{t} + N_{t}, M_{t} + N_{t}⟩ − ⟨M_{t} − N_{t}, M_{t} − N_{t}⟩] and you can see that M_{t}N_{t} − ⟨M_{t}, N_{t}⟩ is always a martingale. So we have the formula

dM_{t}N_{t} = M_{t}dN_{t} + N_{t}dM_{t} + d⟨M_{t}, N_{t}⟩.

If X_{t} is the solution of the stochastic differential equation

dX_{t} = σdW_{t} + bdt

then

X_{t} − ∫_{0}^{t} b(X_{s}, s)ds

is a martingale (since it is a stochastic integral ∫_{0}^{t} σdW_{s}) and its quadratic variation is

⟨X_{t}, X_{t}⟩ = ∫_{0}^{t} a(X_{s}, s)ds.

Example: Exponential Brownian motion. The process

X_{t} = exp{σW_{t} + μt}

where σ and μ are constants is often used to model stock prices. Applying Ito’s formula to the function f(x, t) = exp{σx + μt} we find that

dX_{t} = σX_{t}dW_{t} + (μ + \frac{σ^{2}}{2} )dt.

It is not a martingale unless μ = −σ^{2}/2.

Cameron-Martin-Girsanov. For each nice a(x, t), b(x, t) and starting point x_{0} we have a diffusion which we can think of as a measure P_{x_{0}}^{a,b} on the space of continuous functions starting at x_{0}. The question is how these measures are related for different a’s and b’s. Now if we compute the quadratic variation on [0, T], as we did for Brownian motion, we will find that it is

lim_{n→∞} ∑_{j=0}^{⌊2^{n}t⌋−1} |X_{\frac{j+1}{2^{n}}} − X_{\frac{j}{2^{n}}} |^{2} = ∫_{0}^{T} a(X_{s}, s)ds

with probability one under P_{x_{0}}^{a,b}. That means that if ã ≠ a then P_{x_{0}}^{ã,b} is supported on a completely different set than P_{x_{0}}^{a,b}, namely the paths for which the above limit is ∫_{0}^{T} ã(X_{s}, s)ds. We say that the two measures are mutually singular.

On the other hand if we change b to b̃ the new measure we get is absolutely continuous with respect to the old one, or equivalent meaning that P_{x_{0}}^{a,b̃}(A) > 0 if and only if P_{x_{0}}^{a,b}(A) > 0 and the Radon-Nikodym derivative of one process on [0, t] with respect to the other has an explicit formula. To keep it simple we will compute only the case where one of them is zero. The general formula can be obtained easily by multiplying the corresponding Radon-Nikodym derivatives. The Cameron-Martin-Girsanov formula is

\frac{dP_{x_{0}}^{a,b}}{dP_{x_{0}}^{a,0}} = exp \{∫_{0}^{t} a^{−1}bdX_{s} − \frac{1}{2} ∫_{0}^{t} a^{−1}b^{2}ds\} .

Here’s why: We’ll do it in one dimension so that the notation isn’t too bad. What we want to show is if we define a measure

Q(A) = ∫_{A} exp \{∫_{0}^{t} a^{−1}bdX_{s} − \frac{1}{2} ∫_{0}^{t} a^{−1}b^{2}ds\} dP_{x_{0}}^{a,0}  (1)

then Q is a diffusion with parameters a and b, in other words, for each λ,

exp\{λ(X_{t} − X_{0} − ∫ bds) − \frac{λ^{2}}{2} ∫_{0}^{t} ads\}  (2)

is a martingale with respect to Q. Let

Y_{t} = ∫_{0}^{t}(λ + a^{−1}b)dX_{s} = ∫_{0}^{t}(λ + a^{−1}b)σdW_{s}.

Then we know that for any θ,

exp\{θ(Y_{t} − Y_{0}) − \frac{θ^{2}}{2} ∫_{0}^{t} a(λ + a^{−1}b)^{2}ds\}

is a martingale with respect to P_{x_{0}}^{a,0}. Choose θ = 1. Rearranging things a bit and using that Y_{t} = λX_{t} + ∫_{0}^{t} a^{−1}bdX_{s} we have that

exp\{λ(X_{t} − X_{0} − ∫ bds) − \frac{λ^{2}}{2} ∫_{0}^{t} ads + ∫_{0}^{t} a^{−1}bdX_{s} − \frac{1}{2} ∫_{0}^{t} a^{−1}b^{2}ds\}

is a martingale with respect to P_{x_{0}}^{a,0}. But that is the same thing as (1), (2).

There is an important point to make here. The Radon-Nikodym derivative is with respect to P_{x_{0}}^{a,b} so the X_{s} in the integral is the solution of dX_{s} = σdW_{t} + bdt where W_{t} is Brownian motion. On the other hand with respect to P_{x_{0}}^{a,b̃}, dX_{s} = σdW̃_{t} + b̃dt where W̃_{t} is Brownian motion. So what gives? The point is that it is not the same thing to be a Brownian motion with respect to P_{x_{0}}^{a,b̃} and with respect to P_{x_{0}}^{a,b}. In fact if W_{t} is a Brownian motion with respect to P_{x_{0}}^{a,b} then W̃_{t} = W_{t} + ∫_{0}^{t} σ^{−1}(b̃ − b)ds is a Brownian motion with respect to P_{x_{0}}^{a,b̃}. To see this note that P_{x_{0}}^{a,b} is the distribution of

X_{t} = ∫_{0}^{t} σdW_{s} + ∫_{0}^{t} bds

and P_{x_{0}}^{a,b̃} is the distribution of

Y_{t} = ∫_{0}^{t} σdW_{s} + ∫_{0}^{t} b̃ds.

In other words one obtains Brownian motion as

W_{s} = ∫_{0}^{t} σ^{−1}dX_{s} + ∫_{0}^{t} σ^{−1}bds under P_{x_{0}}^{a,b}

W̃_{s} = ∫_{0}^{t} σ^{−1}dX_{s} + ∫_{0}^{t} σ^{−1}b̃ds under P_{x_{0}}^{a,b̃}.

Problem. Consider the linear stochastic differential equation

dX_{t} = [A(t)X_{t} + a(t)]dt + σ(t)dW_{t}, X_{0} = x,

where W_{t} is an r-dimensional Brownian motion independent of the initial vector x ∈ R^{d} and the d × d, d × 1 and d × r matrices A(t), a(t) and σ(t) are non-random.

Show that the solution is given by

X_{t} = Φ(t)[x + ∫_{0}^{t} Φ^{−1}(s)a(s)ds + ∫_{0}^{t} Φ^{−1}(s)σ(s)dW_{s}]

where Φ is the d × d matrix solution of

Φ̇(t) = A(t)Φ(t), Φ(0) = I.

(I is the identity matrix with 1’s on the diagonal and 0’s elsewhere.)