Simplified Stochastic Calculus With Applications in Economics and Finance

For reasons of tractability, there is a preponderance of continuous-time stochastic models based on Brownian motion. Traditional stochastic calculus reflects this historical bias. As an example, consider a stochastic model for two economic variables, capital $K$ and labour $L$,

Here $W$ and $\widehat{W}$ are two independent Brownian motions. The inputs in this model are $\mu_{K}$, $\mu_{L}$, $\sigma_{K}>0$, $\sigma_{L}>0$, and $\rho_{KL}\in[-1,1]$ describing the correlation between the changes in $K$ and $L$. Informally, the ‘drift part’ $\mu_{K}\mathrm{d}t$ represents expected change, while the ‘noise’ $\sigma_{K}\mathrm{d}W_{t}$ is loosely interpreted as a shock with mean zero and variance $\sigma_{K}^{2}\mathrm{d}t$.

The symbol $\mathrm{d}K_{t}$ represents an increase in capital over an infinitesimal time period $\mathrm{d}t$. The left-hand side of (1.1) signifies percentage change in capital over the same period. The percentage change per unit of time is not well-defined because the derivative $\nicefrac{{\mathrm{d}W_{t}}}{{\mathrm{d}t}}$ does not exist. However, the expected percentage change per unit of time is finite and equal to $\mu_{K}$. This means the expected proportional increase in capital over a fixed time horizon $T$ equals

In the terminology of Samuelson (1965), $K$ and $L$ are geometric Brownian motions with drift rates $\mu_{K}$ and $\mu_{L}$, respectively.

Suppose we are interested in the evolution of the capital-labour ratio, $\nicefrac{{K}}{{L}}$. The standard Itô calculus (Itô 1951, Theorem 6; Karatzas and Shreve 1991, Theorem 3.6; Duffie 2001, p. 95; Björk 2009, Theorem 4.16) yields, after simplifications,

One can make two observations at this point:

1. (1)

   The formula (1.4) is not easy to decipher — the calculations are involved.

2. (2)

   The formula is ‘misleading’ — the processes $K$ and $L$ are already given, therefore the object on the left-hand side is defined path by path as the ratio $\nicefrac{{K_{T}(\omega)}}{{L_{T}(\omega)}}$ and cannot depend on the reference probability measure. In contrast, some of the objects on the right-hand side are strongly measure-dependent: certainly, if we change the reference probability measure, there is no guarantee that $W$ and $\widehat{W}$ will still be Brownian motions under the new measure.

McKean (1969) addresses (1) and informally also (2) by rewriting (1.4) in the form

where $\mathrm{d}K_{t}\mathrm{d}L_{t}$ is understood to stand for $\mathrm{d}[K,L]_{t}$ and $[K,L]$ is the quadratic covariation of the processes $K$ and $L$. In the present case we have

Let us make two further observations:

1. (3)

   Formula (1.5) is not only measure-independent, it is also model-free in the sense that it holds for any two continuous semimartingales $K$ and $L$ such that the integrals of $\nicefrac{{\mathrm{d}K_{t}}}{{K_{t}}}$ and $\nicefrac{{\mathrm{d}L_{t}}}{{L_{t}}}$ are well-defined.

2. (4)

   McKean (1969, p. 33) observes that one can obtain (1.5) much more directly without passing through (1.1)–(1.2) and (1.4), simply by writing down a second-order Taylor expansion for $f(K,L)=\nicefrac{{K}}{{L}}$ in the form

   $$\begin{split}\mathrm{d}f(K_{t},L_{t})&=\tfrac{\partial f}{\partial k}(K_{t},L_{t})\mathrm{d}K_{t}+\tfrac{\partial f}{\partial\ell}(K_{t},L_{t})\mathrm{d}L_{t}\\ &\ +\frac{1}{2}\left(\tfrac{\partial^{2}f}{\partial k^{2}}(K_{t},L_{t})(\mathrm{d}K_{t})^{2}+2\tfrac{\partial^{2}f}{\partial k\partial\ell}(K_{t},L_{t})\mathrm{d}K_{t}\mathrm{d}L_{t}+\tfrac{\partial^{2}f}{\partial\ell^{2}}(K_{t},L_{t})(\mathrm{d}L_{t})^{2}\right).\end{split}$$

   (1.6)

Suppose now we wish to evaluate the expected value of the capital–labour ratio. Here the formula (1.4) is very helpful because it tells us immediately that $\nicefrac{{K}}{{L}}$ is a geometric Brownian motion with the drift rate

With this coefficient in hand one swiftly concludes, in analogy to (1.3),

The need for equation (1.4) is only illusory, however. One can obtain formula (1.7) equally easily from the measure-independent formula (1.5) by inserting the expected rate of change of $K,L$ and their quadratic (co)variations on the right-hand side of (1.5) as implied by (1.1)–(1.2),

Note that apart from the initial values $K_{0}$, $L_{0}$ and the time horizon $T$, the calculation requires five characteristics of the capital and labour processes: $\mu_{K}$, $\mu_{L}$, $\sigma_{K}$, $\sigma_{L}$, and $\rho_{KL}$.

