Itô's lemma
In mathematics, Itô's lemma is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for  option values. Motivation. Suppose we are given the stochastic differential equation formula_1 where is a Wiener process and the functions formula_2 are deterministic (not stochastic) functions of time. In general, it's not possible to write a solution formula_3 directly in terms of formula_4 However, we can formally write an integral solution formula_5 This expression lets us easily read off the mean and variance of formula_3 (which has no higher moments). First, notice that every formula_7 individually has mean 0, so the expectation value of formula_3 is simply the integral of the drift function: formula_9 Similarly, because the formula_10 terms  have variance 1 and no correlation with one another, the variance of formula_3 is simply the integral of the variance of each infinitesimal step in the random walk: formula_12 However, sometimes we are faced with a stochastic differential equation for a more complex process formula_13 in which the process appears on both sides of the differential equation. That is, say formula_14 for some functions formula_15 and formula_16 In this case, we cannot immediately write a formal solution as we did for the simpler case above. Instead, we hope to write the process formula_17 as a function of a simpler  process formula_3 taking the form above. That is, we want to identify three functions formula_19 and formula_20 such that formula_21 and formula_22 In practice, Ito's lemma is used in order to find this transformation. Finally, once we have transformed the problem into the simpler type of problem, we can determine the mean and higher moments of the process. Informal derivation. A formal proof of the lemma relies on taking the limit of a sequence of random variables. This approach is not presented here since it involves a number of technical details. Instead, we give a sketch of how one  can derive Itô's lemma by expanding a Taylor series and applying the rules of stochastic calculus. Suppose is an Itô drift-diffusion process that satisfies the stochastic differential equation where is a Wiener process. If is a twice-differentiable scalar function, its expansion in a Taylor series is Substituting for and therefore for gives In the limit , the terms and tend to zero faster than , which is . Setting the and terms to zero, substituting for (due to the quadratic variation of a Wiener process), and collecting the and terms, we obtain as required. Mathematical formulation of Itô's lemma . In the following subsections we discuss versions of Itô's lemma for different types of stochastic processes. Itô drift-diffusion processes (due to: Kunita–Watanabe). In its simplest form, Itô's lemma states the following: for an Itô drift-diffusion process and any twice differentiable scalar function of two real variables and , one has This immediately implies that is itself an Itô drift-diffusion process. In higher dimensions, if formula_29 is a vector of Itô processes such that for a vector formula_31 and matrix formula_32, Itô's lemma then states that where formula_34 is the gradient of w.r.t. , is the  Hessian matrix of w.r.t. , and is the trace operator. Poisson jump processes. We may also define functions on discontinuous stochastic processes. Let be the jump intensity. The Poisson process model for jumps is that the probability of one jump in the interval is plus higher order terms. could be a constant, a deterministic function of time, or a stochastic process. The survival probability is the probability that no jump has occurred in the interval . The change in the survival probability is So Let be a discontinuous stochastic process. Write formula_37 for the value of "S" as we approach "t" from  the left. Write formula_38 for the non-infinitesimal change in as a result of a jump. Then Let "z" be the magnitude of the jump and let formula_40 be the distribution of "z". The expected magnitude of the jump is Define formula_42, a compensated process and martingale, as Then Consider a function formula_45 of the jump process . If jumps by then jumps by . is drawn from distribution formula_46 which may depend on formula_47, "dg" and formula_37. The jump part of formula_49 is If formula_51 contains drift, diffusion and jump parts, then Itô's Lemma for formula_45 is Itô's lemma  for a process which is the sum of a drift-diffusion process and a jump process is just the sum of the Itô's lemma for the individual parts. Non-continuous semimartingales. Itô's lemma can also be applied to general -dimensional semimartingales, which need not be continuous. In general, a semimartingale is a càdlàg process, and an additional term needs to be added to the formula to ensure that the jumps of the process are correctly given by Itô's lemma. For any cadlag process , the left limit in is denoted by , which is a left-continuous process. The  jumps are written as . Then, Itô's lemma states that if is a -dimensional semimartingale and "f" is a twice continuously differentiable real valued function on then "f"("X") is a semimartingale, and This differs from the formula for continuous semi-martingales by the additional term summing over the jumps of "X", which ensures that the jump of the right hand side at time is Δ"f"("X"). Multiple non-continuous jump processes. There is also a version of this for a twice-continuously differentiable in space once in time function f evaluated at (potentially different) non-continuous semi-martingales which may be  written as follows: Examples. Geometric Brownian motion. A process S is said to follow a geometric Brownian motion with constant volatility "σ" and constant drift "μ" if it satisfies the stochastic differential equation formula_56, for a Brownian motion "B". Applying Itô's lemma with formula_57 gives It follows that exponentiating gives the expression for "S", The correction term of corresponds to the difference between the median and mean of the log-normal distribution, or equivalently for this distribution, the geometric mean and arithmetic mean, with the median (geometric mean) being lower. This is due to the AM–GM inequality, and  corresponds to the logarithm being concave (or convex upwards), so the correction term can accordingly be interpreted as a convexity correction. This is an infinitesimal version of the fact that the annualized return is less than the average return, with the difference proportional to the variance. See geometric moments of the log-normal distribution for further discussion. The same factor of appears in the "d" and "d" auxiliary variables of the Black–Scholes formula, and can be interpreted as a consequence of Itô's lemma. Doléans-Dade exponential. The Doléans-Dade exponential (or stochastic exponential) of a continuous semimartingale "X  can be defined as the solution to the SDE with initial condition . It is sometimes denoted by . Applying Itô's lemma with f"("Y") = log("Y") gives Exponentiating gives the solution Black–Scholes formula. Itô's lemma can be used to derive the Black–Scholes equation for an option. Suppose a stock price follows a geometric Brownian motion given by the stochastic differential equation . Then, if the value of an option at time is "f"("t", "S"), Itô's lemma gives The term represents the change in value in time "dt" of the trading strategy consisting of holding an amount of the  stock. If this trading strategy is followed, and any cash held is assumed to grow at the risk free rate "r", then the total value "V" of this portfolio satisfies the SDE This strategy replicates the option if "V" = "f"("t","S"). Combining these equations gives the celebrated Black–Scholes equation Product rule for Itô processes. Let formula_66 be a two-dimensional Ito process with SDE: Then we can use the multi-dimensional form of Ito's lemma to find an expression for formula_68. We have formula_69 and formula_70. We set formula_71 and observe that formula_72 and formula_73 Substituting these values  in the multi-dimensional version of the lemma gives us: This is a generalisation of Leibniz's product rule to Ito processes, which are non-differentiable. Further, using the second form of the multidimensional version above gives us so we see that the product formula_76 is itself an Itô drift-diffusion process.