Tag: Stochastic Differential Equations

Stochastic Differential Equations

Stochastic differential equations (SDEs) form a large and very important part of the theory of stochastic calculus. Much like ordinary differential equations (ODEs), they describe the behaviour of a dynamical system over infinitesimal time increments, and their solutions show how the system evolves over time. The difference with SDEs is that they include a source of random noise., typically given by a Brownian motion. Since Brownian motion has many pathological properties, such as being everywhere nondifferentiable, classical differential techniques are not well equipped to handle such equations. Standard results regarding the existence and uniqueness of solutions to ODEs do not apply in the stochastic case, and cannot readily describe what it even means to solve such as system. I will make some posts explaining how the theory of stochastic calculus applies to systems described by an SDE.

Consider a stochastic differential equation describing the evolution of a real-valued process {Xt}t≥0,

\displaystyle dX_t = \sigma(X_t)\,dW_t + b(X_t)\,dt (1)

which can be specified along with an initial condition X0 = x0. Here, b is the drift specifying how X moves on average across the dt time, σ is a volatility term giving the amplitude of the random noise and W is a driving Brownian motion providing the source of the randomness. There are numerous situations where equations such as (1) are used, with applications in physics, finance, filtering theory, and many other areas.

In the case where σ is zero, (1) is just an ordinary differential equation dX/dt = b(X). In the general case, we can informally think of dividing through by dt to give an ODE plus an additional noise term

\displaystyle \frac{dX_t}{dt}=b(X_t)+\sigma(X_t)\xi_t. (2)

I have set ξt = dWt/dt which can be thought of as a process whose values at each time are independent zero-mean random variables. As mentioned above, though, Brownian motion is not differentiable so this does not exist in the usual sense. While it can be described by a kind of random distribution, even distribution theory is not well-equipped to handle such equations involving multiplying by the nondifferentiable process σ(Xt). Instead, (1) can be integrated to obtain

\displaystyle X_t=X_0+\int_0^t\sigma(X_s)\,dW_s+\int_0^tb(X_s)\,ds, (3)

where the right-hand-side is interpreted using stochastic integration with respect to the semimartingale W. Likewise, X will be a semimartingale, and such solutions are often referred to as diffusions.

The differential form (1) can be interpreted as a shorthand for the integral expression (3), which I will do in these notes. It can be generalized to n-dimensional processes by allowing b to take values in ℝn, a(x) to be an n × m matrix, and W to be an m-dimensional Brownian motion. That is, W = (W1, …, Wm) where Wi are independent Brownian motions. I will sometimes write this as

\displaystyle dX^t_i=\sigma_{ij}(X_t)dW^j_t+b_i(X_t)dt

where the summation convention is being applied, with subscripts or superscripts occuring more than once in a single term being summed from 1 to n.

Unlike ODEs, when dealing with SDEs we need to consider what underlying probability space the solution is defined with respect to. This leads to the existence of different classes of solutions.

  • Strong solutions where X can be expressed as a measurable function of the Brownian motion W or, equivalently, X is adapted to its natural filtration.
  • Weak solutions where X need not be a function of W. Such cases may require additional randomness so may not exist on the probability space with respect to which the Brownian motion W is defined. It can be necessary to extend the filtered probability space to construct these solutions.

Likewise, when considering uniqueness of solutions, there are different ways this occurs.

  • Pathwise uniqueness where, up to indistinguishability, there is only one solution X. This should hold not just on one specific space containing a Brownian motion W, but on all such spaces. That is, weak solutions should be unique.
  • Uniqueness in law where there may be multiple pathwise solutions, but their distribution is uniquely determined by the SDE.

There are various general conditions under which strong solutions and pathwise uniqueness are guaranteed for SDE (1) , such as the Itô result for Lipschitz continuous coefficients. I covered this situation in a previous post.

Other than using the SDE (1), such systems can also be described by an associated differential operator. For the n-dimensional case set a(x) = σ(x)σ(x)T, which is an n × n positive semidefinite matrix. Then, the second order operator L can be defined

\displaystyle Lf(x)=\frac12a_{ij}(x)f_{,ij}(x)+b_{i}(x)f_{,i}(x)

operating on twice continuously differentiable functions f: ℝn → ℝ. Being able to effortlessly switch between descriptions using the SDE (1) and the operator L is a huge benefit when working with such systems. There are several different ways in which the operator can be used to describe a stochastic process, all of which relate to weak solutions and uniqueness in law of the SDE.