Let us now modify (1.1)–(1.2) by adding two jump components $J^{K}$ and $J^{L}$ that jointly form a Lévy process,

Here $N$ is a Poisson random measure and $\Pi$ the corresponding Lévy measure. In particular, when $(J^{K},J^{L})$ form a compound Poisson process, i.e., when $\Pi(\mathbb{R}^{2})<\infty$, the quantity $\Pi(\mathbb{R}^{2})$ is the arrival intensity of jumps and

is the bivariate cumulative distribution of jump sizes. The complicated expression in (1.14) is required to accommodate the models where the small jumps of $L$ have infinite variation while, at the same time, the large jumps of $L$ have infinite mean.

In the presence of jumps, by convention, the paths of $K$ and $L$ are assumed to be right-continuous with left limits. That means the value of capital before a jump is given by the left limit $K_{t-}$ while $K_{t}$ is the value after a jump. The jump is naturally defined as the difference between the two, $\Delta K_{t}=K_{t}-K_{t-}$ and likewise for $L$; see Figure 1.1.

Very little has changed between (1.1)–(1.2) and (1.11)–(1.12); we have replaced one process with time-homogeneous independent increments by another. But unlike in the Brownian case, it is now mathematically possible that the right-hand side of (1.12) — the percentage change in labour supply $L$ — does not have a finite first moment, while the mean of $\nicefrac{{K}}{{L}}$ is nevertheless finite.¹¹1From a modelling point of view it is not realistic to believe that $L$ has infinite mean. We are simply stating that the calculus must be general enough to entertain such possibility. There are other circumstances where infinite mean arises more naturally, for example stock price $S$ will typically have finite mean but $\ln S$ may plausibly have mean equal to $-\infty$ in a model with jumps if returns close to -100% occur frequently enough.

The main takeaway message is that decomposing a stochastic integral into ‘signal’ and ‘noise’ as suggested by (1.1)–(1.2) is, in general, not straightforward. One possibility is to split $J^{L}$ into two components containing the small and large jumps, respectively, and decompose only the small jump component into signal and noise as shown in (1.14). This makes (1.12) look more like (1.2) and largely represents the current practice in applications where jumps of the most general type are considered, see Kallsen (2000), Fujiwara and Miyahara (2003), Hubalek et al. (2006), Jeanblanc et al. (2007), Øksendal and Sulem (2007), Bender and Niethammer (2008), and Applebaum (2009).

In this paper, we do handle arbitrary jumps but we interpret the difficulty with signal–noise decomposition very differently, which is to say we refrain from using such a decomposition altogether and instead look for a measure-invariant representation of $\nicefrac{{K}}{{L}}$. The sought expression must reduce to the McKean calculus (1.6) when $K$ and $L$ are continuous. At the same time, it must correctly account for changes due to jumps in $K$ and $L$. Just such a formula, suitably reinterpreted, can be traced to Émery (1978, Section 3). Let us now describe what we mean, first informally and then rigorously.

In the present example we seek the representation of $\nicefrac{{K}}{{L}}$. This will be written symbolically as

which informally leads to an expression for percentage changes

For Émery, the right-hand side of (1.16) represents a deterministic, time-constant function

that acts on the increments $\nicefrac{{\mathrm{d}K_{t}}}{{K_{t-}}}$ and $\nicefrac{{\mathrm{d}L_{t}}}{{L_{t-}}}$. The real meaning of the expression $\xi(\mathrm{d}X_{t})$ for a generic deterministic time-constant $\mathcal{C}^{2}$ function $\xi$ with $\xi(0)=0$ and any semimartingale $X$ is supplied by the Émery formula²²2The symbols $D\xi$ and $D^{2}\xi$ stand for first and second partial derivatives of $\xi$. Émery (1978) considers only the case when $\xi$ is deterministic and constant in time. We explicitly allow $\xi$ to be predictable in order to develop a calculus that includes stochastic integration and Itô’s formula.

where the last term yields absolutely summable jumps of finite variation and $[\cdot,\cdot]^{c}$ stands for the continuous part of quadratic covariation. An easy way to memorize the Émery formula is suggested in equation (2.12) and the subsequent paragraph.

