In the broad field of probability I am interested in the following topics, their interconnections and their applications:

• Lévy processes
• Stochastic differential equations
• Backward stochastic differential equations (particularly those driven by Lévy processes)
• Malliavin calculus (particularly in the case of Lévy processes)

### Lévy processes

Lévy processes are stochastic processes which generalize the model of Brownian motion: Applications demanded a mathematical description of a process that evolves in time and possesses increments which are independent over different time intervals. If we assume that the process at a fixed point in time is normally distributed, then the model results in a continuous process called 'Brownian motion'. Brownian motion became ubiquitous in (continuous) stochastic analysis and was fundamental for the famous model of Black and Scholes in mathematical finance.
However, the assumption of underlying normal distributions fails for many real-world applications. Thus, one had to extend the theory of processes with independent increments to a more general class, one that includes a broader set of distributions (infinitely divisible distributions). The resulting processes are called Lévy-processes. They possess properties not only similar to Brownian motion but also newer, interesting ones. For example, Lévy processes may be discontinuous - they possess jumps of random heights at random times. This complicates the model on the one hand - but on the other it makes these processes appropriate for applications that contain random jumps. Such are insurance models (the Poisson processs, a famous accident-counting process is a Lévy process), income models with random loss, market models with shocks and many more. Another obstacle that appears is that Lévy processes do not have moments of any order.
Despite these difficulties, Lévy processes are in some sense not too far away from Brownian motion: The theorem known as 'Lévy-Itô decomposition' states that every Lévy process may be represented as sum of a linear function, a Brownian motion and a part that produces the random jumps. This is a key theorem which makes the theory of Lévy processes a theory of Brownian motion with random jumps and led to great developments within the theory. Still many open problems - already for basic questions - remain: One does not even know yet the distribution of the maximum of a (general) Lévy process over an interval.
Nowadays, because of their versatility (even the assumption of independence of the increments may be weakend) to fit models, this type of stochastic processes is widely used in mathematical descriptions coming from physics, biology, insurances and finance.

Lecture notes by A. Kyprianou

### Stochastic differential equations

Defining stochastic processes with certain distributions is not enough. Usually, stochastic models are considered, when random quantities, disturbances or 'noises' enter a system whose deterministic dynamics were - principally - known. Let us assume, that a mathematical model results in a differential equation. If now, a random noise term is added to the equation, also its solution will be expected to be a stochastic process. So, the deterministic differential equation becomes a stochastic differential equation, an SDE. SDEs may emerge not only from a 'noise' influencing a differential equation. The model may be stochastic from the very fundamentals. This is the case e.g. for stock prices or interest rates, which are modeled as solutions of SDEs, since the underlying movement or fluctuations (of the market, economy) do not permit any obvious deterministic structure.
SDEs are mathematically more difficult to handle than deterministic ones. The main problem is that basic stochastic processes that may be inherent to the models (such as Brownian motion or the Poisson process) are not differentiable. As a response to that fact, a first reaction could be to change the differential formulation of the equation to an integral-integrator formulation. Even then, a first naive formulation in this way fails due to the unbounded variation of some of the involved processes. The solution: A meaningful formulation of stochastic differential equations needed an appropriate integration theory for such processes - stochastic integrals. The development of stochastic integration theory led to an intuitive calculus, fit to treat and solve SDEs in a satisfying way. Main tools of this calculus are the Itô-isometry and Itô's formula (a type of differentiation formula for stochastic processes).
SDEs play no lesser role in stochastic models from various sciences, especially finance, than their underlying processes do. Moreover, for the sake of flexible modeling, the types of processes and SDEs may even be combined: SDEs can be considered with Brownian motion as driving 'noise', 'Lévy noise', depending whether one wants to include jumps in the model or not, or an even more general, unifying type of processes describing the random influence (called semimartingales).

SDEs in Finance by A. Papanicolaou

### Backward stochastic differential equations (BSDEs)

From the early '90s onward there has been growing interest in backward stochastic differential equations. As a result, BSDE theory developed into a broad area of research nowadays. A backward stochastic equation may be viewed as stochastic differential equation but - as the name suggests - backward in time. This may look trivial from a deterministic viewpoint, but stochastics complicates the situation. In a deterministic setting, a terminal value problem is equivalent to an initial value problem since at both time points (the beginning and the end), we have deterministic values. A stochastic initial value problem starts deterministically and as time goes onward, randomness grows. This is where the backward situation behaves differently: We have a random terminal value and want to find a deterministic value in the past. Hence, methods to treat BSDEs are substantially different from the (forward-)SDE situation. In general, it is impossible to attain a deterministic value in the past only by dynamics of the differential equation. To solve this, one needs a process that 'compensates' randomness of the driving (Brownian or Lévy) process backward in time. The solution of a BSDE therefore consists of two processes, the Y-process that satisfies the equation dynamics and the Z-process, which works as coefficient for the random noise of the backward equation. Applications of BSDEs are manifold in finance. Among some mathematicians in finance there exists an unwritten meta-theorem suggesting that every problem in mathematical finance is - in some form - solution to a type of BSDEs. Another valuable aspect is that with help of 'Feynman-Kac theory', BSDEs provide solutions to deterministic partial differential equations. This leads to the possibility to numerically solve partial differential equations by stochastic simulations via Monte-Carlo methods. Following this way, BSDEs in the Lévy process case yield even the possibility to solve partial differential-integral equations, which are otherwise hard to treat.

A very useful introduction to BSDEs by Bruno Bouchard

### Malliavin calculus

Malliavin calculus is a type of differential calculus for random variables. Let us consider random variables whose randomness emerges from an underlying Brownian motion or a Lévy process. Such random variables obey an important theorem, the 'predictable representation property'. The theorem basically states that, if a random variable of this type possesses a finite variance, we can represent it as sum of its expectation plus a stochastic integral over a predictable process, $\xi=\mathbb{E}\xi+\int_0^T H_s(\xi) dW_s$. The stochastic integral in the Brownian case is taken with respect to Brownian motion itself over time (as written above). In the Lévy case the predictable representation is only possible with respect to a stochastic integral relying on a jump measure that involves time and the size of the jumps and is therefore two-dimensional. This theoretical difference of representation theorems is the reason why markets underlying a (non-Brownian) Lévy process are, in general, 'incomplete', some may even allow arbitrage situations.
One basic intention of Malliavin calculus is to compute the integrand that appears in the predictable representation theorem. Therefore the terms 'differential calculus' and 'Malliavin derivative' are justified. One of the main results of Malliavin calculus is the Clark-Ocone formula which presents an explicit formula for the requested integrand.
Another aspect of Malliavin calculus is the chaos decomposition of random variables. We may represent square-integrable random variables not only as sum of a number and a stochastic integral, moreover we may write them as sum of their expectation and a (possibly) infinite series of iterated stochastic integrals over deterministic functions. The $n$-th iterated stochastic integral is referred to as $n$-th chaos. This 'chaos decomposition' has many applications for approximating and simulating certain types of random variables and is closely interconnected to the Malliavin derivative.
Malliavin calculus became a useful tool for treating SDEs as well directly in finance. It allows direct access to 'Greeks' (coefficients and solutions of SDEs from mathematical finance) as well as direct access to the Z-process of BSDEs, delivering e.g. a trading strategy.

An introduction to Malliavin calculus by M. Kunze