Markov Generator: A Markov process is a weak solution to the SDE (1) if its infinitesimal generator is L. That is, if the transition function is Pt then,

\displaystyle \lim_{t\rightarrow0}t^{-1}(P_tf-f)=Lf

for suitably regular functions f.

Backwards Equation: For a function f: ℝn × ℝ+ → ℝ, f(t, Xt) is a local martingale if and only if it solves the partial differential equation (PDE)

\displaystyle \frac{\partial f}{\partial t}+Lf=0.

Consequently, for any time t > 0 and function g: ℝd → ℝ, if we let f be a solution to the PDE above with boundary condition f(x, t) = g(x) then, assuming integrability conditions, the conditional expectations at times s < t are

\displaystyle {\mathbb E}[g(X_t)\;\vert\mathcal F_s]=f(X_s,s).

If the conditions are satisfied, this describes a Markov process and gives its transition probabilities, describing the distribution of X and implying uniqueness in law.

Forward Equation: Assuming that it is sufficiently smooth, the probability density p(t, x) of Xt satisfies the PDE

\displaystyle \frac{\partial p}{\partial t}=L^Tf.

where LT is the transpose of operator L

\displaystyle L^Tp=\frac12(a_{ij}p)_{,ij}+(b_ip)_{,i}.

If this PDE has a unique solution for given initial distribution, then this uniquely determines the distribution of Xt. So, if unique solutions to the forward equation exist starting at every future time, it gives uniqueness in law for X.

Martingale problem: Any weak solution to SDE (1) satisfies the property that

\displaystyle f(X_t)-\int_0^t Lf(X_s)\,ds

is a local martingale for twice continuously differentiable functions f: ℝn → ℝ. This approach, which was pioneered by Stroock and Varadhan, has many benefits over the other applications of operator L described above, since it applies much more generally. We do not need to a-priori impose any properties on X such as being Markov, and as the test functions f are chosen at will, they automatically satisfy the necessary regularity properties. As well as being a very general way to describe solutions to a stochastic dynamical system, it turns out to be very fruitful. The striking and far-reaching Stroock–Varadhan uniqueness theorem, in particular, guarantees existence and uniqueness in law so long as a is continuous and positive definite and b is locally bounded.

Zero-Hitting and Failure of the Martingale Property

For nonnegative local martingales, there is an interesting symmetry between the failure of the martingale property and the possibility of hitting zero, which I will describe now. I will also give a necessary and sufficient condition for solutions to a certain class of stochastic differential equations to hit zero in finite time and, using the aforementioned symmetry, infer a necessary and sufficient condition for the processes to be proper martingales. It is often the case that solutions to SDEs are clearly local martingales, but is hard to tell whether they are proper martingales. So, the martingale condition, given in Theorem 4 below, is a useful result to know. The method described here is relatively new to me, only coming up while preparing the previous post. Applying a hedging argument, it was noted that the failure of the martingale property for solutions to the SDE {dX=X^c\,dB} for {c>1} is related to the fact that, for {c<1}, the process hits zero. This idea extends to all continuous and nonnegative local martingales. The Girsanov transform method applied here is essentially the same as that used by Carlos A. Sin (Complications with stochastic volatility models, Adv. in Appl. Probab. Volume 30, Number 1, 1998, 256-268) and B. Jourdain (Loss of martingality in asset price models with lognormal stochastic volatility, Preprint CERMICS, 2004-267).

Consider nonnegative solutions to the stochastic differential equation

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dX=a(X)X\,dB,\smallskip\\ &\displaystyle X_0=x_0, \end{array} (1)

where {a\colon{\mathbb R}_+\rightarrow{\mathbb R}}, B is a Brownian motion and the fixed initial condition {x_0} is strictly positive. The multiplier X in the coefficient of dB ensures that if X ever hits zero then it stays there. By time-change methods, uniqueness in law is guaranteed as long as a is nonzero and {a^{-2}} is locally integrable on {(0,\infty)}. Consider also the following SDE,

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dY=\tilde a(Y)Y\,dB,\smallskip\\ &\displaystyle Y_0=y_0,\smallskip\\ &\displaystyle \tilde a(y) = a(y^{-1}),\ y_0=x_0^{-1} \end{array} (2)

Being integrals with respect to Brownian motion, solutions to (1) and (2) are local martingales. It is possible for them to fail to be proper martingales though, and they may or may not hit zero at some time. These possibilities are related by the following result.

Theorem 1 Suppose that (1) and (2) satisfy uniqueness in law. Then, X is a proper martingale if and only if Y never hits zero. Similarly, Y is a proper martingale if and only if X never hits zero.