Having assigned meaning to the right-hand side of (1.16) via the formula (1.18) with $\xi=\eta$ from (1.17), equality (1.16) is no longer an informal expression but a theorem whose validity one needs to establish. To accomplish this goal and build the simplified calculus, one can begin with the observation that (1.15), too, represents a function that acts on the increments of the underlying processes; in this case it acts on $\mathrm{d}K_{t}$ and $\mathrm{d}L_{t}$. One important difference is that the function in question is no longer deterministic, i.e., we have

with

Another practical consideration is that the predictable function $\tilde{\eta}$ in (1.19) is not finite-valued at the point where $x_{2}=-L_{-}$.

To accommodate functions with restricted domain, we say that

It is formally straightforward to apply the formula (1.18) to a predictable function $\xi$. However, it is a priori not clear that the formula will be well-defined; in particular, for $\xi=\tilde{\eta}$ in (1.19) it is not clear that one obtains

Ideally, we would like to have a calculus that gives rise only to those predictable functions $\xi$ for which the integral $\int_{0}^{\cdot}\xi_{t}(\mathrm{d}X_{t})$ is always well-defined so we do not have to manually check admissibility every time a new $\xi$ arises. The precise formulation of two such subclasses, whose elements will be called universal representing functions, is given in Appendix A. We denote by $\mathfrak{I}_{0\mathbb{R}}^{d,n}$ the subset of universal representing functions that map $\mathbb{R}^{d}$–valued processes to $\mathbb{R}^{n}$–dimensional processes. We also set $\mathfrak{I}_{0\mathbb{R}}=\bigcup_{d,n\in\mathbb{N}}\mathfrak{I}_{0\mathbb{R}}^{d,n}$. For computations involving characteristic functions, it is convenient to generalize the Émery formula (1.18) to a subset of complex predictable functions by interpreting $D\xi$ and $D^{2}\xi$ as complex derivatives. The subset of all universal representing functions that are complex-differentiable near the origin will be denoted by $\mathfrak{I}_{0\mathbb{C}}$.

Propositions A.3–A.5 in Appendix A show that the class $\mathfrak{I}_{0\mathbb{R}}$ is self-contained; if we use standard real-valued operations we are guaranteed to stay within $\mathfrak{I}_{0\mathbb{R}}$. The class $\mathfrak{I}_{0\mathbb{C}}$ is also self-contained provided the transformations we apply are complex-differentiable, as will be the case in all our examples involving complex numbers. These results guarantee that we will only ever encounter functions in $\mathfrak{I}_{0\mathbb{R}}$ (resp., $\mathfrak{I}_{0\mathbb{C}}$) and so will never have to check the conditions of Definition A.1 manually.

Just as the McKean calculus of Subsection 1.1, the simplified stochastic calculus is most intuitive when expressed in differential form, such as (1.15) and (1.16). When one wishes to speak of the integrated process whose increments are equal to $\xi_{t}(\mathrm{d}X_{t})$, one typically just introduces a new label, say $Y$, writing $\mathrm{d}Y_{t}=\xi_{t}(\mathrm{d}X_{t})$. There is nothing wrong with the relabelling approach; it does deliver all the immediate benefits of the simplified calculus and helps to keep technicalities to a minimum.

Side-by-side with the intuitive differential approach, we want to offer the reader an alternative ‘high-level’ view of the calculus where the roles of $\xi$ and $X$ are acknowledged explicitly. Accordingly, the process with increments $\xi_{t}(\mathrm{d}X_{t})$, starting at 0, will be denoted by $\xi\circ X$, i.e.,

The high-level notation may seem a little abstract at first, but it offers distinct benefits such as compactness and flexibility. For example, in the integral notation one can write

Here we let ${\operatorname{id}}$ denote the identity function;

Below we use ${\operatorname{id}}_{1}(x)=x_{1}$ for the first component, ${\operatorname{id}}_{2}$ for the second, etc., where required.

The notation $\xi\circ X$ also emphasizes the universality of the transformation $X\mapsto\xi\circ X$. In the same way that $[X,X]$ is well-defined for any semimartingale $X$, the process $|{\operatorname{id}}|^{\alpha}\circ X$ is well-defined for any semimartingale $X$ and any real $\alpha\geq 2$. This brings us to other universal transformations that are commonly used in the literature. For example, provided that $X$ and $X_{-}$ are different from zero, the literature defines $\mathcal{L}(X)$ as the process of cumulative percentage change in $X$, i.e.,

Thus, in the integral notation formula (1.16) reads

Provided that the cumulative percentage changes in $K$ and $L$ are well-defined, formula (1.21) holds for arbitrary semimartingales $K$ and $L$; it is new in this generality as far as we know.

