Brownian interpolation of stochastic differential equations

Many applications require knowledge of the state vector at intermediate sample times
that are initially unavailable. One way to approximate these intermediate states is to
perform a deterministic interpolation. However, deterministic interpolation techniques fail
to capture the correct probability distribution at these intermediate times. Brownian (or
stochastic) interpolation captures the correct joint distribution by sampling from a
conditional Gaussian distribution. This sampling technique is sometimes referred to as a
*Brownian Bridge*.

The default stochastic interpolation technique is designed to interpolate into an
existing time series and ignore new interpolated states as additional information becomes
available. This technique is the usual notion of interpolation, which is called
*Interpolation without refinement*.

Alternatively, the interpolation technique may insert new interpolated states into the
existing time series upon which subsequent interpolation is based, by that means refining
information available at subsequent interpolation times. This technique is called
*interpolation with refinement*.

Interpolation without refinement is a more traditional technique, and is most useful when the input series is closely spaced in time. In this situation, interpolation without refinement is a good technique for inferring data in the presence of missing information, but is inappropriate for extrapolation. Interpolation with refinement is more suitable when the input series is widely spaced in time, and is useful for extrapolation.

The stochastic interpolation method is available to any model. It is best illustrated,
however, by way of a constant-parameter Brownian motion process. Consider a correlated,
bivariate Brownian motion (`BM`

) model of the form:

$$\begin{array}{l}d{X}_{1t}=0.3dt+0.2d{W}_{1t}-0.1d{W}_{2t}\\ d{X}_{2t}=0.4dt+0.1d{W}_{1t}-0.2d{W}_{2t}\\ E[d{W}_{1t}d{W}_{2t}]=\rho dt=0.5dt\end{array}$$

Create a

`bm`

object to represent the bivariate model:`mu = [0.3; 0.4]; sigma = [0.2 -0.1; 0.1 -0.2]; rho = [1 0.5; 0.5 1]; obj = bm(mu,sigma,'Correlation',rho);`

Assuming that the drift (

`Mu`

) and diffusion (`Sigma`

) parameters are annualized, simulate a single Monte Carlo trial of daily observations for one calendar year (250 trading days):rng default % make output reproducible dt = 1/250; % 1 trading day = 1/250 years [X,T] = simulate(obj,250,'DeltaTime',dt);

It is helpful to examine a small interval in detail.

Interpolate into the simulated time series with a Brownian bridge:

`t = ((T(1) + dt/2):(dt/2):(T(end) - dt/2)); x = interpolate(obj,t,X,'Times',T);`

Plot both the simulated and interpolated values:

plot(T,X(:,1),'.-r',T,X(:,2),'.-b') grid on; hold on; plot(t,x(:,1),'or',t,x(:,2),'ob') hold off; xlabel('Time (Years)') ylabel('State') title('Bi-Variate Brownian Motion: \rho = 0.5') axis([0.4999 0.6001 0.25 0.4])

In this plot:

The solid red and blue dots indicate the simulated states of the bivariate model.

The straight lines that connect the solid dots indicate intermediate states that would be obtained from a deterministic linear interpolation.

Open circles indicate interpolated states.

Open circles associated with every other interpolated state encircle solid dots associated with the corresponding simulated state. However, interpolated states at the midpoint of each time increment typically deviate from the straight line connecting each solid dot.

You can gain additional insight into the behavior of stochastic interpolation by regarding a Brownian bridge as a Monte Carlo simulation of a conditional Gaussian distribution.

This example examines the behavior of a Brownian bridge over a single time increment.

Divide a single time increment of length

`dt`

into 10 subintervals:mu = [0.3; 0.4]; sigma = [0.2 -0.1; 0.1 -0.2]; rho = [1 0.5; 0.5 1]; obj = bm(mu,sigma,'Correlation',rho); rng default; % make output reproducible dt = 1/250; % 1 trading day = 1/250 years [X,T] = simulate(obj,250,'DeltaTime',dt); n = 125; % index of simulated state near middle times = (T(n):(dt/10):T(n + 1)); nTrials = 25000; % # of Trials at each time

In each subinterval, take 25000 independent draws from a Gaussian distribution, conditioned on the simulated states to the left, and right:

average = zeros(length(times),1); variance = zeros(length(times),1); for i = 1:length(times) t = times(i); x = interpolate(obj,t(ones(nTrials,1)),... X,'Times',T); average(i) = mean(x(:,1)); variance(i) = var(x(:,1)); end