Continue reading “Zero-Hitting and Failure of the Martingale Property” →

Failure of the Martingale Property

In this post, I give an example of a class of processes which can be expressed as integrals with respect to Brownian motion, but are not themselves martingales. As stochastic integration preserves the local martingale property, such processes are guaranteed to be at least local martingales. However, this is not enough to conclude that they are proper martingales. Whereas constructing examples of local martingales which are not martingales is a relatively straightforward exercise, such examples are often slightly contrived and the martingale property fails for obvious reasons (e.g., double-loss betting strategies). The aim here is to show that the martingale property can fail for very simple stochastic differential equations which are likely to be met in practice, and it is not always obvious when this situation arises.

Consider the following stochastic differential equation

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dX = aX^c\,dB +b X dt,\smallskip\\ &\displaystyle X_0=x, \end{array} (1)

for a nonnegative process X. Here, B is a Brownian motion and a,b,c,x are positive constants. This a common SDE appearing, for example, in the constant elasticity of variance model for option pricing. Now consider the following question: what is the expected value of X at time t?

The obvious answer seems to be that {{\mathbb E}[X_t]=xe^{bt}}, based on the idea that X has growth rate b on average. A more detailed argument is to write out (1) in integral form

\displaystyle X_t=x+\int_0^t\,aX^c\,dB+ \int_0^t bX_s\,ds. (2)

The next step is to note that the first integral is with respect to Brownian motion, so has zero expectation. Therefore,

\displaystyle {\mathbb E}[X_t]=x+\int_0^tb{\mathbb E}[X_s]\,ds.

This can be differentiated to obtain the ordinary differential equation {d{\mathbb E}[X_t]/dt=b{\mathbb E}[X_t]}, which has the unique solution {{\mathbb E}[X_t]={\mathbb E}[X_0]e^{bt}}.

In fact this argument is false. For {c\le1} there is no problem, and {{\mathbb E}[X_t]=xe^{bt}} as expected. However, for all {c>1} the conclusion is wrong, and the strict inequality {{\mathbb E}[X_t]<xe^{bt}} holds.

The point where the argument above falls apart is the statement that the first integral in (2) has zero expectation. This would indeed follow if it was known that it is a martingale, as is often assumed to be true for stochastic integrals with respect to Brownian motion. However, stochastic integration preserves the local martingale property and not, in general, the martingale property itself. If {c>1} then we have exactly this situation, where only the local martingale property holds. The first integral in (2) is not a proper martingale, and has strictly negative expectation at all positive times. The reason that the martingale property fails here for {c>1} is that the coefficient {aX^c} of dB grows too fast in X.

In this post, I will mainly be concerned with the special case of (1) with a=1 and zero drift.

\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dX=X^c\,dB,\smallskip\\ &\displaystyle X_0=x. \end{array} (3)

The general form (1) can be reduced to this special case, as I describe below. SDEs (1) and (3) do have unique solutions, as I will prove later. Then, as X is a nonnegative local martingale, if it ever hits zero then it must remain there (0 is an absorbing boundary).

The solution X to (3) has the following properties, which will be proven later in this post.

  • If {c\le1} then X is a martingale and, for {c<1}, it eventually hits zero with probability one.
  • If {c>1} then X is a strictly positive local martingale but not a martingale. In fact, the following inequality holds
    \displaystyle {\mathbb E}[X_t\mid\mathcal{F}_s]<X_s (4)

    (almost surely) for times {s<t}. Furthermore, for any positive constant {p<2c-1}, {{\mathbb E}[X_t^p]} is bounded over {t\ge0} and tends to zero as {t\rightarrow\infty}.

Continue reading “Failure of the Martingale Property” →

SDEs Under Changes of Time and Measure

The previous two posts described the behaviour of standard Brownian motion under stochastic changes of time and equivalent changes of measure. I now demonstrate some applications of these ideas to the study of stochastic differential equations (SDEs). Surprisingly strong results can be obtained and, in many cases, it is possible to prove existence and uniqueness of solutions to SDEs without imposing any continuity constraints on the coefficients. This is in contrast to most standard existence and uniqueness results for both ordinary and stochastic differential equations, where conditions such as Lipschitz continuity is required. For example, consider the following SDE for measurable coefficients {a,b\colon{\mathbb R}\rightarrow{\mathbb R}} and a Brownian motion B

\displaystyle dX_t=a(X_t)\,dB_t+b(X_t)\,dt. (1)

If a is nonzero, {a^{-2}} is locally integrable and b/a is bounded then we can show that this has weak solutions satisfying uniqueness in law for any specified initial distribution of X. The idea is to start with X being a standard Brownian motion and apply a change of time to obtain a solution to (1) in the case where the drift term b is zero. Then, a Girsanov transformation can be used to change to a measure under which X satisfies the SDE for nonzero drift b. As these steps are invertible, every solution can be obtained from a Brownian motion in this way, which uniquely determines the distribution of X.