The simplified stochastic calculus rests on a sequential application of Propositions A.3–A.5. In practice, one would not use these propositions directly but instead combine them into composite rules that transform a represented process $Y=Y_{0}+\xi\circ X$ into another represented process $Z$. In differential notation, we have $\mathrm{d}Y_{t}=\xi_{t}(\mathrm{d}X_{t})$ and the first two rules read as follows.

Stochastic integration with respect to $Y$:

For a locally bounded process $\zeta$ and the integral $Z=\int_{0}^{\cdot}\zeta_{t}\mathrm{d}Y_{t}$ one has

$$\mathrm{d}Z_{t}=\zeta_{t}\xi_{t}(\mathrm{d}X_{t}).$$

Itô formula for $Z=f(Y)$:

For a suitably smooth function $f$ such that $Y$ and $Y_{-}$ lie in the interior of the domain of $f$ one has

$$\mathrm{d}Z_{t}=\mathrm{d}f(Y_{t})=f(Y_{t-}+\mathrm{d}Y_{t})-f(Y_{t-})=f(Y_{t-}+\xi_{t}(\mathrm{d}X_{t}))-f(Y_{t-}).$$

We now restate the above fully rigorously in integral notation.

Let us outline a general scheme of how the two composite rules are applied in practice. At the outset, one will designate the primitive input to the problem at hand; this input process is thereafter labeled $X$. For example, in Subsections 1.1 and 1.2 it is natural to start from the bivariate process

Observe that $X$ is always representable with respect to itself thanks to Proposition A.3 with $\zeta$ equal to the $d\times d$ identity matrix. Starting off from the trivial representation $X=X_{0}+{\operatorname{id}}\circ X$, i.e., taking $Y=X$ and $\xi(x)=x$ in Corollary 2.1, one applies smooth transformation or stochastic integration as required to obtain the first intermediate result $Z$. This intermediate result (relabeled $Y$) becomes the input to the next application of Corollary 2.1 producing the next intermediate output $Z$. The $Y\to Z$ pattern is repeated until one reaches the desired output $Z$; in our example the goal is

Table 2.1 illustrates the steps required in the transition from (2.3) to (2.4).

The discussion above concerns formal calculations where the rules of the simplified calculus are applied mechanically. Many users will prefer a more intuitive approach whose main idea is apparent in the last column of Table 2.1. Here one observes that the calculus traces the behaviour of jumps and so one may effectively pretend that $X$, $Y$, and $Z$ are finite-variation pure-jump processes. In this way, it is possible to arrive at the correct $\xi$ even without applying formal rules. In the context of (1.21), for example, suppose $K$ increases by 50% and $L$ increases by 20%. The percentage change in $\nicefrac{{K}}{{L}}$ is then precisely

Therefore, formula (1.21), among other things, describes jump transformations: every time $\mathcal{L}(K)$ jumps by $x_{1}$ (e.g., 0.5) and $\mathcal{L}(L)$ jumps by $x_{2}$ (e.g., 0.2) the process $\mathcal{L}(\nicefrac{{K}}{{L}})$ jumps by

As a further example, let us see how the rules of the simplified calculus can be used to obtain the representation of the logarithmic return in terms of the rate of return.

As the final introductory example consider the representation of quadratic covariation.

Section 3 contains many more explicit representations that are useful in practice. Some of these are well known in the specialist literature, while others are new. Proposition A.5, in particular, is a powerful tool for obtaining new representations from old ones. We summarize it here in the form of a composition rule. Observe that in the differential notation the rule is completely natural; it asserts that

The composition rule allows the user to store some common calculations and ‘recycle’ them later without having to revisit their detailed derivation. Suppose, for instance, that we are given the evolution of $(\ln K,\ln L)$ as the primitive input. Thanks to Corollary 2.4, there is no need to calculate everything afresh all the way from $(\ln K,\ln L)$ to $\mathcal{L}(\nicefrac{{K}}{{L}})$. One only computes the passage from $(\ln K,\ln L)$ to $(\mathcal{L}(K),\mathcal{L}(L))$ which yields (see equation (3.4) below)

while the passage from $(\mathcal{L}(K),\mathcal{L}(L))$ to $\mathcal{L}(\nicefrac{{K}}{{L}})$ can be recycled from (1.21). The two results composed together give

In differential notation,

substituted into (1.16) yields

which is the differential equivalent of formula (2.7).