Plot the sample mean and variance of each state variable:

### Note

The following graph plots the sample statistics of the first state variable only, but similar results hold for any state variable.

subplot(2,1,1); hold on; grid on; plot([T(n) T(n + 1)],[X(n,1) X(n + 1,1)],'.-b') plot(times, average, 'or') hold off; title('Brownian Bridge without Refinement: Sample Mean') ylabel('Mean') limits = axis; axis([T(n) T(n + 1) limits(3:4)]); subplot(2,1,2) hold on; grid on; plot(T(n),0,'.-b',T(n + 1),0,'.-b') plot(times, variance, '.-r') hold('off'); title('Brownian Bridge without Refinement: Sample Variance') xlabel('Time (Years)') ylabel('Variance') limits = axis; axis([T(n) T(n + 1) limits(3:4)]);

The Brownian interpolation within the chosen interval,

*dt*, illustrates the following:The conditional mean of each state variable lies on a straight-line segment between the original simulated states at each endpoint.

The conditional variance of each state variable is a quadratic function. This function attains its maximum midway between the interval endpoints, and is zero at each endpoint.

The maximum variance, although dependent upon the actual model diffusion-rate function

*G(t,X)*, is the variance of the sum of`NBrowns`

correlated Gaussian variates scaled by the factor*dt/4*.

The previous plot highlights interpolation without refinement, in that none of the interpolated states take into account new information as it becomes available. If you had performed interpolation with refinement, new interpolated states would have been inserted into the time series and made available to subsequent interpolations on a trial-by-trial basis. In this case, all random draws for any given interpolation time would be identical. Also, the plot of the sample mean would exhibit greater variability, but would still cluster around the straight-line segment between the original simulated states at each endpoint. The plot of the sample variance, however, would be zero for all interpolation times, exhibiting no variability.

This function performs a Brownian interpolation into a user-specified time series array, based on a piecewise-constant Euler sampling approach.

Consider a vector-valued SDE of the form:

$$d{X}_{t}=F(t,{X}_{t})dt+G(t,{X}_{t})d{W}_{t}$$

where:

*X*is an*NVars*-by-`1`

state vector.*F*is an*NVars*-by-`1`

drift-rate vector-valued function.*G*is an*NVars*-by-*NBrowns*diffusion-rate matrix-valued function.*W*is an*NBrowns*-by-`1`

Brownian motion vector.

Given a user-specified time series array associated with this equation, this function
performs a Brownian (stochastic) interpolation by sampling from a conditional Gaussian
distribution. This sampling technique is sometimes called a *Brownian
bridge*.

Unlike simulation methods, the `interpolation`

function does not
support user-specified noise processes.

The

`interpolate`

function assumes that all model parameters are piecewise-constant, and evaluates them from the most recent observation time in`Times`

that precedes a specified interpolation time in`T`

. This is consistent with the Euler approach of Monte Carlo simulation.When an interpolation time falls outside the interval specified by

`Times`

, a Euler simulation extrapolates the time series by using the nearest available observation.The user-defined time series

`Paths`

and corresponding observation`Times`

must be fully observed (no missing observations denoted by`NaN`

s).The

`interpolate`

function assumes that the user-specified time series array`Paths`

is associated with the`sde`

object. For example, the`Times`

and`Paths`

input pair are the result of an initial course-grained simulation. However, the interpolation ignores the initial conditions of the`sde`

object (`StartTime`

and`StartState`

), allowing the user-specified`Times`

and`Paths`

input series to take precedence.

[1] Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest
Rate.” *The Review of Financial Studies*, Spring 1996, Vol. 9, No.
2, pp. 385–426.

[2] Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other Nonlinear
Diffusions.” *The Journal of Finance*, Vol. 54, No. 4, August
1999.

[3] Glasserman, P. *Monte Carlo Methods in Financial Engineering.*
New York, Springer-Verlag, 2004.

[4] Hull, J. C. *Options, Futures, and Other Derivatives*, 5th ed.
Englewood Cliffs, NJ: Prentice Hall, 2002.

[5] Johnson, N. L., S. Kotz, and N. Balakrishnan. *Continuous Univariate
Distributions.* Vol. 2, 2nd ed. New York, John Wiley & Sons,
1995.

[6] Shreve, S. E. *Stochastic Calculus for Finance II: Continuous-Time
Models.* New York: Springer-Verlag, 2004.