A standard example demonstrating the concept of weak solutions and uniqueness in law is provided by Tanaka’s SDE

\displaystyle dX_t={\rm sgn}(X_t)\,dB_t (2)

Continue reading “SDEs Under Changes of Time and Measure” →

SDEs with Locally Lipschitz Coefficients

In the previous post it was shown how the existence and uniqueness of solutions to stochastic differential equations with Lipschitz continuous coefficients follows from the basic properties of stochastic integration. However, in many applications, it is necessary to weaken this condition a bit. For example, consider the following SDE for a process X

\displaystyle dX_t =\sigma \vert X_{t-}\vert^{\alpha}\,dZ_t,

where Z is a given semimartingale and {\sigma,\alpha} are fixed real numbers. The function {f(x)\equiv\sigma\vert x\vert^\alpha} has derivative {f^\prime(x)=\sigma\alpha {\rm sgn}(x)|x|^{\alpha-1}} which, for {\alpha>1}, is bounded on bounded subsets of the reals. It follows that f is Lipschitz continuous on such bounded sets. However, the derivative of f diverges to infinity as x goes to infinity, so f is not globally Lipschitz continuous. Similarly, if {\alpha<1} then f is Lipschitz continuous on compact subsets of {{\mathbb R}\setminus\{0\}}, but not globally Lipschitz. To be more widely applicable, the results of the previous post need to be extended to include such locally Lipschitz continuous coefficients.

In fact, uniqueness of solutions to SDEs with locally Lipschitz continuous coefficients follows from the global Lipschitz case. However, solutions need only exist up to a possible explosion time. This is demonstrated by the following simple non-stochastic differential equation

\displaystyle dX= X^2\,dt.

For initial value {X_0=x>0}, this has the solution {X_t=(x^{-1}-t)^{-1}}, which explodes at time {t=x^{-1}}. Continue reading “SDEs with Locally Lipschitz Coefficients” →

Existence of Solutions to Stochastic Differential Equations

A stochastic differential equation, or SDE for short, is a differential equation driven by one or more stochastic processes. For example, in physics, a Langevin equation describing the motion of a point {X=(X^1,\ldots,X^n)} in n-dimensional phase space is of the form

\displaystyle \frac{dX^i}{dt} = \sum_{j=1}^m a_{ij}(X)\eta^j(t) + b_i(X). (1)

The dynamics are described by the functions {a_{ij},b_i\colon{\mathbb R}^n\rightarrow{\mathbb R}}, and the problem is to find a solution for X, given its value at an initial time. What distinguishes this from an ordinary differential equation are random noise terms {\eta^j} and, consequently, solutions to the Langevin equation are stochastic processes. It is difficult to say exactly how {\eta^j} should be defined directly, but we can suppose that their integrals {B^j_t=\int_0^t\eta^j(s)\,ds} are continuous with independent and identically distributed increments. A candidate for such a process is standard Brownian motion and, up to constant scaling factor and drift term, it can be shown that this is the only possibility. However, Brownian motion is nowhere differentiable, so the original noise terms {\eta^j=dB^j_t/dt} do not have well defined values. Instead, we can rewrite equation (1) is terms of the Brownian motions. This gives the following SDE for an n-dimensional process {X=(X^1,\ldots,X^n)}

\displaystyle dX^i_t = \sum_{j=1}^m a_{ij}(X_t)\,dB^j_t + b_i(X_t)\,dt (2)

where {B^1,\ldots,B^m} are independent Brownian motions. This is to be understood in terms of the differential notation for stochastic integration. It is known that if the functions {a_{ij}, b_i} are Lipschitz continuous then, given any starting value for X, equation (2) has a unique solution. In this post, I give a proof of this using the basic properties of stochastic integration as introduced over the past few posts.

First, in keeping with these notes, equation (2) can be generalized by replacing the Brownian motions {B^j} and time t by arbitrary semimartingales. As always, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}. In integral form, the general SDE for a cadlag adapted process {X=(X^1,\ldots,X^n)} is as follows,

\displaystyle X^i = N^i + \sum_{j=1}^m\int a_{ij}(X)\,dZ^j. (3)

Continue reading “Existence of Solutions to Stochastic Differential Equations” →