Having mastered the art of representing one process by means of another, we would like to obtain an analogon of (1.9)–(1.10). Our task is to express the drift of the represented process with the help of the characteristics of the representing process.

To begin with, we collect the predictable characteristics (i.e., the drift rate, the covariance matrix of the associated Brownian motion, and the Lévy measure) of the input process $X=(\mathcal{L}(K),\mathcal{L}(L))$ in equations (1.11)–(1.14) in the more compact form

Here $X[h^{1}]$ denotes the process $X$ with jumps greater than 1 in absolute value removed,

Observe that this precisely matches the decomposition of jumps appearing in (1.13)–(1.14) and ensures that the drift of $X[h^{1}]$ is finite.³³3In contrast, the drift of $X$ need not be well-defined in general; see also Footnote 1. More generally, we will denote by $X[h]$ the process containing the small jumps of $X$ as given by a specific truncation function $h$ and observe that $X[h^{1}]$ corresponds to the choice

where $d$ is the dimension of $X$. The mechanics of truncation are described in Definition B.1.

The reader must be warned that $X[0]$, the continuous part of $X$, is not always well-defined. For this reason, $X[0]$ cannot be universally represented in contrast to $X[h^{1}]$, whose universal representation appears in (2.9). Nonetheless, the situations where $X[0]$ exists do arise in practice, for example, in the Merton (1976) jump-diffusion model. In such models we may write

This, when substituted into (1.18), leads to the simplified expression

which offers a valuable insight into the nature of the Émery formula. We observe that the first two terms of (2.12) correspond to the McKean calculus for the continuous part of $X$ while the last term accounts for the jumps in $X$. The two components do not interact and can be treated separately. In the most general situation where the decomposition (2.11) does not exist, one can make (2.12) rigorous by adding small jumps to the first term and subtracting them in the last term to obtain

The original Émery formula (1.18) corresponds simply to the case where all the jumps of $X$ have been added to the first term and subtracted in the last term of (2.12).

We thus come to understand the Émery formula as a spectrum of equivalent expressions where one can dial the truncation function $h$ all the way down to $0$ or all the way up to $h(x)=x$. In this sense, the truncation is unimportant – we can always choose $h$ to suit our needs. In a univariate case, one would thus pick $h=0$ if jumps of $X$ have finite variation as in the Merton jump-diffusion model, failing that, $h={\operatorname{id}}$ if the drift of $X$ exists, and finally $h=h^{1}={\operatorname{id}}\mathbf{1}_{|{\operatorname{id}}|\leq 1}$ in all remaining cases. In a multivariate case this choice can be performed component-wise. With such choice of $h$, the drift of each contributing term in (2.13) is guaranteed to be finite.

We can now perform the feat previously achieved on a smaller scale in (1.9)–(1.10). By matching each term in (2.13) with its drift contribution, one obtains

Formula (2.14) is proved in Theorem B.6.

Specifically, with $\xi$ given in (1.17) we obtain

For the specific input parameters in (2.8) and the corresponding truncation function in (2.10) the drift conversion formula (2.14) yields

which is the appropriate generalization of (1.10) provided the integral in (2.15) converges.⁴⁴4We might consider, for example, a model where the jumps in $\mathcal{L}(K)$ and $\mathcal{L}(L)$ are independent, in which case $$\Pi(\mathrm{d}x_{1},\mathrm{d}x_{2})=\mathbf{1}_{x_{2}=0}\Pi^{K}(\mathrm{d}x_{1})+\mathbf{1}_{x_{1}=0}\Pi^{L}(\mathrm{d}x_{2}),$$ meaning capital and labour do not jump simultaneously. We may take $\Pi^{K}(\mathrm{d}x_{1})$ to be lognormal so that $\int_{0}^{\infty}x_{1}\Pi^{K}(\mathrm{d}x_{1})<\infty$ and $K$ has finite mean. The choice $\Pi^{L}(\mathrm{d}x_{2})=x_{2}^{-2}\mathbf{1}_{x_{2}>0}\mathrm{d}x_{2}$ then provides an example where $L$ has infinite variation (even with $\sigma^{L}=0$) and infinite mean while the mean of $K/L$ remains finite (see also Theorem B.4). Formula (1.8) continues to hold with this choice of $b$; see Theorems B.4 and B.6 below.